{"id":3676,"date":"2018-12-11T13:57:55","date_gmt":"2018-12-11T18:57:55","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/use-the-complex-number-system\/"},"modified":"2018-12-11T13:57:55","modified_gmt":"2018-12-11T18:57:55","slug":"use-the-complex-number-system","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/use-the-complex-number-system\/","title":{"raw":"Use the Complex Number System","rendered":"Use the Complex Number System"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Evaluate the square root of a negative number<\/li><li>Add and subtract complex numbers<\/li><li>Multiply complex numbers<\/li><li>Divide complex numbers<\/li><li>Simplify powers of \\(i\\)<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1169147958916\" class=\"be-prepared\"><p id=\"fs-id1169148197895\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1169147826247\" type=\"1\"><li>Given the numbers \\(-4,-\\sqrt{7},0.\\stackrel{\u2013}{5},\\frac{7}{3},3,\\sqrt{81},\\) list the <span class=\"token\">\u24d0<\/span> rational numbers, <span class=\"token\">\u24d1<\/span> irrational numbers, <span class=\"token\">\u24d2<\/span> real numbers.<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/99b2296a-9957-4380-aff4-248abadc862b#fs-id1167836546317\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Multiply: \\(\\left(x-3\\right)\\left(2x+5\\right).\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836544266\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Rationalize the denominator:\\(\\frac{\\sqrt{5}}{\\sqrt{5}-\\sqrt{3}}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836392219\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169148123144\"><h3 data-type=\"title\">Evaluate the Square Root of a Negative Number<\/h3><p id=\"fs-id1169147830975\">Whenever we have a situation where we have a square root of a negative number we say there is no real number that equals that square root. For example, to simplify \\(\\sqrt{-1},\\) we are looking for a real number <em data-effect=\"italics\">x<\/em> so that <em data-effect=\"italics\">x<\/em><sup>2<\/sup> = \u20131. Since all real numbers squared are positive numbers, there is no real number that equals \u20131 when squared.<\/p><p id=\"fs-id1169147737576\">Mathematicians have often expanded their numbers systems as needed. They added 0 to the counting numbers to get the whole numbers. When they needed negative balances, they added negative numbers to get the integers. When they needed the idea of parts of a whole they added fractions and got the rational numbers. Adding the irrational numbers allowed numbers like \\(\\sqrt{5}.\\) All of these together gave us the real numbers and so far in your study of mathematics, that has been sufficient.<\/p><p id=\"fs-id1169147736362\">But now we will expand the real numbers to include the square roots of negative numbers. We start by defining the <span data-type=\"term\">imaginary unit<\/span> \\(i\\) as the number whose square is \u20131.<\/p><div data-type=\"note\" id=\"fs-id1169147854180\"><div data-type=\"title\">Imaginary Unit<\/div><p id=\"fs-id1169145619988\">The <strong data-effect=\"bold\">imaginary unit<\/strong> <em data-effect=\"italics\">i<\/em> is the number whose square is \u20131.<\/p><div data-type=\"equation\" id=\"fs-id1169148060192\" class=\"unnumbered\" data-label=\"\">\\({i}^{2}=-1\\phantom{\\rule{0.2em}{0ex}}\\text{or}\\phantom{\\rule{0.2em}{0ex}}i=\\sqrt{-1}\\)<\/div><\/div><p id=\"fs-id1169148126453\">We will use the imaginary unit to simplify the square roots of negative numbers.<\/p><div data-type=\"note\" id=\"fs-id1169147837254\"><div data-type=\"title\">Square Root of a Negative Number<\/div><p id=\"fs-id1169148129753\">If <em data-effect=\"italics\">b<\/em> is a positive real number, then<\/p><div data-type=\"equation\" id=\"fs-id1169147851238\" class=\"unnumbered\" data-label=\"\">\\(\\sqrt{\\text{\u2212}b}=\\sqrt{b}\\phantom{\\rule{0.2em}{0ex}}i\\)<\/div><\/div><p id=\"fs-id1169147846598\">We will use this definition in the next example. Be careful that it is clear that the <em data-effect=\"italics\">i<\/em> is not under the radical. Sometimes you will see this written as \\(\\sqrt{\\text{\u2212}b}=i\\sqrt{b}\\) to emphasize the <em data-effect=\"italics\">i<\/em> is not under the radical. But the \\(\\sqrt{\\text{\u2212}b}=\\sqrt{b}\\phantom{\\rule{0.2em}{0ex}}i\\) is considered standard form.<\/p><div data-type=\"example\" id=\"fs-id1169145730176\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148206264\"><div data-type=\"problem\"><p id=\"fs-id1169147751427\">Write each expression in terms of <em data-effect=\"italics\">i<\/em> and simplify if possible:<\/p><p id=\"fs-id1169143316387\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt{-25}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt{-7}\\)<span class=\"token\">\u24d2<\/span>\\(\\sqrt{-12}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147744813\"><p id=\"fs-id1169141447233\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{8.5em}{0ex}}\\sqrt{-25}\\hfill \\\\ \\begin{array}{c}\\text{Use the definition of the square root of}\\hfill \\\\ \\text{negative numbers.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{8.5em}{0ex}}\\sqrt{25}\\phantom{\\rule{0.2em}{0ex}}i\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{8.5em}{0ex}}5i\\hfill \\end{array}\\)<p id=\"fs-id1169143576332\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\sqrt{-7}\\hfill \\\\ \\begin{array}{c}\\text{Use the definition of the square root of}\\hfill \\\\ \\text{negative numbers.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\sqrt{7}i\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\begin{array}{c}\\hfill \\text{Be careful that it is clear that}\\phantom{\\rule{0.2em}{0ex}}i\\phantom{\\rule{0.2em}{0ex}}\\text{is not under the}\\hfill \\\\ \\text{radical sign.}\\hfill \\end{array}\\hfill \\end{array}\\)<p id=\"fs-id1169147768810\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{8.5em}{0ex}}\\sqrt{-12}\\hfill \\\\ \\begin{array}{c}\\text{Use the definition of the square root of}\\hfill \\\\ \\text{negative numbers.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{8.5em}{0ex}}\\sqrt{12}\\phantom{\\rule{0.2em}{0ex}}i\\hfill \\\\ \\text{Simplify}\\phantom{\\rule{0.2em}{0ex}}\\sqrt{12}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{8.5em}{0ex}}2\\sqrt{3}\\phantom{\\rule{0.2em}{0ex}}i\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169141188298\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145526150\"><div data-type=\"problem\" id=\"fs-id1169145977084\"><p id=\"fs-id1169148105109\">Write each expression in terms of <em data-effect=\"italics\">i<\/em> and simplify if possible:<\/p><p id=\"fs-id1169147840176\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt{-81}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt{-5}\\)<span class=\"token\">\u24d2<\/span>\\(\\sqrt{-18}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143459048\"><p id=\"fs-id1169147958539\"><span class=\"token\">\u24d0<\/span>\\(9i\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt{5}i\\)<span class=\"token\">\u24d2<\/span>\\(3\\sqrt{2}i\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147983632\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147826095\"><div data-type=\"problem\" id=\"fs-id1169148211494\"><p id=\"fs-id1169148114426\">Write each expression in terms of <em data-effect=\"italics\">i<\/em> and simplify if possible:<\/p><p id=\"fs-id1169147961162\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt{-36}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt{-3}\\)<span class=\"token\">\u24d2<\/span>\\(\\sqrt{-27}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148068950\"><p id=\"fs-id1169148123807\"><span class=\"token\">\u24d0<\/span>\\(6i\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt{3}i\\)<span class=\"token\">\u24d2<\/span>\\(3\\sqrt{3}i\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169147783255\">Now that we are familiar with the imaginary number <em data-effect=\"italics\">i<\/em>, we can expand the real numbers to include imaginary numbers. The <span data-type=\"term\">complex number system<\/span> includes the real numbers and the imaginary numbers. A <span data-type=\"term\">complex number<\/span> is of the form <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bi<\/em>, where <em data-effect=\"italics\">a, b<\/em> are real numbers. We call <em data-effect=\"italics\">a<\/em> the real part and <em data-effect=\"italics\">b<\/em> the imaginary part.<\/p><div data-type=\"note\" id=\"fs-id1169147868416\"><div data-type=\"title\">Complex Number<\/div><p id=\"fs-id1169148217595\">A <strong data-effect=\"bold\">complex number<\/strong> is of the form <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bi<\/em>, where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers.<\/p><span data-type=\"media\" id=\"fs-id1169147771025\" data-alt=\"The image shows the expression a plus b i. The number a is labeled \u201creal part\u201d and the number b i is labeled \u201cimaginary part\u201d.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The image shows the expression a plus b i. The number a is labeled \u201creal part\u201d and the number b i is labeled \u201cimaginary part\u201d.\"><\/span><\/div><p id=\"fs-id1169147821874\">A complex number is in standard form when written as \\(a+bi,\\) where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers.<\/p><p id=\"fs-id1169148248276\">If \\(b=0,\\) then \\(a+bi\\) becomes \\(a+0\u00b7i=a,\\) and is a real number.<\/p><p id=\"fs-id1169147854128\">If \\(b\\ne 0,\\) then \\(a+bi\\) is an imaginary number.<\/p><p id=\"fs-id1169147861742\">If \\(a=0,\\) then \\(a+bi\\) becomes \\(0+bi=bi,\\) and is called a pure imaginary number.<\/p><p id=\"fs-id1169147863085\">We summarize this here.<\/p><table id=\"fs-id1169141357905\" class=\"unnumbered\" summary=\"The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and \u201cReal number\u201d. The third row contains b is not equal to 0, a plus b i, and \u201cImaginary number\u201d. The fourth row contains a = 0, 0 plus b i, and \u201cPure imaginary number\u201d.\"><tbody><tr valign=\"top\"><td><\/td><td data-valign=\"middle\" data-align=\"left\">\\(a+bi\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\">\\(b=0\\)<\/td><td data-valign=\"middle\" data-align=\"left\">\\(\\begin{array}{}\\\\ a+0\u00b7i\\\\ \\\\ \\phantom{\\rule{1.2em}{0ex}}a\\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"left\">Real number<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\">\\(b\\ne 0\\)<\/td><td data-valign=\"middle\" data-align=\"left\">\\(a+bi\\)<\/td><td data-valign=\"middle\" data-align=\"left\">Imaginary number<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\">\\(a=0\\)<\/td><td data-valign=\"middle\" data-align=\"left\">\\(\\begin{array}{l}0+bi\\\\ \\\\ \\\\ \\phantom{\\rule{0.8em}{0ex}}bi\\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"left\">Pure imaginary number<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169145519328\">The standard form of a complex number is \\(a+bi,\\) so this explains why the preferred form is \\(\\sqrt{\\text{\u2212}b}=\\sqrt{b}i\\) when \\(b&gt;0.\\)<\/p><p id=\"fs-id1169147847796\">The diagram helps us visualize the complex number system. It is made up of both the real numbers and the imaginary numbers.<\/p><span data-type=\"media\" id=\"fs-id1169145621336\" data-alt=\"The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and \u201cReal number\u201d. The third row contains b is not equal to 0, a plus b i, and \u201cImaginary number\u201d. The fourth row contains a = 0, 0 plus b i, and \u201cPure imaginary number\u201d.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_002_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and \u201cReal number\u201d. The third row contains b is not equal to 0, a plus b i, and \u201cImaginary number\u201d. The fourth row contains a = 0, 0 plus b i, and \u201cPure imaginary number\u201d.\"><\/span><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169142605352\"><h3 data-type=\"title\">Add or Subtract Complex Numbers<\/h3><p id=\"fs-id1169148123410\">We are now ready to perform the operations of addition, subtraction, multiplication and division on the complex numbers\u2014just as we did with the real numbers.<\/p><p id=\"fs-id1169145622397\">Adding and subtracting complex numbers is much like adding or subtracting like terms. We add or subtract the real parts and then add or subtract the imaginary parts. Our final result should be in standard form.<\/p><div data-type=\"example\" id=\"fs-id1169147935028\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147847594\"><div data-type=\"problem\" id=\"fs-id1169145591200\"><p id=\"fs-id1169148101553\">Add: \\(\\sqrt{-12}+\\sqrt{-27}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145988331\"><p id=\"fs-id1169148062911\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{-12}+\\sqrt{-27}\\hfill \\\\ \\begin{array}{c}\\text{Use the definition of the square root of}\\hfill \\\\ \\text{negative numbers.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{12}\\phantom{\\rule{0.2em}{0ex}}i+\\sqrt{27}\\phantom{\\rule{0.2em}{0ex}}i\\hfill \\\\ \\text{Simplify the square roots.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}2\\sqrt{3}\\phantom{\\rule{0.2em}{0ex}}i+3\\sqrt{3}\\phantom{\\rule{0.2em}{0ex}}i\\hfill \\\\ \\text{Add.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}5\\sqrt{3}\\phantom{\\rule{0.2em}{0ex}}i\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147822009\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145495053\"><div data-type=\"problem\" id=\"fs-id1169148105509\"><p id=\"fs-id1169143318310\">Add: \\(\\sqrt{-8}+\\sqrt{-32}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1169145729376\">\\(6\\sqrt{2}i\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148208039\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145696301\"><div data-type=\"problem\" id=\"fs-id1169147810027\"><p id=\"fs-id1169147830354\">Add: \\(\\sqrt{-27}+\\sqrt{-48}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147874000\"><p id=\"fs-id1169147859230\">\\(7\\sqrt{3}i\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169148078266\">Remember to add both the real parts and the imaginary parts in this next example.<\/p><div data-type=\"example\" id=\"fs-id1169147793628\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148225075\"><div data-type=\"problem\" id=\"fs-id1169143572966\"><p id=\"fs-id1169147977713\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\left(4-3i\\right)+\\left(5+6i\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(2-5i\\right)-\\left(5-2i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169142138982\"><p id=\"fs-id1169148069557\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\left(4-3i\\right)+\\left(5+6i\\right)\\hfill \\\\ \\begin{array}{c}\\text{Use the Associative Property to put the real}\\hfill \\\\ \\text{parts and the imaginary parts together.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\left(4+5\\right)+\\left(-3i+6i\\right)\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}9+3i\\hfill \\end{array}\\)<p id=\"fs-id1169147960428\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\left(2-5i\\right)-\\left(5-2i\\right)\\hfill \\\\ \\text{Distribute.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}2-5i-5+2i\\hfill \\\\ \\begin{array}{c}\\text{Use the Associative Property to put the real}\\hfill \\\\ \\text{parts and the imaginary parts together.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}2-5-5i+2i\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}-3-3i\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147846393\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1169148080567\"><p id=\"fs-id1169147776135\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\left(2+7i\\right)+\\left(4-2i\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(8-4i\\right)-\\left(2-i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147869982\"><p id=\"fs-id1169148048099\"><span class=\"token\">\u24d0<\/span>\\(6+5i\\)<span class=\"token\">\u24d1<\/span>\\(6-3i\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147846302\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147710394\"><div data-type=\"problem\" id=\"fs-id1169147725798\"><p id=\"fs-id1169145665290\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\left(3-2i\\right)+\\left(-5-4i\\right)\\) <span class=\"token\">\u24d1<\/span> \\(\\left(4+3i\\right)-\\left(2-6i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147810568\"><p id=\"fs-id1169143516718\"><span class=\"token\">\u24d0<\/span>\\(-2-6i\\)<span class=\"token\">\u24d1<\/span>\\(2+9i\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169148054042\"><h3 data-type=\"title\">Multiply Complex Numbers<\/h3><p id=\"fs-id1169147747655\">Multiplying complex numbers is also much like multiplying expressions with coefficients and variables. There is only one special case we need to consider. We will look at that after we practice in the next two examples.<\/p><div data-type=\"example\" id=\"fs-id1169147907196\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147854723\"><div data-type=\"problem\" id=\"fs-id1169147751104\"><p id=\"fs-id1169147726338\">Multiply: \\(2i\\left(7-5i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143686657\"><p id=\"fs-id1169148086081\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{11em}{0ex}}2i\\left(7-5i\\right)\\hfill \\\\ \\text{Distribute.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{11em}{0ex}}14i-10{i}^{2}\\hfill \\\\ \\text{Simplify}\\phantom{\\rule{0.2em}{0ex}}{i}^{2}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{11em}{0ex}}14i-10\\left(-1\\right)\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{11em}{0ex}}14i+10\\hfill \\\\ \\text{Write in standard form.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{11em}{0ex}}10+14i\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145732335\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147746940\"><div data-type=\"problem\" id=\"fs-id1169147829425\"><p id=\"fs-id1169147760715\">Multiply: \\(4i\\left(5-3i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147731418\"><p>\\(12+20i\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147866899\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147751305\"><div data-type=\"problem\" id=\"fs-id1169147988364\"><p id=\"fs-id1169147771313\">Multiply: \\(-3i\\left(2+4i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145536453\"><p id=\"fs-id1169145785706\">\\(12+6i\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169147981016\">In the next example, we multiply the binomials using the <span data-type=\"term\" class=\"no-emphasis\">Distributive Property<\/span> or <span data-type=\"term\" class=\"no-emphasis\">FOIL<\/span>.<\/p><div data-type=\"example\" id=\"fs-id1169143511338\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169145576905\"><div data-type=\"problem\" id=\"fs-id1169147961150\"><p id=\"fs-id1169145498324\">Multiply: \\(\\left(3+2i\\right)\\left(4-3i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147805831\"><p id=\"fs-id1169147794243\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{6.5em}{0ex}}\\left(3+2i\\right)\\left(4-3i\\right)\\hfill \\\\ \\text{Use FOIL.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{6.5em}{0ex}}12-9i+8i-6{i}^{2}\\hfill \\\\ \\text{Simplify}\\phantom{\\rule{0.2em}{0ex}}{i}^{2}\\phantom{\\rule{0.2em}{0ex}}\\text{and combine like terms.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{6.5em}{0ex}}12-i-6\\left(-1\\right)\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{6.5em}{0ex}}12-i+6\\hfill \\\\ \\text{Combine the real parts.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{6.5em}{0ex}}18-i\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147758224\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148099754\"><div data-type=\"problem\" id=\"fs-id1169147849540\"><p id=\"fs-id1169147833881\">Multiply: \\(\\left(5-3i\\right)\\left(-1-2i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147906922\"><p>\\(-11-7i\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145601980\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147742026\"><div data-type=\"problem\" id=\"fs-id1169147935573\"><p id=\"fs-id1169147866757\">Multiply: \\(\\left(-4-3i\\right)\\left(2+i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148224477\"><p id=\"fs-id1169147839828\">\\(-5-10i\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169147835113\">In the next example, we could use FOIL or the <span data-type=\"term\" class=\"no-emphasis\">Product of Binomial Squares Pattern<\/span>.<\/p><div data-type=\"example\" id=\"fs-id1169145685376\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169143659210\"><div data-type=\"problem\" id=\"fs-id1169145567304\"><p id=\"fs-id1169148069276\">Multiply: \\({\\left(3+2i\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148070995\"><table id=\"fs-id1169147863045\" class=\"unnumbered unstyled\" summary=\"Use the Binomial Squares Pattern formula the quantity a plus b in parentheses squared equals a squared plus 2 a b plus b squared. Applied to this example we get the expression 3 squared plus 2 times 3 times 2 i plus the quantity 2 i in parentheses squared. Simplifying we get 9 plus 12 i plus 4 i squared. Simplifying further we get 9 plus 12 i plus 4 times negative 1. The final simplified version is 5 plus 12 i.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169147962139\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Use the Product of Binomial Squares Pattern, \\({\\left(a+b\\right)}^{2}={a}^{2}+2ab+{b}^{2}.\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169145660655\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169147860309\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify \\({i}^{2}.\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169147802783\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169143518464\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169143492286\"><div data-type=\"problem\" id=\"fs-id1169147805544\"><p id=\"fs-id1169147715424\">Multiply using the Binomial Squares pattern: \\({\\left(-2-5i\\right)}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147840934\"><p id=\"fs-id1169145733710\">\\(-21-20i\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169143295681\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145657632\"><div data-type=\"problem\"><p id=\"fs-id1169141030754\">Multiply using the Binomial Squares pattern: \\({\\left(-5+4i\\right)}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147808386\"><p id=\"fs-id1169145642404\">\\(9-40i\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169148105017\">Since the square root of a negative number is not a real number, we cannot use the Product Property for Radicals. In order to multiply square roots of negative numbers we should first write them as complex numbers, using \\(\\sqrt{\\text{\u2212}b}=\\sqrt{b}i.\\) This is one place students tend to make errors, so be careful when you see multiplying with a negative square root.<\/p><div data-type=\"example\" id=\"fs-id1169143266846\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1169147866217\"><p>Multiply: \\(\\sqrt{-36}\u00b7\\sqrt{-4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145491768\"><p id=\"fs-id1169147803148\">To multiply square roots of negative numbers, we first write them as complex numbers.<\/p><p>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{-36}\u00b7\\sqrt{-4}\\hfill \\\\ \\text{Write as complex numbers using}\\phantom{\\rule{0.2em}{0ex}}\\sqrt{\\text{\u2212}b}=\\sqrt{b}i.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{36}\\phantom{\\rule{0.2em}{0ex}}i\u00b7\\sqrt{4}\\phantom{\\rule{0.2em}{0ex}}i\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}6i\u00b72i\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}12{i}^{2}\\hfill \\\\ \\text{Simplify}\\phantom{\\rule{0.2em}{0ex}}{i}^{2}\\phantom{\\rule{0.2em}{0ex}}\\text{and multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}-12\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147870800\"><div data-type=\"problem\"><p id=\"fs-id1169147768510\">Multiply: \\(\\sqrt{-49}\u00b7\\sqrt{-4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148097432\"><p id=\"fs-id1169145498758\">\\(-14\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148250411\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169143661264\"><div data-type=\"problem\" id=\"fs-id1169141473006\"><p>Multiply: \\(\\sqrt{-36}\u00b7\\sqrt{-81}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147833959\"><p id=\"fs-id1169145643813\">\\(-54\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165927630506\">In the next example, each binomial has a square root of a negative number. Before multiplying, each square root of a negative number must be written as a complex number.<\/p><div data-type=\"example\" id=\"fs-id1169145622189\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169145606346\"><div data-type=\"problem\" id=\"fs-id1169148064048\"><p id=\"fs-id1169147979154\">Multiply: \\(\\left(3-\\sqrt{-12}\\right)\\left(5+\\sqrt{-27}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145663400\"><p id=\"fs-id1169147824279\">To multiply square roots of negative numbers, we first write them as complex numbers.<\/p><p id=\"fs-id1169145520026\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\left(3-\\sqrt{-12}\\right)\\left(5+\\sqrt{-27}\\right)\\hfill \\\\ \\text{Write as complex numbers using}\\phantom{\\rule{0.2em}{0ex}}\\sqrt{\\text{\u2212}b}=\\sqrt{b}i.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\left(3-2\\sqrt{3}\\phantom{\\rule{0.2em}{0ex}}i\\right)\\left(5+3\\sqrt{3}\\phantom{\\rule{0.2em}{0ex}}i\\right)\\hfill \\\\ \\text{Use FOIL.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}15+9\\sqrt{3}\\phantom{\\rule{0.2em}{0ex}}i-10\\sqrt{3}\\phantom{\\rule{0.2em}{0ex}}i-6\u00b73{i}^{2}\\hfill \\\\ \\text{Combine like terms and simplify}\\phantom{\\rule{0.2em}{0ex}}{i}^{2}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}15-\\sqrt{3}\\phantom{\\rule{0.2em}{0ex}}i-6\u00b7\\left(-3\\right)\\hfill \\\\ \\text{Multiply and combine like terms.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}33-\\sqrt{3}\\phantom{\\rule{0.2em}{0ex}}i\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147979332\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169143280628\"><div data-type=\"problem\" id=\"fs-id1169147856967\"><p id=\"fs-id1169147746775\">Multiply: \\(\\left(4-\\sqrt{-12}\\right)\\left(3-\\sqrt{-48}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147830796\"><p id=\"fs-id1169147776931\">\\(-12-22\\sqrt{3}i\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145775017\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145658407\"><div data-type=\"problem\" id=\"fs-id1169148230802\"><p>Multiply: \\(\\left(-2+\\sqrt{-8}\\right)\\left(3-\\sqrt{-18}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169141298078\"><p id=\"fs-id1169143581121\">\\(6+12\\sqrt{2}i\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169147909818\">We first looked at conjugate pairs when we studied polynomials. We said that a pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a <em data-effect=\"italics\">conjugate pair<\/em> and is of the form \\(\\left(a-b\\right),\\left(a+b\\right).\\)<\/p><p id=\"fs-id1169143572992\">A <span data-type=\"term\">complex conjugate pair<\/span> is very similar. For a complex number of the form \\(a+bi,\\) its conjugate is \\(a-bi.\\) Notice they have the same first term and the same last term, but one is a sum and one is a difference.<\/p><div data-type=\"note\" id=\"fs-id1169145572553\"><div data-type=\"title\">Complex Conjugate Pair<\/div><p id=\"fs-id1169148200251\">A <strong data-effect=\"bold\">complex conjugate pair<\/strong> is of the form \\(a+bi,\\)\\(a-bi.\\)<\/p><\/div><p id=\"fs-id1169147837933\">We will multiply a complex conjugate pair in the next example.<\/p><div data-type=\"example\" id=\"fs-id1169147851512\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169143316403\"><div data-type=\"problem\" id=\"fs-id1169148080502\"><p id=\"fs-id1169147700609\">Multiply: \\(\\left(3-2i\\right)\\left(3+2i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143662126\"><p id=\"fs-id1169147877692\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\left(3-2i\\right)\\left(3+2i\\right)\\hfill \\\\ \\text{Use FOIL.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}9+6i-6i-4{i}^{2}\\hfill \\\\ \\text{Combine like terms and simplify}\\phantom{\\rule{0.2em}{0ex}}{i}^{2}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}9-4\\left(-1\\right)\\hfill \\\\ \\text{Multiply and combine like terms.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}13\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147709375\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169141322580\"><div data-type=\"problem\" id=\"fs-id1169145607061\"><p id=\"fs-id1169147770009\">Multiply: \\(\\left(4-3i\\right)\u00b7\\left(4+3i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147949644\"><p id=\"fs-id1169147740020\">25<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148048174\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147868111\"><div data-type=\"problem\" id=\"fs-id1169147784324\"><p id=\"fs-id1169147744297\">Multiply: \\(\\left(-2+5i\\right)\u00b7\\left(-2-5i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147736562\"><p id=\"fs-id1169147846906\">29<\/p><\/div><\/div><\/div><p id=\"fs-id1169147764265\">From our study of polynomials, we know the product of conjugates is always of the form \\(\\left(a-b\\right)\\left(a+b\\right)={a}^{2}-{b}^{2}.\\) The result is called a <span data-type=\"term\" class=\"no-emphasis\">difference of squares<\/span>. We can multiply a complex conjugate pair using this pattern.<\/p><p id=\"fs-id1169147739518\">The last example we used FOIL. Now we will use the <span data-type=\"term\" class=\"no-emphasis\">Product of Conjugates Pattern<\/span>.<\/p><span data-type=\"media\" id=\"fs-id1169147862786\" data-alt=\"The quantity a minus b in parentheses times the quantity a plus b in parentheses is written above the expression showing the product of 3 minus 2 i in parentheses and 3 plus 2 i in parentheses. In the next line a squared minus b squared is written above the expression 3 squared minus the quantity 2 i in parentheses squared. Simplifying we get 9 minus 4 i squared. This is equal to 9 minus 4 times negative 1. The final result is 13.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The quantity a minus b in parentheses times the quantity a plus b in parentheses is written above the expression showing the product of 3 minus 2 i in parentheses and 3 plus 2 i in parentheses. In the next line a squared minus b squared is written above the expression 3 squared minus the quantity 2 i in parentheses squared. Simplifying we get 9 minus 4 i squared. This is equal to 9 minus 4 times negative 1. The final result is 13.\"><\/span><p id=\"fs-id1169147865956\">Notice this is the same result we found in <a href=\"#fs-id1169147851512\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><p id=\"fs-id1169148218790\">When we multiply complex conjugates, the product of the last terms will always have an \\({i}^{2}\\) which simplifies to \\(-1.\\)<\/p><div data-type=\"equation\" id=\"fs-id1169147775175\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\hfill \\left(a-bi\\right)\\left(a+bi\\right)\\hfill \\\\ \\hfill {a}^{2}-{\\left(bi\\right)}^{2}\\hfill \\\\ \\hfill {a}^{2}-{b}^{2}{i}^{2}\\hfill \\\\ \\hfill {a}^{2}-{b}^{2}\\left(-1\\right)\\hfill \\\\ \\hfill {a}^{2}+{b}^{2}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1169148037437\">This leads us to the Product of Complex Conjugates Pattern: \\(\\left(a-bi\\right)\\left(a+bi\\right)={a}^{2}+{b}^{2}\\)<\/p><div data-type=\"note\" id=\"fs-id1169147706826\"><div data-type=\"title\">Product of Complex Conjugates<\/div><p id=\"fs-id1169145496922\">If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers, then<\/p><div data-type=\"equation\" id=\"fs-id1169147848448\" class=\"unnumbered\" data-label=\"\">\\(\\left(a-bi\\right)\\left(a+bi\\right)={a}^{2}+{b}^{2}\\)<\/div><\/div><div data-type=\"example\" id=\"fs-id1169147867029\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148071873\"><div data-type=\"problem\" id=\"fs-id1169147846363\"><p id=\"fs-id1169145715989\">Multiply using the Product of Complex Conjugates Pattern: \\(\\left(8-2i\\right)\\left(8+2i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145667175\"><table id=\"fs-id1169143550228\" class=\"unnumbered unstyled can-break\" summary=\"The quantity a minus b i in parentheses times the quantity a plus b i in parentheses is written above the expression showing the product of 8 minus 2 i in parentheses and 8 plus 2 i in parentheses. In the next line a squared plus b squared is written above the expression 8 squared plus the quantity 2 squared. Simplifying we get 64 plus 4. The final result is 68.\" data-label=\"\"><tbody><tr><td><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169143573503\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Use the Product of Complex Conjugates Pattern,<div data-type=\"newline\"><br><\/div>\\(\\left(a-bi\\right)\\left(a+bi\\right)={a}^{2}+{b}^{2}.\\)<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169143431827\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify the squares.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169147808213\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Add.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169148250711\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148227244\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145526008\"><div data-type=\"problem\" id=\"fs-id1169148205847\"><p id=\"fs-id1169148227408\">Multiply using the Product of Complex Conjugates Pattern: \\(\\left(3-10i\\right)\\left(3+10i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147966173\"><p id=\"fs-id1169143520545\">109<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147776926\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148115972\"><div data-type=\"problem\" id=\"fs-id1169147737029\"><p id=\"fs-id1169147979756\">Multiply using the Product of Complex Conjugates Pattern: \\(\\left(-5+4i\\right)\\left(-5-4i\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147735634\"><p id=\"fs-id1169147804823\">41<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147873364\"><h3 data-type=\"title\">Divide Complex Numbers<\/h3><p id=\"fs-id1169143518290\">Dividing complex numbers is much like rationalizing a denominator. We want our result to be in standard form with no imaginary numbers in the denominator.<\/p><div data-type=\"example\" id=\"fs-id1169147851054\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Divide Complex Numbers<\/div><div data-type=\"exercise\" id=\"fs-id1169147759739\"><div data-type=\"problem\" id=\"fs-id1169141522111\"><p id=\"fs-id1169145519620\">Divide: \\(\\frac{4+3i}{3-4i}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147730871\"><span data-type=\"media\" id=\"fs-id1169147808080\" data-alt=\"Step 1 is to write both the numerator and denominator in standard form. For this example they are both in standard form.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_006a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write both the numerator and denominator in standard form. For this example they are both in standard form.\"><\/span><span data-type=\"media\" id=\"fs-id1169143662469\" data-alt=\"Step 2 is to multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 3 minus 4 i is 3 plus 4 i. The resulting expression is the quantity 4 plus 3 i in parentheses times the quantity 3 plus 4 i in parentheses divided by the product of 3 minus 4 i in parentheses and the quantity 3 plus 4 i in parentheses.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_006b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 3 minus 4 i is 3 plus 4 i. The resulting expression is the quantity 4 plus 3 i in parentheses times the quantity 3 plus 4 i in parentheses divided by the product of 3 minus 4 i in parentheses and the quantity 3 plus 4 i in parentheses.\"><\/span><span data-type=\"media\" id=\"fs-id1169145729564\" data-alt=\"Step 3 is to simplify and write the result in standard form. Use the pattern the quantity a plus b i in parentheses equals a squared plus b squared in the denominator. The expression for this example then becomes the quantity 12 plus 16 i plus 9 i plus 12 i squared in parentheses divided by the sum of 9 and 16. Combining like terms we get the quantity 12 plus 25 i minus 12 in parentheses divided by 25. Simplifying we get 25 i divided by 25. Write the result in standard form. The result is i.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_006c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to simplify and write the result in standard form. Use the pattern the quantity a plus b i in parentheses equals a squared plus b squared in the denominator. The expression for this example then becomes the quantity 12 plus 16 i plus 9 i plus 12 i squared in parentheses divided by the sum of 9 and 16. Combining like terms we get the quantity 12 plus 25 i minus 12 in parentheses divided by 25. Simplifying we get 25 i divided by 25. Write the result in standard form. The result is i.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169143317947\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147880316\"><div data-type=\"problem\" id=\"fs-id1169147849308\"><p id=\"fs-id1169147846945\">Divide: \\(\\frac{2+5i}{5-2i}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147767699\"><p id=\"fs-id1169147737517\"><em data-effect=\"italics\">i<\/em><\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145643886\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147737214\"><div data-type=\"problem\" id=\"fs-id1169147803413\"><p id=\"fs-id1169145658419\">Divide: \\(\\frac{1+6i}{6-i}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147751252\"><p id=\"fs-id1169145536152\"><em data-effect=\"italics\">i<\/em><\/p><\/div><\/div><\/div><p id=\"fs-id1169143578648\">We summarize the steps here.<\/p><div data-type=\"note\" id=\"fs-id1169147726518\" class=\"howto\"><div data-type=\"title\">How to divide complex numbers.<\/div><ol id=\"fs-id1169148237090\" type=\"1\" class=\"stepwise\"><li>Write both the numerator and denominator in standard form.<\/li><li>Multiply the numerator and denominator by the complex conjugate of the denominator.<\/li><li>Simplify and write the result in standard form.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1169147878899\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148231374\"><div data-type=\"problem\" id=\"fs-id1169142122806\"><p id=\"fs-id1169147862468\">Divide, writing the answer in standard form: \\(\\frac{-3}{5+2i}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148071478\"><p id=\"fs-id1169143525632\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{-3}{5+2i}\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\text{Multiply the numerator and denominator by the}\\hfill \\\\ \\text{complex conjugate of the denominator.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{-3\\left(5-2i\\right)}{\\left(5+2i\\right)\\left(5-2i\\right)}\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\text{Multiply in the numerator and use the Product of}\\hfill \\\\ \\text{Complex Conjugates Pattern in the denominator.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{-15+6i}{{5}^{2}+{2}^{2}}\\hfill \\\\ \\\\ \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{-15+6i}{29}\\hfill \\\\ \\\\ \\\\ \\text{Write in standard form.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}-\\frac{15}{29}+\\frac{6}{29}i\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148053850\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145971630\"><div data-type=\"problem\" id=\"fs-id1169147847176\"><p id=\"fs-id1169142281226\">Divide, writing the answer in standard form: \\(\\frac{4}{1-4i}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147767469\"><p id=\"fs-id1169148185370\">\\(\\frac{4}{17}+\\frac{16}{17}i\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147862734\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147727903\"><div data-type=\"problem\" id=\"fs-id1169147709749\"><p id=\"fs-id1169147832655\">Divide, writing the answer in standard form: \\(\\frac{-2}{-1+2i}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148233919\"><p id=\"fs-id1169145843850\">\\(\\frac{2}{5}+\\frac{4}{5}i\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169147794219\">Be careful as you find the conjugate of the denominator.<\/p><div data-type=\"example\" id=\"fs-id1169148224121\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169147848442\"><div data-type=\"problem\" id=\"fs-id1169143748194\"><p id=\"fs-id1169147807705\">Divide: \\(\\frac{5+3i}{4i}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145607677\"><p id=\"fs-id1169148076300\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{5+3i}{4i}\\hfill \\\\ \\text{Write the denominator in standard form.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{5+3i}{0+4i}\\hfill \\\\ \\begin{array}{c}\\text{Multiply the numerator and denominator by}\\hfill \\\\ \\text{the complex conjugate of the denominator.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\left(5+3i\\right)\\left(0-4i\\right)}{\\left(0+4i\\right)\\left(0-4i\\right)}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\left(5+3i\\right)\\left(-4i\\right)}{\\left(4i\\right)\\left(-4i\\right)}\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{-20i-12{i}^{2}}{-16{i}^{2}}\\hfill \\\\ \\text{Simplify the}\\phantom{\\rule{0.2em}{0ex}}{i}^{2}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{-20i+12}{16}\\hfill \\\\ \\text{Rewrite in standard form.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{12}{16}-\\frac{20}{16}i\\hfill \\\\ \\text{Simplify the fractions.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{3}{4}-\\frac{5}{4}i\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147838114\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148070414\"><div data-type=\"problem\" id=\"fs-id1169147862015\"><p id=\"fs-id1169145575732\">Divide: \\(\\frac{3+3i}{2i}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147746802\"><p id=\"fs-id1169147707150\">\\(\\frac{3}{2}-\\frac{3}{2}i\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147852048\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147784202\"><div data-type=\"problem\" id=\"fs-id1169145640113\"><p id=\"fs-id1169143579172\">Divide: \\(\\frac{2+4i}{5i}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148225190\"><p id=\"fs-id1169147946732\">\\(\\frac{4}{5}-\\frac{2}{5}i\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169148190703\"><h3 data-type=\"title\">Simplify Powers of <em data-effect=\"italics\">i<\/em><\/h3><p id=\"fs-id1169147850832\">The powers of \\(i\\) make an interesting pattern that will help us simplify higher powers of <em data-effect=\"italics\">i<\/em>. Let\u2019s evaluate the powers of \\(i\\) to see the pattern.<\/p><div data-type=\"equation\" id=\"fs-id1169143728354\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccccccc}\\hfill {i}^{1}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{2}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{3}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{4}\\hfill \\\\ \\hfill i\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}-1\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{2}\u00b7i\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{2}\u00b7{i}^{2}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}-1\u00b7i\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\left(-1\\right)\\left(-1\\right)\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}-i\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}1\\hfill \\\\ \\\\ \\\\ \\hfill {i}^{5}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{6}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{7}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{8}\\hfill \\\\ \\hfill {i}^{4}\u00b7i\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{4}\u00b7{i}^{2}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{4}\u00b7{i}^{3}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{4}\u00b7{i}^{4}\\hfill \\\\ \\hfill 1\u00b7i\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}1\u00b7{i}^{2}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}1\u00b7{i}^{3}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}1\u00b71\\hfill \\\\ i\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{2}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}{i}^{3}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}1\\hfill \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}-1\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}-i\\hfill \\end{array}\\)<\/div><p id=\"fs-id1169148072161\">We summarize this now.<\/p><div data-type=\"equation\" id=\"fs-id1169147741472\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccccc}\\hfill {i}^{1}&amp; =\\hfill &amp; i\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}{i}^{5}&amp; =\\hfill &amp; i\\hfill \\\\ \\hfill {i}^{2}&amp; =\\hfill &amp; -1\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}{i}^{6}&amp; =\\hfill &amp; -1\\hfill \\\\ \\hfill {i}^{3}&amp; =\\hfill &amp; \\text{\u2212}i\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}{i}^{7}&amp; =\\hfill &amp; \\text{\u2212}i\\hfill \\\\ \\hfill {i}^{4}&amp; =\\hfill &amp; 1\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}{i}^{8}&amp; =\\hfill &amp; 1\\hfill \\end{array}\\)<\/div><p id=\"fs-id1169143573199\">If we continued, the pattern would keep repeating in blocks of four. We can use this pattern to help us simplify powers of <em data-effect=\"italics\">i<\/em>. Since <em data-effect=\"italics\">i<\/em><sup>4<\/sup> = 1, we rewrite each power, <em data-effect=\"italics\">i<sup>n<\/sup><\/em>, as a product using <em data-effect=\"italics\">i<\/em><sup>4<\/sup> to a power and another power of <em data-effect=\"italics\">i<\/em>.<\/p><p id=\"fs-id1169145662274\">We rewrite it in the form \\({i}^{n}={\\left({i}^{4}\\right)}^{q}\u00b7{i}^{r},\\) where the exponent, <em data-effect=\"italics\">q<\/em>, is the quotient of <em data-effect=\"italics\">n<\/em> divided by 4 and the exponent, <em data-effect=\"italics\">r<\/em>, is the remainder from this division. For example, to simplify <em data-effect=\"italics\">i<\/em><sup>57<\/sup>, we divide 57 by 4 and we get 14 with a remainder of 1. In other words, \\(57=4\u00b714+1.\\) So we write \\({i}^{57}={\\left({1}^{4}\\right)}^{14}\u00b7{i}^{1}\\) and then simplify from there.<\/p><span data-type=\"media\" id=\"fs-id1168040486190\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"example\" id=\"fs-id1169141376582\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1169145641420\"><p id=\"fs-id1169145641422\">Simplify: \\({i}^{86}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147963111\"><p>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{i}^{86}\\hfill \\\\ \\begin{array}{c}\\text{Divide 86 by 4 and rewrite}\\phantom{\\rule{0.2em}{0ex}}{i}^{86}\\phantom{\\rule{0.2em}{0ex}}\\text{in the}\\hfill \\\\ {i}^{n}={\\left({i}^{4}\\right)}^{q}\u00b7{i}^{r}\\phantom{\\rule{0.2em}{0ex}}\\text{form.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{\\left({1}^{4}\\right)}^{21}\u00b7{i}^{2}\\hfill \\end{array}\\)<\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1168040462857\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccccccccccc}\\text{Simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{10em}{0ex}}{\\left(1\\right)}^{21}\u00b7\\left(-1\\right)\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\phantom{\\rule{10em}{0ex}}\u20131\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147766271\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145748075\"><div data-type=\"problem\" id=\"fs-id1169145748077\"><p id=\"fs-id1169142437136\">Simplify: \\({i}^{75}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148063916\"><p id=\"fs-id1169148063919\">\\(\\text{\u2212}i\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169145733988\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169145716787\"><div data-type=\"problem\" id=\"fs-id1169145737890\"><p id=\"fs-id1169145737892\">Simplify: \\({i}^{92}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147846616\"><p id=\"fs-id1169147960608\">\\(1\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169147965255\" class=\"media-2\"><p id=\"fs-id1169145645015\">Access these online resources for additional instruction and practice with the complex number system.<\/p><ul id=\"fs-id1169143332448\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37CompNumb1\">Expressing Square Roots of Negative Numbers with i<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37CompNumb2\">Subtract and Multiply Complex Numbers<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37CompNumb3\">Dividing Complex Numbers<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37CompNumb4\">Rewriting Powers of i<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169145732671\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1169147850757\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Square Root of a Negative Number<\/strong><ul id=\"fs-id1169145716315\" data-bullet-style=\"open-circle\"><li>If <em data-effect=\"italics\">b<\/em> is a positive real number, then \\(\\sqrt{\\text{\u2212}b}=\\sqrt{b}i\\)<div data-type=\"newline\"><br><\/div> <table id=\"fs-id1169147827697\" summary=\"The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and \u201cReal number\u201d. The third row contains b is not equal to 0, a plus b i, and \u201cImaginary number\u201d. The fourth row contains a = 0, 0 plus b i, and \u201cPure imaginary number\u201d.\"><tbody><tr valign=\"top\"><td><\/td><td data-valign=\"middle\" data-align=\"left\">\\(a+bi\\)<\/td><td><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\">\\(b=0\\)<\/td><td data-valign=\"middle\" data-align=\"left\">\\(\\begin{array}{}\\\\ a+0\u00b7i\\\\ \\\\ \\phantom{\\rule{1.2em}{0ex}}a\\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"left\">Real number<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\">\\(b\\ne 0\\)<\/td><td data-valign=\"middle\" data-align=\"left\">\\(a+bi\\)<\/td><td data-valign=\"middle\" data-align=\"left\">Imaginary number<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\">\\(a=0\\)<\/td><td data-valign=\"middle\" data-align=\"left\">\\(\\begin{array}{l}0+bi\\\\ \\\\ \\\\ \\phantom{\\rule{0.8em}{0ex}}bi\\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"left\">Pure imaginary number<\/td><\/tr><\/tbody><\/table><\/li><li>A complex number is in <strong data-effect=\"bold\">standard form<\/strong> when written as <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bi<\/em>, where <em data-effect=\"italics\">a, b<\/em> are real numbers.<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1169143534059\" data-alt=\"The diagram has a rectangle with the labels \u201cComplex Numbers\u201d and a plus b i. A second rectangle has the labels \u201cReal Numbers\u201d, a plus b i, b = 0. A third rectangle has the labels \u201cImaginary Numbers\u201d, a plus b i, b not equal to 0. Arrows go from the Real Numbers rectangle and Imaginary Numbers rectangle and point toward the Complex Numbers rectangle.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The diagram has a rectangle with the labels \u201cComplex Numbers\u201d and a plus b i. A second rectangle has the labels \u201cReal Numbers\u201d, a plus b i, b = 0. A third rectangle has the labels \u201cImaginary Numbers\u201d, a plus b i, b not equal to 0. Arrows go from the Real Numbers rectangle and Imaginary Numbers rectangle and point toward the Complex Numbers rectangle.\"><\/span><\/li><\/ul><\/li><li><strong data-effect=\"bold\">Product of Complex Conjugates<\/strong><ul id=\"fs-id1169145496103\" data-bullet-style=\"open-circle\"><li>If <em data-effect=\"italics\">a, b<\/em> are real numbers, then<div data-type=\"newline\"><br><\/div> \\(\\left(a-bi\\right)\\left(a+bi\\right)={a}^{2}+{b}^{2}\\)<\/li><\/ul><\/li><li><strong data-effect=\"bold\">How to Divide Complex Numbers<\/strong><ol id=\"fs-id1169142400660\" type=\"1\" class=\"stepwise\"><li>Write both the numerator and denominator in standard form.<\/li><li>Multiply the numerator and denominator by the complex conjugate of the denominator.<\/li><li>Simplify and write the result in standard form.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169143332524\"><h3 data-type=\"title\">Section Exercises<\/h3><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169147966479\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1169148227286\"><strong data-effect=\"bold\">Evaluate the Square Root of a Negative Number<\/strong><\/p><p id=\"fs-id1169145728436\">In the following exercises, write each expression in terms of <em data-effect=\"italics\">i<\/em> and simplify if possible.<\/p><div data-type=\"exercise\" id=\"fs-id1169145667504\"><div data-type=\"problem\" id=\"fs-id1169147829166\"><p id=\"fs-id1169147829168\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{-16}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt{-11}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\sqrt{-8}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169147865256\"><p id=\"fs-id1169147878400\"><span class=\"token\">\u24d0<\/span>\\(4i\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt{11}i\\)<span class=\"token\">\u24d2<\/span>\\(2\\sqrt{2}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147797185\"><div data-type=\"problem\" id=\"fs-id1169147797187\"><p id=\"fs-id1169147841507\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{-121}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt{-1}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\sqrt{-20}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143614499\"><div data-type=\"problem\" id=\"fs-id1169143614501\"><p id=\"fs-id1169145496257\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{-100}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt{-13}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\sqrt{-45}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169145491586\"><p id=\"fs-id1169147855122\"><span class=\"token\">\u24d0<\/span>\\(10i\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt{13}i\\)<span class=\"token\">\u24d2<\/span>\\(3\\sqrt{5}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145664846\"><div data-type=\"problem\" id=\"fs-id1169145644765\"><p id=\"fs-id1169145644767\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{-49}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt{-15}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\sqrt{-75}\\)<\/div><\/div><p id=\"fs-id1169145519944\"><strong data-effect=\"bold\">Add or Subtract Complex Numbers<\/strong> In the following exercises, add or subtract.<\/p><div data-type=\"exercise\" id=\"fs-id1169147961706\"><div data-type=\"problem\" id=\"fs-id1169147783916\"><p id=\"fs-id1169147783918\">\\(\\sqrt{-75}+\\sqrt{-48}\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1169148098416\">\\(9\\sqrt{3}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145844083\"><div data-type=\"problem\" id=\"fs-id1169145844085\"><p id=\"fs-id1169143580639\">\\(\\sqrt{-12}+\\sqrt{-75}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147751002\"><div data-type=\"problem\" id=\"fs-id1169145669417\"><p id=\"fs-id1169145669419\">\\(\\sqrt{-50}+\\sqrt{-18}\\)<\/p><\/div><div data-type=\"solution\"><p>\\(8\\sqrt{2}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143580475\"><div data-type=\"problem\" id=\"fs-id1169143580477\"><p id=\"fs-id1169147966380\">\\(\\sqrt{-72}+\\sqrt{-8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145505198\"><div data-type=\"problem\" id=\"fs-id1169145696480\"><p id=\"fs-id1169145696482\">\\(\\left(1+3i\\right)+\\left(7+4i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145505873\"><p id=\"fs-id1169143688226\">\\(8+7i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147978369\"><div data-type=\"problem\" id=\"fs-id1169148103770\"><p id=\"fs-id1169148103772\">\\(\\left(6+2i\\right)+\\left(3-4i\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148211974\"><div data-type=\"problem\" id=\"fs-id1169147740283\"><p id=\"fs-id1169147740285\">\\(\\left(8-i\\right)+\\left(6+3i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147839316\"><p id=\"fs-id1169147839318\">\\(14+2i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148071796\"><div data-type=\"problem\" id=\"fs-id1169148071799\"><p id=\"fs-id1169147843469\">\\(\\left(7-4i\\right)+\\left(-2-6i\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143728397\"><div data-type=\"problem\" id=\"fs-id1169143728399\"><p id=\"fs-id1169147720896\">\\(\\left(1-4i\\right)-\\left(3-6i\\right)\\)<\/p><\/div><div data-type=\"solution\"><p>\\(-2+2i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145661457\"><div data-type=\"problem\" id=\"fs-id1169145670340\"><p id=\"fs-id1169145670343\">\\(\\left(8-4i\\right)-\\left(3+7i\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145663778\"><div data-type=\"problem\" id=\"fs-id1169145505214\"><p id=\"fs-id1169145505217\">\\(\\left(6+i\\right)-\\left(-2-4i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148229777\"><p id=\"fs-id1169148229779\">\\(8+5i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148233529\"><div data-type=\"problem\" id=\"fs-id1169148064110\"><p id=\"fs-id1169148064112\">\\(\\left(-2+5i\\right)-\\left(-5+6i\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143643797\"><div data-type=\"problem\" id=\"fs-id1169143643799\"><p id=\"fs-id1169145722644\">\\(\\left(5-\\sqrt{-36}\\right)+\\left(2-\\sqrt{-49}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147848260\"><p id=\"fs-id1169147848263\">\\(7-13i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147878950\"><div data-type=\"problem\" id=\"fs-id1169148229531\"><p id=\"fs-id1169148229533\">\\(\\left(-3+\\sqrt{-64}\\right)+\\left(5-\\sqrt{-16}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147982815\"><div data-type=\"problem\" id=\"fs-id1169145670355\"><p id=\"fs-id1169145670357\">\\(\\left(-7-\\sqrt{-50}\\right)-\\left(-32-\\sqrt{-18}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148207078\"><p id=\"fs-id1169148207080\">\\(25-2\\sqrt{2}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169141298513\"><div data-type=\"problem\" id=\"fs-id1169141298515\"><p id=\"fs-id1169145532866\">\\(\\left(-5+\\sqrt{-27}\\right)-\\left(-4-\\sqrt{-48}\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1169145737955\"><strong data-effect=\"bold\">Multiply Complex Numbers<\/strong><\/p><p id=\"fs-id1169147775327\">In the following exercises, multiply.<\/p><div data-type=\"exercise\" id=\"fs-id1169147775330\"><div data-type=\"problem\" id=\"fs-id1169147775332\"><p id=\"fs-id1169147965722\">\\(4i\\left(5-3i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145670301\"><p id=\"fs-id1169145670303\">\\(12+20i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147982473\"><div data-type=\"problem\" id=\"fs-id1169145720264\"><p id=\"fs-id1169145720266\">\\(2i\\left(-3+4i\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147746296\"><div data-type=\"problem\" id=\"fs-id1169143534249\"><p id=\"fs-id1169143534251\">\\(-6i\\left(-3-2i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147863494\"><p id=\"fs-id1169147863496\">\\(-12+18i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147841097\"><div data-type=\"problem\" id=\"fs-id1169147841099\"><p id=\"fs-id1169147841101\">\\(\\text{\u2212}i\\left(6+5i\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145639745\"><div data-type=\"problem\" id=\"fs-id1169145639748\"><p id=\"fs-id1169145639750\">\\(\\left(4+3i\\right)\\left(-5+6i\\right)\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1169141298387\">\\(-38++9i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147739873\"><div data-type=\"problem\" id=\"fs-id1169147739876\"><p id=\"fs-id1169148233360\">\\(\\left(-2-5i\\right)\\left(-4+3i\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147783892\"><div data-type=\"problem\" id=\"fs-id1169147783894\"><p id=\"fs-id1169147783896\">\\(\\left(-3+3i\\right)\\left(-2-7i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145506798\"><p id=\"fs-id1169145506800\">\\(27+15i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147878970\"><div data-type=\"problem\" id=\"fs-id1169147878972\"><p id=\"fs-id1169147878974\">\\(\\left(-6-2i\\right)\\left(-3-5i\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1169147739869\">In the following exercises, multiply using the Product of Binomial Squares Pattern.<\/p><div data-type=\"exercise\" id=\"fs-id1169148077956\"><div data-type=\"problem\" id=\"fs-id1169148077958\"><p id=\"fs-id1169147963850\">\\({\\left(3+4i\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169142133163\"><p id=\"fs-id1169143520481\">\\(-7+24i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147852519\"><div data-type=\"problem\" id=\"fs-id1169147852521\"><p id=\"fs-id1169145733973\">\\({\\left(-1+5i\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147803530\"><div data-type=\"problem\" id=\"fs-id1169145577064\"><p id=\"fs-id1169145577066\">\\({\\left(-2-3i\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147804412\"><p id=\"fs-id1169147804414\">\\(-5-12i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143573850\"><div data-type=\"problem\" id=\"fs-id1169143533452\"><p id=\"fs-id1169143533454\">\\({\\left(-6-5i\\right)}^{2}\\)<\/p><\/div><\/div><p id=\"fs-id1169148123174\">In the following exercises, multiply.<\/p><div data-type=\"exercise\" id=\"fs-id1169148123177\"><div data-type=\"problem\" id=\"fs-id1169148123179\"><p id=\"fs-id1169147874530\">\\(\\sqrt{-25}\u00b7\\sqrt{-36}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169141298052\"><p>\\(30i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147741809\"><div data-type=\"problem\" id=\"fs-id1169147741811\"><p id=\"fs-id1169148097690\">\\(\\sqrt{-4}\u00b7\\sqrt{-16}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147840679\"><div data-type=\"problem\" id=\"fs-id1169147840681\"><p id=\"fs-id1169147840683\">\\(\\sqrt{-9}\u00b7\\sqrt{-100}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143442627\"><p id=\"fs-id1169143442630\">\\(-30\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147875254\"><div data-type=\"problem\" id=\"fs-id1169147875256\"><p id=\"fs-id1169147875258\">\\(\\sqrt{-64}\u00b7\\sqrt{-9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143728425\"><div data-type=\"problem\" id=\"fs-id1169145605886\"><p id=\"fs-id1169145605888\">\\(\\left(-2-\\sqrt{-27}\\right)\\left(4-\\sqrt{-48}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169142416307\"><p id=\"fs-id1169142416309\">\\(-44+4\\sqrt{3}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147764582\"><div data-type=\"problem\" id=\"fs-id1169147764584\"><p id=\"fs-id1169147764587\">\\(\\left(5-\\sqrt{-12}\\right)\\left(-3+\\sqrt{-75}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145662602\"><div data-type=\"problem\" id=\"fs-id1169145662604\"><p id=\"fs-id1169143517978\">\\(\\left(2+\\sqrt{-8}\\right)\\left(-4+\\sqrt{-18}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147965152\"><p id=\"fs-id1169145519425\">\\(-20-2\\sqrt{2}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145601711\"><div data-type=\"problem\" id=\"fs-id1169145598899\"><p id=\"fs-id1169145598901\">\\(\\left(5+\\sqrt{-18}\\right)\\left(-2-\\sqrt{-50}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147768224\"><div data-type=\"problem\" id=\"fs-id1169147768226\"><p id=\"fs-id1169147768228\">\\(\\left(2-i\\right)\\left(2+i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143525861\"><p id=\"fs-id1169145606349\">5<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145606354\"><div data-type=\"problem\" id=\"fs-id1169141473728\"><p id=\"fs-id1169141473730\">\\(\\left(4-5i\\right)\\left(4+5i\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p>\\(\\left(7-2i\\right)\\left(7+2i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145597852\"><p id=\"fs-id1169145597854\">53<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148225589\"><div data-type=\"problem\" id=\"fs-id1169148225591\"><p id=\"fs-id1169143586050\">\\(\\left(-3-8i\\right)\\left(-3+8i\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1169148252283\">In the following exercises, multiply using the Product of Complex Conjugates Pattern.<\/p><div data-type=\"exercise\" id=\"fs-id1169145785940\"><div data-type=\"problem\" id=\"fs-id1169145785942\"><p id=\"fs-id1169141409123\">\\(\\left(7-i\\right)\\left(7+i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147837530\"><p id=\"fs-id1169148125806\">50<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147771371\"><div data-type=\"problem\" id=\"fs-id1169147771373\"><p id=\"fs-id1169147771375\">\\(\\left(6-5i\\right)\\left(6+5i\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145670789\"><div data-type=\"problem\" id=\"fs-id1169145670791\"><p id=\"fs-id1169148218572\">\\(\\left(9-2i\\right)\\left(9+2i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147855189\"><p id=\"fs-id1169147855191\">85<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148227215\"><div data-type=\"problem\" id=\"fs-id1169148190446\"><p id=\"fs-id1169148190448\">\\(\\left(-3-4i\\right)\\left(-3+4i\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1169148097665\"><strong data-effect=\"bold\">Divide Complex Numbers<\/strong><\/p><p id=\"fs-id1169148097671\">In the following exercises, divide.<\/p><div data-type=\"exercise\" id=\"fs-id1169148125820\"><div data-type=\"problem\" id=\"fs-id1169148125822\"><p id=\"fs-id1169145622251\">\\(\\frac{3+4i}{4-3i}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147815810\"><p id=\"fs-id1169147815812\"><em data-effect=\"italics\">i<\/em><\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143579887\"><div data-type=\"problem\" id=\"fs-id1169143579889\"><p id=\"fs-id1169145747976\">\\(\\frac{5-2i}{2+5i}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145598404\"><div data-type=\"problem\" id=\"fs-id1169145598406\"><p id=\"fs-id1169145598408\">\\(\\frac{2+i}{3-4i}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145578886\"><p id=\"fs-id1169145578888\">\\(\\frac{2}{25}+\\frac{11}{25}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148248849\"><div data-type=\"problem\" id=\"fs-id1169148248851\"><p id=\"fs-id1169148248853\">\\(\\frac{3-2i}{6+i}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145670320\"><div data-type=\"problem\" id=\"fs-id1169145670322\"><p id=\"fs-id1169148218590\">\\(\\frac{3}{2-3i}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148125778\"><p id=\"fs-id1169148125780\">\\(\\frac{6}{13}+\\frac{9}{13}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148079849\"><div data-type=\"problem\" id=\"fs-id1169148079851\"><p id=\"fs-id1169148079853\">\\(\\frac{2}{4-5i}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145716408\"><div data-type=\"problem\" id=\"fs-id1169145716410\"><p id=\"fs-id1169145716412\">\\(\\frac{-4}{3-2i}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145607991\"><p id=\"fs-id1169145748060\">\\(-\\frac{12}{13}-\\frac{8}{13}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145608611\"><div data-type=\"problem\" id=\"fs-id1169145608613\"><p id=\"fs-id1169145608616\">\\(\\frac{-1}{3+2i}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143520334\"><div data-type=\"problem\" id=\"fs-id1169143520336\"><p id=\"fs-id1169141376159\">\\(\\frac{1+4i}{3i}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145640198\"><p id=\"fs-id1169145640200\">\\(\\frac{4}{3}-\\frac{1}{3}i\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1169143614240\"><p id=\"fs-id1169145622261\">\\(\\frac{4+3i}{7i}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148248922\"><div data-type=\"problem\" id=\"fs-id1169148248924\"><p id=\"fs-id1169148248926\">\\(\\frac{-2-3i}{4i}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143579814\"><p id=\"fs-id1169143579816\">\\(-\\frac{3}{4}+\\frac{1}{2}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143385633\"><div data-type=\"problem\" id=\"fs-id1169143385635\"><p id=\"fs-id1169143385637\">\\(\\frac{-3-5i}{2i}\\)<\/p><\/div><\/div><p id=\"fs-id1169141030520\"><strong data-effect=\"bold\">Simplify Powers of <em data-effect=\"italics\">i<\/em><\/strong><\/p><p id=\"fs-id1169145575830\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1169145575833\"><div data-type=\"problem\" id=\"fs-id1169145575835\"><p id=\"fs-id1169145575837\">\\({i}^{41}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148233571\"><p id=\"fs-id1169148233573\"><em data-effect=\"italics\">i<\/em><\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169141036435\"><div data-type=\"problem\" id=\"fs-id1169141036438\"><p id=\"fs-id1169141036440\">\\({i}^{39}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145666672\"><div data-type=\"problem\" id=\"fs-id1169145666674\"><p id=\"fs-id1169145666676\">\\({i}^{66}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143687470\"><p id=\"fs-id1169143687472\">\\(-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145716645\"><div data-type=\"problem\" id=\"fs-id1169145716647\"><p id=\"fs-id1169145716649\">\\({i}^{48}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147849782\"><div data-type=\"problem\" id=\"fs-id1169147849784\"><p id=\"fs-id1169147849786\">\\({i}^{128}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143637803\"><p id=\"fs-id1169143637805\">1<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148077997\"><div data-type=\"problem\" id=\"fs-id1169148077999\"><p id=\"fs-id1169148078001\">\\({i}^{162}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p>\\({i}^{137}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143280974\"><p id=\"fs-id1169143280976\"><em data-effect=\"italics\">i<\/em><\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143280983\"><div data-type=\"problem\" id=\"fs-id1169145663864\"><p id=\"fs-id1169145663866\">\\({i}^{255}\\)<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1169142417355\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1169141472919\"><div data-type=\"problem\" id=\"fs-id1169141472921\"><p id=\"fs-id1169141472924\">Explain the relationship between real numbers and complex numbers.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169141472928\"><p id=\"fs-id1169143637743\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143637748\"><div data-type=\"problem\" id=\"fs-id1169143637750\"><p id=\"fs-id1169143637752\">Aniket multiplied as follows and he got the wrong answer. What is wrong with his reasoning?<\/p><p id=\"fs-id1169143637755\">\\(\\begin{array}{c}\\hfill \\sqrt{-7}\u00b7\\sqrt{-7}\\hfill \\\\ \\hfill \\sqrt{49}\\hfill \\\\ \\hfill 7\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147878746\"><div data-type=\"problem\" id=\"fs-id1169147878748\"><p id=\"fs-id1169147878750\">Why is \\(\\sqrt{-64}=8i\\) but \\(\\sqrt[3]{-64}=-4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147878773\"><p id=\"fs-id1169147878775\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169141298444\"><div data-type=\"problem\" id=\"fs-id1169141298446\"><p id=\"fs-id1169141298448\">Explain how dividing complex numbers is similar to rationalizing a denominator.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169147841323\"><h4 data-type=\"title\">Self Check<\/h4><p><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1169147841336\" data-alt=\"The table has 4 columns and 4 rows. The first row is a header row with the headers \u201cI can\u2026\u201d, \u201cConfidently\u201d, \u201cWith some help.\u201d, and \u201cNo \u2013 I don\u2019t get it!\u201d. The first column contains the phrases \u201cevaluate the square root of a negative number\u201d, \u201cadd or subtract complex numbers\u201d, \u201cmultiply complex numbers\u201d, \u201cdivide complex numbers\u201d, and \u201csimplify powers of i\u201d. The other columns are left blank so the learner can indicate their level of understanding.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The table has 4 columns and 4 rows. The first row is a header row with the headers \u201cI can\u2026\u201d, \u201cConfidently\u201d, \u201cWith some help.\u201d, and \u201cNo \u2013 I don\u2019t get it!\u201d. The first column contains the phrases \u201cevaluate the square root of a negative number\u201d, \u201cadd or subtract complex numbers\u201d, \u201cmultiply complex numbers\u201d, \u201cdivide complex numbers\u201d, and \u201csimplify powers of i\u201d. The other columns are left blank so the learner can indicate their level of understanding.\"><\/span><p id=\"fs-id1169143581464\"><span class=\"token\">\u24d1<\/span> On a scale of \\(1-10,\\) how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p><\/div><\/div><div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1169145519825\"><h3 data-type=\"title\">Chapter Review Exercises<\/h3><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169145519828\"><h4 data-type=\"title\"><a href=\"\/contents\/eb8b8413-de16-4b16-8555-f29918cf1207\" class=\"target-chapter\">Simplify Expressions with Roots<\/a><\/h4><p id=\"fs-id1169145494084\"><strong data-effect=\"bold\">Simplify Expressions with Roots<\/strong><\/p><p id=\"fs-id1169145494089\">In the following exercises, simplify.<\/p><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1169145494096\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt{225}\\)<span class=\"token\">\u24d1<\/span>\\(\\text{\u2212}\\sqrt{16}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143533387\"><p id=\"fs-id1169143533389\"><span class=\"token\">\u24d0<\/span> 15 <span class=\"token\">\u24d1<\/span> \\(-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148125835\"><div data-type=\"problem\" id=\"fs-id1169148125837\"><p id=\"fs-id1169148125840\"><span class=\"token\">\u24d0<\/span>\\(\\text{\u2212}\\sqrt{169}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt{-8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145519880\"><div data-type=\"problem\" id=\"fs-id1169145519883\"><p id=\"fs-id1169145519885\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt[3]{8}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[4]{81}\\)<span class=\"token\">\u24d2<\/span>\\(\\sqrt[5]{243}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145785915\"><p id=\"fs-id1169145785917\"><span class=\"token\">\u24d0<\/span> 2 <span class=\"token\">\u24d1<\/span> 3 <span class=\"token\">\u24d2<\/span> 3<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169141015057\"><div data-type=\"problem\" id=\"fs-id1169141015059\"><p id=\"fs-id1169141015061\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt[3]{-512}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[4]{-81}\\)<span class=\"token\">\u24d2<\/span>\\(\\sqrt[5]{-1}\\)<\/p><\/div><\/div><p id=\"fs-id1169148218622\"><strong data-effect=\"bold\">Estimate and Approximate Roots<\/strong><\/p><p id=\"fs-id1169148218628\">In the following exercises, estimate each root between two consecutive whole numbers.<\/p><div data-type=\"exercise\" id=\"fs-id1169148218631\"><div data-type=\"problem\" id=\"fs-id1169145716563\"><p id=\"fs-id1169145716565\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt{68}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{84}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148125724\"><p id=\"fs-id1169148125726\"><span class=\"token\">\u24d0<\/span>\\(8&lt;\\sqrt{68}&lt;9\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(4&lt;\\sqrt[3]{84}&lt;5\\)<\/div><\/div><p id=\"fs-id1169148218563\">In the following exercises, approximate each root and round to two decimal places.<\/p><div data-type=\"exercise\" id=\"fs-id1169148218567\"><div data-type=\"problem\" id=\"fs-id1169148218569\"><p id=\"fs-id1169147878704\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt{37}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{84}\\)<span class=\"token\">\u24d2<\/span>\\(\\sqrt[4]{125}\\)<\/p><\/div><\/div><p id=\"fs-id1169141376105\"><strong data-effect=\"bold\">Simplify Variable Expressions with Roots<\/strong><\/p><p id=\"fs-id1169143637764\">In the following exercises, simplify using absolute values as necessary.<\/p><div data-type=\"exercise\" id=\"fs-id1169143637767\"><div data-type=\"problem\" id=\"fs-id1169143637769\"><p id=\"fs-id1169143637772\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt[3]{{a}^{3}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt[7]{{b}^{7}}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169148117671\"><p id=\"fs-id1169148117673\"><span class=\"token\">\u24d0<\/span><em data-effect=\"italics\">a<\/em><span class=\"token\">\u24d1<\/span>\\(|b|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169141376123\"><div data-type=\"problem\" id=\"fs-id1169141376125\"><p id=\"fs-id1169141376127\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{{a}^{14}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt{{w}^{24}}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169141373734\"><div data-type=\"problem\" id=\"fs-id1169141373736\"><p id=\"fs-id1169141373738\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt[4]{{m}^{8}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt[5]{{n}^{20}}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169143579829\"><p id=\"fs-id1169143579832\"><span class=\"token\">\u24d0<\/span>\\({m}^{2}\\)<span class=\"token\">\u24d1<\/span>\\({n}^{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143579862\"><div data-type=\"problem\" id=\"fs-id1169143579864\"><p id=\"fs-id1169143579866\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{121{m}^{20}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\text{\u2212}\\sqrt{64{a}^{2}}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147797054\"><div data-type=\"problem\" id=\"fs-id1169147797057\"><p id=\"fs-id1169147797059\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt[3]{216{a}^{6}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt[5]{32{b}^{20}}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169145573794\"><p id=\"fs-id1169145573796\"><span class=\"token\">\u24d0<\/span>\\(6{a}^{2}\\)<span class=\"token\">\u24d1<\/span>\\(2{b}^{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145748005\"><div data-type=\"problem\" id=\"fs-id1169145748007\"><p id=\"fs-id1169145748010\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{144{x}^{2}{y}^{2}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt{169{w}^{8}{y}^{10}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\sqrt[3]{8{a}^{51}{b}^{6}}\\)<\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169143385610\"><h4 data-type=\"title\"><a href=\"\/contents\/dbb319b9-f421-4570-9953-ca5a39b933dc\" class=\"target-chapter\">Simplify Radical Expressions<\/a><\/h4><p id=\"fs-id1169143385621\"><strong data-effect=\"bold\">Use the Product Property to Simplify Radical Expressions<\/strong><\/p><p id=\"fs-id1169145977397\">In the following exercises, use the Product Property to simplify radical expressions.<\/p><div data-type=\"exercise\" id=\"fs-id1169145977400\"><div data-type=\"problem\" id=\"fs-id1169145977402\"><p id=\"fs-id1169145977404\">\\(\\sqrt{125}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145977414\"><p id=\"fs-id1169145977416\">\\(5\\sqrt{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145977428\"><div data-type=\"problem\" id=\"fs-id1169145977430\"><p id=\"fs-id1169145977432\">\\(\\sqrt{675}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145977455\"><div data-type=\"problem\" id=\"fs-id1169145977458\"><p id=\"fs-id1169145977460\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt[3]{625}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[6]{128}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147784223\"><p id=\"fs-id1169147784225\"><span class=\"token\">\u24d0<\/span>\\(5\\sqrt[3]{5}\\)<span class=\"token\">\u24d1<\/span>\\(2\\sqrt[6]{2}\\)<\/p><\/div><\/div><p id=\"fs-id1169147784260\">In the following exercises, simplify using absolute value signs as needed.<\/p><div data-type=\"exercise\" id=\"fs-id1169147784263\"><div data-type=\"problem\" id=\"fs-id1169147784265\"><p id=\"fs-id1169147784267\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{{a}^{23}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{{b}^{8}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\sqrt[8]{{c}^{13}}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145977715\"><div data-type=\"problem\" id=\"fs-id1169145977717\"><p id=\"fs-id1169145977719\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{80{s}^{15}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt[5]{96{a}^{7}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\sqrt[6]{128{b}^{7}}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169145579099\"><p id=\"fs-id1169145579101\"><span class=\"token\">\u24d0<\/span>\\(4|{s}^{7}|\\sqrt[]{5s}\\)<span class=\"token\">\u24d1<\/span>\\(2a\\sqrt[5]{3{a}^{2}}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(2|b|\\sqrt[6]{2b}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147987538\"><div data-type=\"problem\" id=\"fs-id1169147987540\"><p id=\"fs-id1169147987542\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{96{r}^{3}{s}^{3}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{80{x}^{7}{y}^{6}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\sqrt[4]{80{x}^{8}{y}^{9}}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148063219\"><div data-type=\"problem\" id=\"fs-id1169148063221\"><p id=\"fs-id1169147725423\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt[5]{-32}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt[8]{-1}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169147725454\"><p id=\"fs-id1169147725457\"><span class=\"token\">\u24d0<\/span>\\(-2\\)<span class=\"token\">\u24d1<\/span> not real<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147725477\"><div data-type=\"problem\" id=\"fs-id1169147725479\"><p id=\"fs-id1169147725481\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(8+\\sqrt{96}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{2+\\sqrt{40}}{2}\\)<\/div><\/div><p id=\"fs-id1169148053130\"><strong data-effect=\"bold\">Use the Quotient Property to Simplify Radical Expressions<\/strong><\/p><p id=\"fs-id1169148053136\">In the following exercises, use the Quotient Property to simplify square roots.<\/p><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1169148053142\"><p id=\"fs-id1169148053144\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt{\\frac{72}{98}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{\\frac{24}{81}}\\)<span class=\"token\">\u24d2<\/span>\\(\\sqrt[4]{\\frac{6}{96}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148053203\"><p id=\"fs-id1169148053205\"><span class=\"token\">\u24d0<\/span>\\(\\frac{6}{7}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{2}{3}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{1}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143519248\"><div data-type=\"problem\" id=\"fs-id1169143519250\"><p id=\"fs-id1169143519253\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt{\\frac{{y}^{4}}{{y}^{8}}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[5]{\\frac{{u}^{21}}{{u}^{11}}}\\)<span class=\"token\">\u24d2<\/span>\\(\\sqrt[6]{\\frac{{v}^{30}}{{v}^{12}}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143634779\"><div data-type=\"problem\" id=\"fs-id1169143634782\"><p id=\"fs-id1169143634784\">\\(\\sqrt{\\frac{300{m}^{5}}{64}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143634804\"><p id=\"fs-id1169143634806\">\\(\\frac{10{m}^{2}\\sqrt{3m}}{8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148249775\"><div data-type=\"problem\" id=\"fs-id1169148249777\"><p id=\"fs-id1169148249779\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{\\frac{28{p}^{7}}{{q}^{2}}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{\\frac{81{s}^{8}}{{t}^{3}}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\sqrt[4]{\\frac{64{p}^{15}}{{q}^{12}}}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143459265\"><div data-type=\"problem\" id=\"fs-id1169143459267\"><p id=\"fs-id1169143459269\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{\\frac{27{p}^{2}q}{108{p}^{4}{q}^{3}}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{\\frac{16{c}^{5}{d}^{7}}{250{c}^{2}{d}^{2}}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\sqrt[6]{\\frac{2{m}^{9}{n}^{7}}{128{m}^{3}n}}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169145491841\"><p id=\"fs-id1169145491843\"><span class=\"token\">\u24d0<\/span>\\(\\frac{1}{2|pq|}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{2cd\\sqrt[5]{2{d}^{2}}}{5}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\frac{|mn|\\sqrt[6]{2}}{2}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147834409\"><div data-type=\"problem\" id=\"fs-id1169147834411\"><p id=\"fs-id1169147834413\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{80{q}^{5}}}{\\sqrt{5q}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{-625}}{\\sqrt[3]{5}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\frac{\\sqrt[4]{80{m}^{7}}}{\\sqrt[4]{5m}}\\)<\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169142429740\"><h4 data-type=\"title\"><a href=\"\/contents\/eb676a52-0094-4ffb-95d3-c8eb0596e397\" class=\"target-chapter\">Simplify Rational Exponents<\/a><\/h4><p id=\"fs-id1169142429749\"><strong data-effect=\"bold\">Simplify expressions with \\({a}^{\\frac{1}{n}}\\)<\/strong><\/p><p id=\"fs-id1169142429768\">In the following exercises, write as a radical expression.<\/p><div data-type=\"exercise\" id=\"fs-id1169142429771\"><div data-type=\"problem\" id=\"fs-id1169142429774\"><p id=\"fs-id1169142429776\"><span class=\"token\">\u24d0<\/span>\\({r}^{\\frac{1}{2}}\\)<span class=\"token\">\u24d1<\/span>\\({s}^{\\frac{1}{3}}\\)<span class=\"token\">\u24d2<\/span>\\({t}^{\\frac{1}{4}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145969665\"><p id=\"fs-id1169145969667\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt{r}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{s}\\)<span class=\"token\">\u24d2<\/span>\\(\\sqrt[4]{t}\\)<\/p><\/div><\/div><p id=\"fs-id1169145969708\">In the following exercises, write with a rational exponent.<\/p><div data-type=\"exercise\" id=\"fs-id1169145969712\"><div data-type=\"problem\" id=\"fs-id1169145969714\"><p id=\"fs-id1169145969716\"><span class=\"token\">\u24d0<\/span>\\(\\sqrt{21p}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[4]{8q}\\)<span class=\"token\">\u24d2<\/span>\\(4\\sqrt[6]{36r}\\)<\/p><\/div><\/div><p id=\"fs-id1169145744783\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1169145744786\"><div data-type=\"problem\" id=\"fs-id1169145744788\"><p id=\"fs-id1169145744790\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({625}^{\\frac{1}{4}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({243}^{\\frac{1}{5}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({32}^{\\frac{1}{5}}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169145744850\"><p id=\"fs-id1169145744852\"><span class=\"token\">\u24d0<\/span> 5 <span class=\"token\">\u24d1<\/span> 3 <span class=\"token\">\u24d2<\/span> 2<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145924873\"><div data-type=\"problem\" id=\"fs-id1169145924876\"><p id=\"fs-id1169145924878\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({\\left(-1,000\\right)}^{\\frac{1}{3}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\text{\u2212}{1,000}^{\\frac{1}{3}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({\\left(1,000\\right)}^{-\\frac{1}{3}}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143638122\"><div data-type=\"problem\" id=\"fs-id1169143638124\"><p id=\"fs-id1169143638126\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({\\left(-32\\right)}^{\\frac{1}{5}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({\\left(243\\right)}^{-\\frac{1}{5}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\text{\u2212}{125}^{\\frac{1}{3}}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169143638202\"><p id=\"fs-id1169143638204\"><span class=\"token\">\u24d0<\/span>\\(-2\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{1}{3}\\)<span class=\"token\">\u24d2<\/span>\\(-5\\)<\/p><\/div><\/div><p id=\"fs-id1169148217684\"><strong data-effect=\"bold\">Simplify Expressions with \\({a}^{\\frac{m}{n}}\\)<\/strong><\/p><p id=\"fs-id1169148217703\">In the following exercises, write with a rational exponent.<\/p><div data-type=\"exercise\" id=\"fs-id1169148217707\"><div data-type=\"problem\" id=\"fs-id1169148217709\"><p id=\"fs-id1169148217711\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt[4]{{r}^{7}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({\\left(\\sqrt[5]{2pq}\\right)}^{3}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\sqrt[4]{{\\left(\\frac{12m}{7n}\\right)}^{3}}\\)<\/div><\/div><p id=\"fs-id1169145642785\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1169145642788\"><div data-type=\"problem\" id=\"fs-id1169145642791\"><p id=\"fs-id1169145642793\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({25}^{\\frac{3}{2}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({9}^{-\\frac{3}{2}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({\\left(-64\\right)}^{\\frac{2}{3}}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169145642860\"><p id=\"fs-id1169145642862\"><span class=\"token\">\u24d0<\/span> 125 <span class=\"token\">\u24d1<\/span> \\(\\frac{1}{27}\\) <span class=\"token\">\u24d2<\/span> 16<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143573570\"><div data-type=\"problem\" id=\"fs-id1169143573572\"><p id=\"fs-id1169143573575\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\text{\u2212}{64}^{\\frac{3}{2}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\text{\u2212}{64}^{-\\frac{3}{2}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\({\\left(-64\\right)}^{\\frac{3}{2}}\\)<\/div><\/div><p id=\"fs-id1169143581323\"><strong data-effect=\"bold\">Use the Laws of Exponents to Simplify Expressions with Rational Exponents<\/strong><\/p><p id=\"fs-id1169143581329\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1169143581332\"><div data-type=\"problem\" id=\"fs-id1169143581334\"><p id=\"fs-id1169143581336\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({6}^{\\frac{5}{2}}\u00b7{6}^{\\frac{1}{2}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({\\left({b}^{15}\\right)}^{\\frac{3}{5}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\frac{{w}^{\\frac{2}{7}}}{{w}^{\\frac{9}{7}}}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169148237312\"><p id=\"fs-id1169148237315\"><span class=\"token\">\u24d0<\/span>\\({6}^{3}\\)<span class=\"token\">\u24d1<\/span>\\({b}^{9}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{1}{w}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148237359\"><div data-type=\"problem\" id=\"fs-id1169148237361\"><p id=\"fs-id1169148237363\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\frac{{a}^{\\frac{3}{4}}\u00b7{a}^{-\\frac{1}{4}}}{{a}^{-\\frac{10}{4}}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({\\left(\\frac{27\\text{\u200b}{b}^{\\frac{2}{3}}\\text{\u200b}{c}^{-\\frac{5}{2}}}{{b}^{-\\frac{7}{3}}{c}^{\\frac{1}{2}}}\\right)}^{\\frac{1}{3}}\\)<\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169148115149\"><h4 data-type=\"title\"><a href=\"\/contents\/a38378e8-cc28-44f4-8c29-789cc2550c6a\" class=\"target-chapter\">Add, Subtract and Multiply Radical Expressions<\/a><\/h4><p id=\"fs-id1169147730989\"><strong data-effect=\"bold\">Add and Subtract Radical Expressions<\/strong><\/p><p id=\"fs-id1169147730996\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1169147730999\"><div data-type=\"problem\" id=\"fs-id1169147731001\"><p id=\"fs-id1169147731003\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(7\\sqrt{2}-3\\sqrt{2}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(7\\sqrt[3]{p}+2\\sqrt[3]{p}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(5\\sqrt[3]{x}-3\\sqrt[3]{x}\\)<\/div><div data-type=\"solution\"><p id=\"fs-id1169147731079\"><span class=\"token\">\u24d0<\/span>\\(4\\sqrt{2}\\)<span class=\"token\">\u24d1<\/span>\\(9\\sqrt[3]{p}\\)<span class=\"token\">\u24d2<\/span>\\(2\\sqrt[3]{x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147977766\"><div data-type=\"problem\" id=\"fs-id1169147977768\"><p id=\"fs-id1169147977770\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{11b}-5\\sqrt{11b}+3\\sqrt{11b}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(8\\sqrt[4]{11cd}+5\\sqrt[4]{11cd}-9\\sqrt[4]{11cd}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143686190\"><div data-type=\"problem\" id=\"fs-id1169143686193\"><p id=\"fs-id1169143686195\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{48}+\\sqrt{27}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{54}+\\sqrt[3]{128}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(6\\sqrt[4]{5}-\\frac{3}{2}\\sqrt[4]{320}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169148075655\"><p id=\"fs-id1169148075657\"><span class=\"token\">\u24d0<\/span>\\(7\\sqrt{3}\\)<span class=\"token\">\u24d1<\/span>\\(7\\sqrt[3]{2}\\)<span class=\"token\">\u24d2<\/span>\\(3\\sqrt[4]{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148075705\"><div data-type=\"problem\" id=\"fs-id1169148075708\"><p id=\"fs-id1169148075710\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{80{c}^{7}}-\\sqrt{20{c}^{7}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(2\\sqrt[4]{162{r}^{10}}+4\\sqrt[4]{32{r}^{10}}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145520451\"><div data-type=\"problem\" id=\"fs-id1169145495463\"><p id=\"fs-id1169145495465\">\\(3\\sqrt{75{y}^{2}}+8y\\sqrt{48}-\\sqrt{300{y}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145495504\"><p id=\"fs-id1169145495506\">\\(37y\\sqrt{3}\\)<\/p><\/div><\/div><p id=\"fs-id1169145495520\"><strong data-effect=\"bold\">Multiply Radical Expressions<\/strong><\/p><p id=\"fs-id1169145495526\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1169145495530\"><div data-type=\"problem\" id=\"fs-id1169145495532\"><p id=\"fs-id1169145495534\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(5\\sqrt{6}\\right)\\left(\\text{\u2212}\\sqrt{12}\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(-2\\sqrt[4]{18}\\right)\\left(\\text{\u2212}\\sqrt[4]{9}\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169146012528\"><div data-type=\"problem\" id=\"fs-id1169146012531\"><p id=\"fs-id1169146012533\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(3\\sqrt{2{x}^{3}}\\right)\\left(7\\sqrt{18{x}^{2}}\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(-6\\sqrt[3]{20{a}^{2}}\\right)\\left(-2\\sqrt[3]{16{a}^{3}}\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1169145945831\"><p id=\"fs-id1169145945833\"><span class=\"token\">\u24d0<\/span>\\(126{x}^{2}\\sqrt{2}\\)<span class=\"token\">\u24d1<\/span>\\(48a\\sqrt[3]{{a}^{2}}\\)<\/p><\/div><\/div><p id=\"fs-id1169145945878\"><strong data-effect=\"bold\">Use Polynomial Multiplication to Multiply Radical Expressions<\/strong><\/p><p id=\"fs-id1169148125298\">In the following exercises, multiply.<\/p><div data-type=\"exercise\" id=\"fs-id1169148125302\"><div data-type=\"problem\" id=\"fs-id1169148125304\"><p id=\"fs-id1169148125306\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{11}\\left(8+4\\sqrt{11}\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{3}\\left(\\sqrt[3]{9}+\\sqrt[3]{18}\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145969398\"><div data-type=\"problem\" id=\"fs-id1169145969400\"><p id=\"fs-id1169145969402\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\left(3-2\\sqrt{7}\\right)\\left(5-4\\sqrt{7}\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\left(\\sqrt[3]{x}-5\\right)\\left(\\sqrt[3]{x}-3\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1169145969487\"><p id=\"fs-id1169145969489\"><span class=\"token\">\u24d0<\/span>\\(71-22\\sqrt{7}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{{x}^{2}}-8\\sqrt[3]{x}+15\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145970420\"><div data-type=\"problem\" id=\"fs-id1169145970422\"><p id=\"fs-id1169145970424\">\\(\\left(2\\sqrt{7}-5\\sqrt{11}\\right)\\left(4\\sqrt{7}+9\\sqrt{11}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145970489\"><div data-type=\"problem\" id=\"fs-id1169145491418\"><p id=\"fs-id1169145491420\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\({\\left(4+\\sqrt{11}\\right)}^{2}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\({\\left(3-2\\sqrt{5}\\right)}^{2}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169145491478\"><p id=\"fs-id1169145491481\"><span class=\"token\">\u24d0<\/span>\\(27+8\\sqrt{11}\\)<span class=\"token\">\u24d1<\/span>\\(29-12\\sqrt{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145491521\"><div data-type=\"problem\" id=\"fs-id1169145491523\"><p id=\"fs-id1169145491525\">\\(\\left(7+\\sqrt{10}\\right)\\left(7-\\sqrt{10}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148063307\"><div data-type=\"problem\" id=\"fs-id1169148063309\"><p id=\"fs-id1169148063311\">\\(\\left(\\sqrt[3]{3x}+2\\right)\\left(\\sqrt[3]{3x}-2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148063355\"><p id=\"fs-id1169148063357\">\\(\\sqrt[3]{9{x}^{2}}-4\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169143662135\"><h4 data-type=\"title\"><a href=\"\/contents\/94bbb796-17c2-4a4d-a202-8ead79bc0700\" class=\"target-chapter\">Divide Radical Expressions<\/a><\/h4><p id=\"fs-id1169143662146\"><strong data-effect=\"bold\">Divide Square Roots<\/strong><\/p><p id=\"fs-id1169143662152\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1169143662155\"><div data-type=\"problem\" id=\"fs-id1169143662157\"><p id=\"fs-id1169143662159\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{48}}{\\sqrt{75}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{81}}{\\sqrt[3]{24}}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143662241\"><div data-type=\"problem\" id=\"fs-id1169143662243\"><p id=\"fs-id1169143662245\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{320m{n}^{-5}}}{\\sqrt{45{m}^{-7}{n}^{3}}}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{16{x}^{4}{y}^{-2}}}{\\sqrt[3]{-54{x}^{-2}{y}^{4}}}\\)<\/div><div data-type=\"solution\" id=\"fs-id1169145497025\"><p id=\"fs-id1169145497027\"><span class=\"token\">\u24d0<\/span>\\(\\frac{8{m}^{4}}{3{n}^{4}}\\)<span class=\"token\">\u24d1<\/span>\\(-\\frac{{x}^{2}}{2{y}^{2}}\\)<\/p><\/div><\/div><p id=\"fs-id1169145720654\"><strong data-effect=\"bold\">Rationalize a One Term Denominator<\/strong><\/p><p id=\"fs-id1169145720660\">In the following exercises, rationalize the denominator.<\/p><div data-type=\"exercise\" id=\"fs-id1169145720663\"><div data-type=\"problem\" id=\"fs-id1169145720665\"><p id=\"fs-id1169145720668\"><span class=\"token\">\u24d0<\/span>\\(\\frac{8}{\\sqrt{3}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt{\\frac{7}{40}}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{8}{\\sqrt{2y}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143686314\"><div data-type=\"problem\" id=\"fs-id1169143686316\"><p id=\"fs-id1169143686318\"><span class=\"token\">\u24d0<\/span>\\(\\frac{1}{\\sqrt[3]{11}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{\\frac{7}{54}}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{3}{\\sqrt[3]{3{x}^{2}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145714378\"><p id=\"fs-id1169145714380\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt[3]{121}}{11}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{28}}{6}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{\\sqrt[3]{9x}}{x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145714442\"><div data-type=\"problem\" id=\"fs-id1169145714444\"><p id=\"fs-id1169145714446\"><span class=\"token\">\u24d0<\/span>\\(\\frac{1}{\\sqrt[4]{4}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[4]{\\frac{9}{32}}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{6}{\\sqrt[4]{9{x}^{3}}}\\)<\/p><\/div><\/div><p id=\"fs-id1169145971565\"><strong data-effect=\"bold\">Rationalize a Two Term Denominator<\/strong><\/p><p id=\"fs-id1169145971571\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1169145971574\"><div data-type=\"problem\" id=\"fs-id1169145971576\"><p id=\"fs-id1169145971578\">\\(\\frac{7}{2-\\sqrt{6}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145645732\"><p id=\"fs-id1169145645734\">\\(-\\frac{7\\left(2+\\sqrt{6}\\right)}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145645763\"><div data-type=\"problem\" id=\"fs-id1169145645765\"><p id=\"fs-id1169145645768\">\\(\\frac{\\sqrt{5}}{\\sqrt{n}-\\sqrt{7}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145645824\"><div data-type=\"problem\" id=\"fs-id1169145645826\"><p id=\"fs-id1169145645828\">\\(\\frac{\\sqrt{x}+\\sqrt{8}}{\\sqrt{x}-\\sqrt{8}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169141299944\"><p id=\"fs-id1169141299946\">\\({\\frac{\\left(\\sqrt{x}+2\\sqrt{2}\\right)}{x-8}}^{2}\\)<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169141299984\"><h4 data-type=\"title\"><a href=\"\/contents\/d5923faf-e056-4b0a-b96d-ee6429d48e36\" class=\"target-chapter\">Solve Radical Equations<\/a><\/h4><p id=\"fs-id1169141299994\"><strong data-effect=\"bold\">Solve Radical Equations<\/strong><\/p><p id=\"fs-id1169141300000\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1169141300003\"><div data-type=\"problem\" id=\"fs-id1169141300005\"><p id=\"fs-id1169141300007\">\\(\\sqrt{4x-3}=7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143660081\"><div data-type=\"problem\" id=\"fs-id1169143660084\"><p id=\"fs-id1169143660086\">\\(\\sqrt{5x+1}=-3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143660106\"><p id=\"fs-id1169143660108\">no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143660113\"><div data-type=\"problem\" id=\"fs-id1169143660116\"><p id=\"fs-id1169143660118\">\\(\\sqrt[3]{4x-1}=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143660156\"><div data-type=\"problem\" id=\"fs-id1169143660158\"><p id=\"fs-id1169143660160\">\\(\\sqrt{u-3}+3=u\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143660183\"><p id=\"fs-id1169142416331\">\\(u=3,u=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169142416352\"><div data-type=\"problem\" id=\"fs-id1169142416355\"><p id=\"fs-id1169142416357\">\\(\\sqrt[3]{4x+5}-2=-5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169142416399\"><div data-type=\"problem\" id=\"fs-id1169142416402\"><p id=\"fs-id1169142416404\">\\({\\left(8x+5\\right)}^{\\frac{1}{3}}+2=-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169142416441\"><p id=\"fs-id1169142416443\">\\(x=-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143604687\"><div data-type=\"problem\" id=\"fs-id1169143604689\"><p id=\"fs-id1169143604691\">\\(\\sqrt{y+4}-y+2=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143604733\"><div data-type=\"problem\" id=\"fs-id1169143604735\"><p id=\"fs-id1169143604737\">\\(2\\sqrt{8r+1}-8=2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143604764\"><p id=\"fs-id1169143604766\">\\(r=3\\)<\/p><\/div><\/div><p id=\"fs-id1169143604778\"><strong data-effect=\"bold\">Solve Radical Equations with Two Radicals<\/strong><\/p><p id=\"fs-id1169143604784\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1169143604787\"><div data-type=\"problem\" id=\"fs-id1169143604789\"><p id=\"fs-id1169143539917\">\\(\\sqrt{10+2c}=\\sqrt{4c+16}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143539960\"><div data-type=\"problem\" id=\"fs-id1169143539963\"><p id=\"fs-id1169143539965\">\\(\\sqrt[3]{2{x}^{2}+9x-18}=\\sqrt[3]{{x}^{2}+3x-2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143540015\"><p id=\"fs-id1169143540017\">\\(x=-8,x=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145623314\"><div data-type=\"problem\" id=\"fs-id1169145623316\"><p id=\"fs-id1169145623318\">\\(\\sqrt{r}+6=\\sqrt{r+8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145623349\"><div data-type=\"problem\" id=\"fs-id1169145623351\"><p id=\"fs-id1169145623353\">\\(\\sqrt{x+1}-\\sqrt{x-2}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145623382\"><p id=\"fs-id1169145623384\">\\(x=3\\)<\/p><\/div><\/div><p id=\"fs-id1169145623397\"><strong data-effect=\"bold\">Use Radicals in Applications<\/strong><\/p><p id=\"fs-id1169145623403\">In the following exercises, solve. Round approximations to one decimal place.<\/p><div data-type=\"exercise\" id=\"fs-id1169145623406\"><div data-type=\"problem\" id=\"fs-id1169145623409\"><p id=\"fs-id1169145623411\"><strong data-effect=\"bold\">Landscaping<\/strong> Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula \\(s=\\sqrt{A}\\) to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145577657\"><div data-type=\"problem\" id=\"fs-id1169145577659\"><p id=\"fs-id1169145577661\"><strong data-effect=\"bold\">Accident investigation<\/strong> An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. Use the formula \\(s=\\sqrt{24d}\\) to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145577686\"><p id=\"fs-id1169145577688\">\\(64.8\\) feet<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169145577699\"><h4 data-type=\"title\"><a href=\"\/contents\/d5887b1b-5ca2-40b7-99c3-43f38b1a5425\" class=\"target-chapter\">Use Radicals in Functions<\/a><\/h4><p id=\"fs-id1169145577709\"><strong data-effect=\"bold\">Evaluate a Radical Function<\/strong><\/p><p id=\"fs-id1169145577715\">In the following exercises, evaluate each function.<\/p><div data-type=\"exercise\" id=\"fs-id1169145577718\"><div data-type=\"problem\" id=\"fs-id1169145577721\"><p id=\"fs-id1169145577723\">\\(g\\left(x\\right)=\\sqrt{6x+1},\\) find<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(g\\left(4\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(g\\left(8\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147816061\"><div data-type=\"problem\" id=\"fs-id1169147816063\"><p id=\"fs-id1169147816065\">\\(G\\left(x\\right)=\\sqrt{5x-1},\\) find<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(G\\left(5\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(G\\left(2\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1169145535783\"><p id=\"fs-id1169145535785\"><span class=\"token\">\u24d0<\/span>\\(G\\left(5\\right)=2\\sqrt{6}\\)<span class=\"token\">\u24d1<\/span>\\(G\\left(2\\right)=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145535842\"><div data-type=\"problem\" id=\"fs-id1169141037377\"><p id=\"fs-id1169141037379\">\\(h\\left(x\\right)=\\sqrt[3]{{x}^{2}-4},\\) find<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(h\\left(-2\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(h\\left(6\\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143442804\"><div data-type=\"problem\" id=\"fs-id1169143442806\"><p id=\"fs-id1169143442808\">For the function<\/p><div data-type=\"newline\"><br><\/div>\\(g\\left(x\\right)=\\sqrt[4]{4-4x},\\) find<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(g\\left(1\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> \\(g\\left(-3\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1169143442881\"><p id=\"fs-id1169143442883\"><span class=\"token\">\u24d0<\/span>\\(g\\left(1\\right)=0\\)<span class=\"token\">\u24d1<\/span>\\(g\\left(-3\\right)=2\\)<\/p><\/div><\/div><p id=\"fs-id1169143580727\"><strong data-effect=\"bold\">Find the Domain of a Radical Function<\/strong><\/p><p id=\"fs-id1169143580734\">In the following exercises, find the domain of the function and write the domain in interval notation.<\/p><div data-type=\"exercise\" id=\"fs-id1169143580738\"><div data-type=\"problem\" id=\"fs-id1169143580740\"><p id=\"fs-id1169143580742\">\\(g\\left(x\\right)=\\sqrt{2-3x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143580796\"><div data-type=\"problem\" id=\"fs-id1169143580798\"><p id=\"fs-id1169143580801\">\\(F\\left(x\\right)=\\sqrt{\\frac{x+3}{x-2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169141522401\"><p id=\"fs-id1169141522403\">\\(\\left(2,\\infty \\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169141522422\"><div data-type=\"problem\" id=\"fs-id1169141522424\"><p id=\"fs-id1169141522426\">\\(f\\left(x\\right)=\\sqrt[3]{4{x}^{2}-16}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169141522484\"><div data-type=\"problem\" id=\"fs-id1169142133190\"><p id=\"fs-id1169142133192\">\\(F\\left(x\\right)=\\sqrt[4]{10-7x}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169142133223\"><p id=\"fs-id1169142133225\">\\(\\left[\\frac{7}{10},\\infty \\right)\\)<\/p><\/div><\/div><p id=\"fs-id1169142133249\"><strong data-effect=\"bold\">Graph Radical Functions<\/strong><\/p><p id=\"fs-id1169142133255\">In the following exercises, <span class=\"token\">\u24d0<\/span> find the domain of the function <span class=\"token\">\u24d1<\/span> graph the function <span class=\"token\">\u24d2<\/span> use the graph to determine the range.<\/p><div data-type=\"exercise\" id=\"fs-id1169142133269\"><div data-type=\"problem\" id=\"fs-id1169142133271\"><p id=\"fs-id1169142133273\">\\(g\\left(x\\right)=\\sqrt{x+4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147765384\"><div data-type=\"problem\" id=\"fs-id1169147765386\"><p id=\"fs-id1169147765389\">\\(g\\left(x\\right)=2\\sqrt{x}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147765412\"><p id=\"fs-id1169147765414\"><span class=\"token\">\u24d0<\/span> domain: \\(\\left[0,\\infty \\right)\\)<\/p><div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1169143576361\" data-alt=\"The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 0) and goes through the points (1, 2) and (4, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_302_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 0) and goes through the points (1, 2) and (4, 4).\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> range: \\(\\left[0,\\infty \\right)\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143576399\"><div data-type=\"problem\" id=\"fs-id1169143576401\"><p id=\"fs-id1169143576404\">\\(f\\left(x\\right)=\\sqrt[3]{x-1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143295372\"><div data-type=\"problem\" id=\"fs-id1169143295374\"><p id=\"fs-id1169143295376\">\\(f\\left(x\\right)=\\sqrt[3]{x}+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143295404\"><p id=\"fs-id1169143295406\"><span class=\"token\">\u24d0<\/span> domain: \\(\\left(\\text{\u2212}\\infty ,\\infty \\right)\\)<\/p><div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1169143295437\" data-alt=\"The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a center point at (0, 3) and goes through the points (negative 1, 2) and (1, 4).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a center point at (0, 3) and goes through the points (negative 1, 2) and (1, 4).\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> range: \\(\\left(\\text{\u2212}\\infty ,\\infty \\right)\\)<\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169143658297\"><h4 data-type=\"title\"><a href=\"\/contents\/916a2094-3b51-4f1a-803d-95909a359123\" class=\"target-chapter\">Use the Complex Number System<\/a><\/h4><p id=\"fs-id1169143658308\"><strong data-effect=\"bold\">Evaluate the Square Root of a Negative Number<\/strong><\/p><p id=\"fs-id1169143658314\">In the following exercises, write each expression in terms of <em data-effect=\"italics\">i<\/em> and simplify if possible.<\/p><div data-type=\"exercise\" id=\"fs-id1169143658322\"><div data-type=\"problem\"><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\sqrt{-100}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\sqrt{-13}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\sqrt{-45}\\)<\/div><\/div><p id=\"fs-id1169148225776\"><strong data-effect=\"bold\">Add or Subtract Complex Numbers<\/strong><\/p><p id=\"fs-id1168037345963\">In the following exercises, add or subtract.<\/p><div data-type=\"exercise\" id=\"fs-id1169148225784\"><div data-type=\"problem\" id=\"fs-id1169148225786\"><p id=\"fs-id1169148225788\">\\(\\sqrt{-50}+\\sqrt{-18}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148225804\"><p id=\"fs-id1169148225806\">\\(8\\sqrt{2}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148225820\"><div data-type=\"problem\" id=\"fs-id1169148225822\"><p id=\"fs-id1169148225824\">\\(\\left(8-i\\right)+\\left(6+3i\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148225876\"><div data-type=\"problem\" id=\"fs-id1169143600008\"><p id=\"fs-id1169143600010\">\\(\\left(6+i\\right)-\\left(-2-4i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143600046\"><p id=\"fs-id1169143600048\">\\(8+5i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143600063\"><div data-type=\"problem\" id=\"fs-id1169143600065\"><p id=\"fs-id1169143600067\">\\(\\left(-7-\\sqrt{-50}\\right)-\\left(-32-\\sqrt{-18}\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1169143600125\"><strong data-effect=\"bold\">Multiply Complex Numbers<\/strong><\/p><p id=\"fs-id1169147910490\">In the following exercises, multiply.<\/p><div data-type=\"exercise\" id=\"fs-id1169147910493\"><div data-type=\"problem\" id=\"fs-id1169147910495\"><p id=\"fs-id1169147910497\">\\(\\left(-2-5i\\right)\\left(-4+3i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147910532\"><p id=\"fs-id1169147910535\">\\(23+14i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147910550\"><div data-type=\"problem\" id=\"fs-id1169147910552\"><p id=\"fs-id1169147910554\">\\(-6i\\left(-3-2i\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147910595\"><div data-type=\"problem\" id=\"fs-id1169147910598\"><p id=\"fs-id1169147910600\">\\(\\sqrt{-4}\u00b7\\sqrt{-16}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147948334\"><p id=\"fs-id1169147948336\">\\(-6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147948344\"><div data-type=\"problem\" id=\"fs-id1169147948346\"><p id=\"fs-id1169147948348\">\\(\\left(5-\\sqrt{-12}\\right)\\left(-3+\\sqrt{-75}\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1169147948404\">In the following exercises, multiply using the Product of Binomial Squares Pattern.<\/p><div data-type=\"exercise\" id=\"fs-id1169147948407\"><div data-type=\"problem\" id=\"fs-id1169147948409\"><p id=\"fs-id1169147948411\">\\({\\left(-2-3i\\right)}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147948436\"><p id=\"fs-id1169147948438\">\\(-5-12i\\)<\/p><\/div><\/div><p id=\"fs-id1169148206456\">In the following exercises, multiply using the Product of Complex Conjugates Pattern.<\/p><div data-type=\"exercise\" id=\"fs-id1169148206459\"><div data-type=\"problem\" id=\"fs-id1169148206461\"><p id=\"fs-id1169148206463\">\\(\\left(9-2i\\right)\\left(9+2i\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1169148206509\"><strong data-effect=\"bold\">Divide Complex Numbers<\/strong><\/p><p id=\"fs-id1169148206515\">In the following exercises, divide.<\/p><div data-type=\"exercise\" id=\"fs-id1169148206519\"><div data-type=\"problem\" id=\"fs-id1169148206521\"><p id=\"fs-id1169148206523\">\\(\\frac{2+i}{3-4i}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148206546\"><p id=\"fs-id1169148206548\">\\(\\frac{2}{25}+\\frac{11}{25}i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148071897\"><div data-type=\"problem\" id=\"fs-id1169148071899\"><p id=\"fs-id1169148071901\">\\(\\frac{-4}{3-2i}\\)<\/p><\/div><\/div><p id=\"fs-id1169148071949\"><strong data-effect=\"bold\">Simplify Powers of <em data-effect=\"italics\">i<\/em><\/strong><\/p><p id=\"fs-id1169148071958\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1169148071962\"><div data-type=\"problem\" id=\"fs-id1169148071964\"><p id=\"fs-id1169148071966\">\\({i}^{48}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148071978\"><p id=\"fs-id1169148071980\">1<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148071985\"><div data-type=\"problem\" id=\"fs-id1169148071987\"><p id=\"fs-id1169148071989\">\\({i}^{255}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1169145548148\"><h3 data-type=\"title\">Practice Test<\/h3><p id=\"fs-id1169145548155\">In the following exercises, simplify using absolute values as necessary.<\/p><div data-type=\"exercise\" id=\"fs-id1169145548158\"><div data-type=\"problem\" id=\"fs-id1169145548160\"><p id=\"fs-id1169145548162\">\\(\\sqrt[3]{125{x}^{9}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145548180\"><p id=\"fs-id1169145548182\">\\(5{x}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145548196\"><div data-type=\"problem\" id=\"fs-id1169145548198\"><p id=\"fs-id1169145548200\">\\(\\sqrt{169{x}^{8}{y}^{6}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145548241\"><div data-type=\"problem\" id=\"fs-id1169145548244\"><p id=\"fs-id1169145548246\">\\(\\sqrt[3]{72{x}^{8}{y}^{4}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143600773\"><p id=\"fs-id1169143600775\">\\(2{x}^{2}{y}^{}\\sqrt[3]{9{x}^{2}y}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143600808\"><div data-type=\"problem\" id=\"fs-id1169143600810\"><p id=\"fs-id1169143600812\">\\(\\sqrt{\\frac{45{x}^{3}{y}^{4}}{180{x}^{5}{y}^{2}}}\\)<\/p><\/div><\/div><p id=\"fs-id1169147948649\">In the following exercises, simplify. Assume all variables are positive.<\/p><div data-type=\"exercise\" id=\"fs-id1169147948652\"><div data-type=\"problem\" id=\"fs-id1169147948655\"><p id=\"fs-id1169147948657\"><span class=\"token\">\u24d0<\/span>\\({216}^{-\\frac{1}{4}}\\)<span class=\"token\">\u24d1<\/span>\\(\\text{\u2212}{49}^{\\frac{3}{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147948701\"><p id=\"fs-id1169147948703\"><span class=\"token\">\u24d0<\/span>\\(\\frac{1}{4}\\)<span class=\"token\">\u24d1<\/span>\\(-343\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147948731\"><div data-type=\"problem\" id=\"fs-id1169147948733\"><p id=\"fs-id1169147948735\">\\(\\sqrt{-45}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147948761\"><div data-type=\"problem\" id=\"fs-id1169147948763\"><p id=\"fs-id1169147948765\">\\(\\frac{{x}^{-\\frac{1}{4}}\u00b7{x}^{\\frac{5}{4}}}{{x}^{-\\frac{3}{4}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145734818\"><p id=\"fs-id1169145734820\">\\({x}^{\\frac{7}{4}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145734836\"><div data-type=\"problem\" id=\"fs-id1169145734838\"><p id=\"fs-id1169145734841\">\\({\\left(\\frac{8\\text{\u200b}{x}^{\\frac{2}{3}}\\text{\u200b}{y}^{-\\frac{5}{2}}}{{x}^{-\\frac{7}{3}}{y}^{\\frac{1}{2}}}\\right)}^{\\frac{1}{3}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147775930\"><div data-type=\"problem\" id=\"fs-id1169147775932\"><p id=\"fs-id1169147775934\">\\(\\sqrt{48{x}^{5}}-\\sqrt{75{x}^{5}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147775961\"><p id=\"fs-id1169147775963\">\\(\\text{\u2212}{x}^{2}\\sqrt{3x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147775983\"><div data-type=\"problem\" id=\"fs-id1169147775985\"><p id=\"fs-id1169147775987\">\\(\\sqrt{27{x}^{2}}-4x\\sqrt{12}+\\sqrt{108{x}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145519492\"><div data-type=\"problem\" id=\"fs-id1169145519494\"><p id=\"fs-id1169145519496\">\\(2\\sqrt{12{x}^{5}}\u00b73\\sqrt{6{x}^{3}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145519527\"><p id=\"fs-id1169145519529\">\\(36{x}^{4}\\sqrt{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145519546\"><div data-type=\"problem\" id=\"fs-id1169145519549\"><p id=\"fs-id1169145519551\">\\(\\sqrt[3]{4}\\left(\\sqrt[3]{16}-\\sqrt[3]{6}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143519724\"><div data-type=\"problem\" id=\"fs-id1169143519726\"><p id=\"fs-id1169143519729\">\\(\\left(4-3\\sqrt{3}\\right)\\left(5+2\\sqrt{3}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143519766\"><p id=\"fs-id1169143519768\">\\(2-7\\sqrt{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143519784\"><div data-type=\"problem\" id=\"fs-id1169143519786\"><p id=\"fs-id1169143519788\">\\(\\frac{\\sqrt[3]{128}}{\\sqrt[3]{54}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169143519826\"><div data-type=\"problem\" id=\"fs-id1169148122896\"><p id=\"fs-id1169148122898\">\\(\\frac{\\sqrt{245x{y}^{-4}}}{\\sqrt{45{x}^{-4}{y}^{3}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148122935\"><p id=\"fs-id1169148122938\">\\(\\frac{7{x}^{5}}{3{y}^{7}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148122962\"><div data-type=\"problem\" id=\"fs-id1169148122964\"><p id=\"fs-id1169148122966\">\\(\\frac{1}{\\sqrt[3]{5}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148123000\"><div data-type=\"problem\" id=\"fs-id1169148123003\"><p id=\"fs-id1169148123005\">\\(\\frac{3}{2+\\sqrt{3}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145760791\"><p id=\"fs-id1169145760793\">\\(3\\left(2-\\sqrt{3}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145760816\"><div data-type=\"problem\" id=\"fs-id1169145760818\"><p id=\"fs-id1169145760820\">\\(\\sqrt{-4}\u00b7\\sqrt{-9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145760847\"><div data-type=\"problem\" id=\"fs-id1169145760849\"><p id=\"fs-id1169145760851\">\\(-4i\\left(-2-3i\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169145760876\"><p id=\"fs-id1169145760878\">\\(-12+8i\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169145760893\"><div data-type=\"problem\" id=\"fs-id1169145760895\"><p id=\"fs-id1169145760897\">\\(\\frac{4+i}{3-2i}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148107267\"><div data-type=\"problem\" id=\"fs-id1169148107269\"><p id=\"fs-id1169148107271\">\\({i}^{172}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148107283\"><p id=\"fs-id1169148107285\">\\(\\text{\u2212}i\\)<\/p><\/div><\/div><p id=\"fs-id1169148107296\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1169148107299\"><div data-type=\"problem\" id=\"fs-id1169148107301\"><p id=\"fs-id1169148107303\">\\(\\sqrt{2x+5}+8=6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148107335\"><div data-type=\"problem\" id=\"fs-id1169148107337\"><p id=\"fs-id1169148107339\">\\(\\sqrt{x+5}+1=x\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148105697\"><p id=\"fs-id1169148105699\">\\(x=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148105712\"><div data-type=\"problem\" id=\"fs-id1169148105714\"><p id=\"fs-id1169148105716\">\\(\\sqrt[3]{2{x}^{2}-6x-23}=\\sqrt[3]{{x}^{2}-3x+5}\\)<\/p><\/div><\/div><p id=\"fs-id1169148105789\">In the following exercise, <span class=\"token\">\u24d0<\/span> find the domain of the function <span class=\"token\">\u24d1<\/span> graph the function <span class=\"token\">\u24d2<\/span> use the graph to determine the range.<\/p><div data-type=\"exercise\" id=\"fs-id1169148105802\"><div data-type=\"problem\" id=\"fs-id1169143517571\"><p id=\"fs-id1169143517574\">\\(g\\left(x\\right)=\\sqrt{x+2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169143517599\"><p id=\"fs-id1169143517601\"><span class=\"token\">\u24d0<\/span> domain: \\(\\left[-2,\\infty \\right)\\)<\/p><div data-type=\"newline\"><br><\/div> <span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1169143517631\" data-alt=\"The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 6. The y-axis runs from 0 to 8. The function has a starting point at (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 6. The y-axis runs from 0 to 8. The function has a starting point at (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).\"><\/span><p><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> range: \\(\\left[0,\\infty \\right)\\)<\/div><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1169143517674\"><dt>complex conjugate pair<\/dt><dd id=\"fs-id1169143517677\">A complex conjugate pair is of the form <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bi<\/em>, <em data-effect=\"italics\">a<\/em> \u2013 <em data-effect=\"italics\">bi<\/em>.<\/dd><\/dl><dl id=\"fs-id1169148236939\"><dt>complex number<\/dt><dd id=\"fs-id1169148236942\">A complex number is of the form <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bi<\/em>, where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers. We call <em data-effect=\"italics\">a<\/em> the real part and <em data-effect=\"italics\">b<\/em> the imaginary part.<\/dd><\/dl><dl id=\"fs-id1169148236978\"><dt>complex number system<\/dt><dd id=\"fs-id1169148236981\">The complex number system is made up of both the real numbers and the imaginary numbers.<\/dd><\/dl><dl id=\"fs-id1169148236986\"><dt>imaginary unit<\/dt><dd id=\"fs-id1169148236989\">The imaginary unit \\(i\\) is the number whose square is \u20131. <em data-effect=\"italics\">i<\/em><sup>2<\/sup> = \u20131 or \\(i=\\sqrt{-1}.\\)<\/dd><\/dl><dl id=\"fs-id1169148237018\"><dt>standard form<\/dt><dd id=\"fs-id1169148237021\">A complex number is in standard form when written as \\(a+bi,\\) where <em data-effect=\"italics\">a, b<\/em> are real numbers.<\/dd><\/dl><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Evaluate the square root of a negative number<\/li>\n<li>Add and subtract complex numbers<\/li>\n<li>Multiply complex numbers<\/li>\n<li>Divide complex numbers<\/li>\n<li>Simplify powers of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-695d9d59bd04859c6c99e7feb11daab6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/><\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147958916\" class=\"be-prepared\">\n<p id=\"fs-id1169148197895\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1169147826247\" type=\"1\">\n<li>Given the numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d145e2537144fcc063f141769e221d5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#44;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#44;&#48;&#46;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;&#44;&#51;&#44;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"184\" style=\"vertical-align: -6px;\" \/> list the <span class=\"token\">\u24d0<\/span> rational numbers, <span class=\"token\">\u24d1<\/span> irrational numbers, <span class=\"token\">\u24d2<\/span> real numbers.\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/99b2296a-9957-4380-aff4-248abadc862b#fs-id1167836546317\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f2bf238bf51948947023acbf6f234a70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836544266\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Rationalize the denominator:<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-796ef0441ab879da355164fcea4f5826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"53\" style=\"vertical-align: -11px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836392219\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169148123144\">\n<h3 data-type=\"title\">Evaluate the Square Root of a Negative Number<\/h3>\n<p id=\"fs-id1169147830975\">Whenever we have a situation where we have a square root of a negative number we say there is no real number that equals that square root. For example, to simplify <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a958132396582f30f248c03dbc6e99fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"40\" style=\"vertical-align: -4px;\" \/> we are looking for a real number <em data-effect=\"italics\">x<\/em> so that <em data-effect=\"italics\">x<\/em><sup>2<\/sup> = \u20131. Since all real numbers squared are positive numbers, there is no real number that equals \u20131 when squared.<\/p>\n<p id=\"fs-id1169147737576\">Mathematicians have often expanded their numbers systems as needed. They added 0 to the counting numbers to get the whole numbers. When they needed negative balances, they added negative numbers to get the integers. When they needed the idea of parts of a whole they added fractions and got the rational numbers. Adding the irrational numbers allowed numbers like <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5732850b12282f9370bf53f13e44a811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -2px;\" \/> All of these together gave us the real numbers and so far in your study of mathematics, that has been sufficient.<\/p>\n<p id=\"fs-id1169147736362\">But now we will expand the real numbers to include the square roots of negative numbers. We start by defining the <span data-type=\"term\">imaginary unit<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-695d9d59bd04859c6c99e7feb11daab6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> as the number whose square is \u20131.<\/p>\n<div data-type=\"note\" id=\"fs-id1169147854180\">\n<div data-type=\"title\">Imaginary Unit<\/div>\n<p id=\"fs-id1169145619988\">The <strong data-effect=\"bold\">imaginary unit<\/strong> <em data-effect=\"italics\">i<\/em> is the number whose square is \u20131.<\/p>\n<div data-type=\"equation\" id=\"fs-id1169148060192\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2c9835b62a9459e7fedc5d60835a6b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#61;&#45;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"151\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169148126453\">We will use the imaginary unit to simplify the square roots of negative numbers.<\/p>\n<div data-type=\"note\" id=\"fs-id1169147837254\">\n<div data-type=\"title\">Square Root of a Negative Number<\/div>\n<p id=\"fs-id1169148129753\">If <em data-effect=\"italics\">b<\/em> is a positive real number, then<\/p>\n<div data-type=\"equation\" id=\"fs-id1169147851238\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcb722e9419b1fdc6e133143215389a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#98;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"77\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169147846598\">We will use this definition in the next example. Be careful that it is clear that the <em data-effect=\"italics\">i<\/em> is not under the radical. Sometimes you will see this written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f35c0e107572580682534746744eba49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#125;&#61;&#105;&#92;&#115;&#113;&#114;&#116;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -2px;\" \/> to emphasize the <em data-effect=\"italics\">i<\/em> is not under the radical. But the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bcb722e9419b1fdc6e133143215389a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#98;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"77\" style=\"vertical-align: -2px;\" \/> is considered standard form.<\/p>\n<div data-type=\"example\" id=\"fs-id1169145730176\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148206264\">\n<div data-type=\"problem\">\n<p id=\"fs-id1169147751427\">Write each expression in terms of <em data-effect=\"italics\">i<\/em> and simplify if possible:<\/p>\n<p id=\"fs-id1169143316387\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7cdc351b2532599665ca0e80e412978_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-858f0f7eb283c92d96a49fe9d13f297d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f04f382d0f3ea42faa262be32b018d9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147744813\">\n<p id=\"fs-id1169141447233\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9a23d8558438197c26ca7c069ead5c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#102;&#105;&#110;&#105;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#114;&#111;&#111;&#116;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"84\" width=\"564\" style=\"vertical-align: -36px;\" \/><\/p>\n<p id=\"fs-id1169143576332\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b68dd0079424a3d174265476df4d0823_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#55;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#102;&#105;&#110;&#105;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#114;&#111;&#111;&#116;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#101;&#32;&#99;&#97;&#114;&#101;&#102;&#117;&#108;&#32;&#116;&#104;&#97;&#116;&#32;&#105;&#116;&#32;&#105;&#115;&#32;&#99;&#108;&#101;&#97;&#114;&#32;&#116;&#104;&#97;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#110;&#111;&#116;&#32;&#117;&#110;&#100;&#101;&#114;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#32;&#115;&#105;&#103;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"106\" width=\"746\" style=\"vertical-align: -47px;\" \/><\/p>\n<p id=\"fs-id1169147768810\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e995a766b0a9ffa0796bd65b7ff86662_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#102;&#105;&#110;&#105;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#114;&#111;&#111;&#116;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"84\" width=\"564\" style=\"vertical-align: -36px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169141188298\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145526150\">\n<div data-type=\"problem\" id=\"fs-id1169145977084\">\n<p id=\"fs-id1169148105109\">Write each expression in terms of <em data-effect=\"italics\">i<\/em> and simplify if possible:<\/p>\n<p id=\"fs-id1169147840176\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-babd90f3ad09b9568e8552ca30752c1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a286fdfc70bf1ea0d82960fcedde2f4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e10c9c9cfcb693b7855bd82da6860f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143459048\">\n<p id=\"fs-id1169147958539\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f2c445e2bc6406156f0e1c35a7241b23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74bedd5895a29e019106e631cb55eb56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"29\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-433fd678979d8b79b09e526f1c683ac5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147983632\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147826095\">\n<div data-type=\"problem\" id=\"fs-id1169148211494\">\n<p id=\"fs-id1169148114426\">Write each expression in terms of <em data-effect=\"italics\">i<\/em> and simplify if possible:<\/p>\n<p id=\"fs-id1169147961162\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e147c65350129481b966194b9306d610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8d170dfac28f03fb9ad2c4759fb303f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5a20db1121150a2e840158bdef95252_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#55;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148068950\">\n<p id=\"fs-id1169148123807\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb4de4a9de0c4c711b557f277e15f76b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20612eb70da44423b9980339651583a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"29\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc480c334bb9ea1d083a573b8ac3861c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147783255\">Now that we are familiar with the imaginary number <em data-effect=\"italics\">i<\/em>, we can expand the real numbers to include imaginary numbers. The <span data-type=\"term\">complex number system<\/span> includes the real numbers and the imaginary numbers. A <span data-type=\"term\">complex number<\/span> is of the form <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bi<\/em>, where <em data-effect=\"italics\">a, b<\/em> are real numbers. We call <em data-effect=\"italics\">a<\/em> the real part and <em data-effect=\"italics\">b<\/em> the imaginary part.<\/p>\n<div data-type=\"note\" id=\"fs-id1169147868416\">\n<div data-type=\"title\">Complex Number<\/div>\n<p id=\"fs-id1169148217595\">A <strong data-effect=\"bold\">complex number<\/strong> is of the form <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bi<\/em>, where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147771025\" data-alt=\"The image shows the expression a plus b i. The number a is labeled \u201creal part\u201d and the number b i is labeled \u201cimaginary part\u201d.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The image shows the expression a plus b i. The number a is labeled \u201creal part\u201d and the number b i is labeled \u201cimaginary part\u201d.\" \/><\/span><\/div>\n<p id=\"fs-id1169147821874\">A complex number is in standard form when written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e80cd255be59b21381715aaa50d79181_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\" \/> where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers.<\/p>\n<p id=\"fs-id1169148248276\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d76691e8cfec706c988d22fb21bd369f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ffcd7d918a83e37de98a41b94384c07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"45\" style=\"vertical-align: -2px;\" \/> becomes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00d0bfeb89d8ec250918926063e56bd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#48;&middot;&#105;&#61;&#97;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/> and is a real number.<\/p>\n<p id=\"fs-id1169147854128\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6a0135701625ac5da01c134efeaebdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ffcd7d918a83e37de98a41b94384c07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"45\" style=\"vertical-align: -2px;\" \/> is an imaginary number.<\/p>\n<p id=\"fs-id1169147861742\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c5e413b70b2e3a32f46a8942b947900_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ffcd7d918a83e37de98a41b94384c07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"45\" style=\"vertical-align: -2px;\" \/> becomes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e66fcbdb1e2100e3f3542cf7a332b507_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#43;&#98;&#105;&#61;&#98;&#105;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\" \/> and is called a pure imaginary number.<\/p>\n<p id=\"fs-id1169147863085\">We summarize this here.<\/p>\n<table id=\"fs-id1169141357905\" class=\"unnumbered\" summary=\"The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and \u201cReal number\u201d. The third row contains b is not equal to 0, a plus b i, and \u201cImaginary number\u201d. The fourth row contains a = 0, 0 plus b i, and \u201cPure imaginary number\u201d.\">\n<tbody>\n<tr valign=\"top\">\n<td><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ffcd7d918a83e37de98a41b94384c07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-597ae6c01abdebf6816c77f3ce1fd9de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43695884e5027a485867e17fb783bfc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#97;&#43;&#48;&middot;&#105;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"37\" style=\"vertical-align: -33px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\">Real number<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1edd53778d94abb1dc1e19acff79e9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ffcd7d918a83e37de98a41b94384c07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\">Imaginary number<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-072bed0ebf929f9d7c14da365c8512a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23c8a64b4f38763727af5268c4a3f8ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#125;&#48;&#43;&#98;&#105;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#105;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"79\" width=\"45\" style=\"vertical-align: -33px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\">Pure imaginary number<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169145519328\">The standard form of a complex number is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e80cd255be59b21381715aaa50d79181_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\" \/> so this explains why the preferred form is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20332daafa6451a81cb3899b64b0211f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#98;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -2px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4943929b99ad104795b668d3dc011dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#62;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1169147847796\">The diagram helps us visualize the complex number system. It is made up of both the real numbers and the imaginary numbers.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169145621336\" data-alt=\"The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and \u201cReal number\u201d. The third row contains b is not equal to 0, a plus b i, and \u201cImaginary number\u201d. The fourth row contains a = 0, 0 plus b i, and \u201cPure imaginary number\u201d.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_002_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and \u201cReal number\u201d. The third row contains b is not equal to 0, a plus b i, and \u201cImaginary number\u201d. The fourth row contains a = 0, 0 plus b i, and \u201cPure imaginary number\u201d.\" \/><\/span><\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169142605352\">\n<h3 data-type=\"title\">Add or Subtract Complex Numbers<\/h3>\n<p id=\"fs-id1169148123410\">We are now ready to perform the operations of addition, subtraction, multiplication and division on the complex numbers\u2014just as we did with the real numbers.<\/p>\n<p id=\"fs-id1169145622397\">Adding and subtracting complex numbers is much like adding or subtracting like terms. We add or subtract the real parts and then add or subtract the imaginary parts. Our final result should be in standard form.<\/p>\n<div data-type=\"example\" id=\"fs-id1169147935028\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147847594\">\n<div data-type=\"problem\" id=\"fs-id1169145591200\">\n<p id=\"fs-id1169148101553\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a00a7b4e82a166989a7b4afc551d216_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#50;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#55;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145988331\">\n<p id=\"fs-id1169148062911\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6967c0fc76b54b0fbfa79758fafcfef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#50;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#55;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#102;&#105;&#110;&#105;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#114;&#111;&#111;&#116;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#116;&#104;&#101;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#114;&#111;&#111;&#116;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#43;&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#100;&#100;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"553\" style=\"vertical-align: -47px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147822009\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145495053\">\n<div data-type=\"problem\" id=\"fs-id1169148105509\">\n<p id=\"fs-id1169143318310\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b45ff316d08f1d1f23b3de11fd167742_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#56;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#51;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1169145729376\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4ef26bc7fb8e8cccf68bfc95bf606f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148208039\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145696301\">\n<div data-type=\"problem\" id=\"fs-id1169147810027\">\n<p id=\"fs-id1169147830354\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-434257c35d35160a469d480b51bd8c60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#55;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147874000\">\n<p id=\"fs-id1169147859230\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cc3ddc909c50120a638bde9320ed1ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148078266\">Remember to add both the real parts and the imaginary parts in this next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1169147793628\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148225075\">\n<div data-type=\"problem\" id=\"fs-id1169143572966\">\n<p id=\"fs-id1169147977713\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76e03af0e9c0ecb079e0b6b71ec94813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#54;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df753d4b62f363d2189ff66c0c67614d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"146\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169142138982\">\n<p id=\"fs-id1169148069557\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68cdefe917d27184f269d3f99acdce0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#54;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#65;&#115;&#115;&#111;&#99;&#105;&#97;&#116;&#105;&#118;&#101;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#32;&#116;&#111;&#32;&#112;&#117;&#116;&#32;&#116;&#104;&#101;&#32;&#114;&#101;&#97;&#108;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#97;&#114;&#116;&#115;&#32;&#97;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#105;&#109;&#97;&#103;&#105;&#110;&#97;&#114;&#121;&#32;&#112;&#97;&#114;&#116;&#115;&#32;&#116;&#111;&#103;&#101;&#116;&#104;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#105;&#43;&#54;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#43;&#51;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"83\" width=\"637\" style=\"vertical-align: -36px;\" \/><\/p>\n<p id=\"fs-id1169147960428\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c118d1cd7cedec8e871009fddc331c87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#115;&#116;&#114;&#105;&#98;&#117;&#116;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#45;&#53;&#105;&#45;&#53;&#43;&#50;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#65;&#115;&#115;&#111;&#99;&#105;&#97;&#116;&#105;&#118;&#101;&#32;&#80;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#32;&#116;&#111;&#32;&#112;&#117;&#116;&#32;&#116;&#104;&#101;&#32;&#114;&#101;&#97;&#108;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#97;&#114;&#116;&#115;&#32;&#97;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#105;&#109;&#97;&#103;&#105;&#110;&#97;&#114;&#121;&#32;&#112;&#97;&#114;&#116;&#115;&#32;&#116;&#111;&#103;&#101;&#116;&#104;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#45;&#53;&#45;&#53;&#105;&#43;&#50;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#51;&#45;&#51;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"105\" width=\"623\" style=\"vertical-align: -47px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147846393\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1169148080567\">\n<p id=\"fs-id1169147776135\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9189c803354329d1986b2e34d1cd2bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#43;&#55;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20ad5d091511601506d72f935e65c458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147869982\">\n<p id=\"fs-id1169148048099\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e4f27ca8b99c0b3ed83d8a813238cde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#43;&#53;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"46\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7052d5da65c6dd420412c8057e61feb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#45;&#51;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"46\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147846302\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147710394\">\n<div data-type=\"problem\" id=\"fs-id1169147725798\">\n<p id=\"fs-id1169145665290\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db082d01f13a422aad2a4391770498dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"152\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9da7810b06a646349b1e3a6dafd9f949_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#43;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#54;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"146\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147810568\">\n<p id=\"fs-id1169143516718\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b61e9471ee19ca1b3b7e028362cf6b36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#45;&#54;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"58\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-407732f07fcad97b14c87b36023e06fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#43;&#57;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"46\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169148054042\">\n<h3 data-type=\"title\">Multiply Complex Numbers<\/h3>\n<p id=\"fs-id1169147747655\">Multiplying complex numbers is also much like multiplying expressions with coefficients and variables. There is only one special case we need to consider. We will look at that after we practice in the next two examples.<\/p>\n<div data-type=\"example\" id=\"fs-id1169147907196\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147854723\">\n<div data-type=\"problem\" id=\"fs-id1169147751104\">\n<p id=\"fs-id1169147726338\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a56eef14c498b35924ab6768040decf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#105;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143686657\">\n<p id=\"fs-id1169148086081\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6736a184468e0e2e3f8245c4a86ee667_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#105;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#115;&#116;&#114;&#105;&#98;&#117;&#116;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#52;&#105;&#45;&#49;&#48;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#52;&#105;&#45;&#49;&#48;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#52;&#105;&#43;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#48;&#43;&#49;&#52;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"104\" width=\"531\" style=\"vertical-align: -46px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145732335\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147746940\">\n<div data-type=\"problem\" id=\"fs-id1169147829425\">\n<p id=\"fs-id1169147760715\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f9a657aa2a876640f057c9b5859a035_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#105;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147731418\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ad95e90d5fd7d81081831329027f831_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#43;&#50;&#48;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147866899\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147751305\">\n<div data-type=\"problem\" id=\"fs-id1169147988364\">\n<p id=\"fs-id1169147771313\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bdf4cabd17d39f3fed33bd8f6c202c1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#105;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#43;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145536453\">\n<p id=\"fs-id1169145785706\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea95317ba48e0e739f7e82985dff3b22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#43;&#54;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"53\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147981016\">In the next example, we multiply the binomials using the <span data-type=\"term\" class=\"no-emphasis\">Distributive Property<\/span> or <span data-type=\"term\" class=\"no-emphasis\">FOIL<\/span>.<\/p>\n<div data-type=\"example\" id=\"fs-id1169143511338\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169145576905\">\n<div data-type=\"problem\" id=\"fs-id1169147961150\">\n<p id=\"fs-id1169145498324\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91090b965c25f593bbc49affe9f22790_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147805831\">\n<p id=\"fs-id1169147794243\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc63563a1688e9b82137a87b5c8bff0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#70;&#79;&#73;&#76;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#50;&#45;&#57;&#105;&#43;&#56;&#105;&#45;&#54;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#99;&#111;&#109;&#98;&#105;&#110;&#101;&#32;&#108;&#105;&#107;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#50;&#45;&#105;&#45;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#50;&#45;&#105;&#43;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#109;&#98;&#105;&#110;&#101;&#32;&#116;&#104;&#101;&#32;&#114;&#101;&#97;&#108;&#32;&#112;&#97;&#114;&#116;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#56;&#45;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"105\" width=\"568\" style=\"vertical-align: -47px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147758224\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148099754\">\n<div data-type=\"problem\" id=\"fs-id1169147849540\">\n<p id=\"fs-id1169147833881\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0713833e2cc5737a8721b9c755f42ed8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147906922\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0deea4af12eba440c8809cbb0138023f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;&#45;&#55;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145601980\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147742026\">\n<div data-type=\"problem\" id=\"fs-id1169147935573\">\n<p id=\"fs-id1169147866757\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-335121de74a8eb96ba92565d5fd15539_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#45;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#43;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148224477\">\n<p id=\"fs-id1169147839828\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5db6c7ce4c920aa67005b952db319f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#45;&#49;&#48;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147835113\">In the next example, we could use FOIL or the <span data-type=\"term\" class=\"no-emphasis\">Product of Binomial Squares Pattern<\/span>.<\/p>\n<div data-type=\"example\" id=\"fs-id1169145685376\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169143659210\">\n<div data-type=\"problem\" id=\"fs-id1169145567304\">\n<p id=\"fs-id1169148069276\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-314642d85810a85c90ef62924b0c1f20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148070995\">\n<table id=\"fs-id1169147863045\" class=\"unnumbered unstyled\" summary=\"Use the Binomial Squares Pattern formula the quantity a plus b in parentheses squared equals a squared plus 2 a b plus b squared. Applied to this example we get the expression 3 squared plus 2 times 3 times 2 i plus the quantity 2 i in parentheses squared. Simplifying we get 9 plus 12 i plus 4 i squared. Simplifying further we get 9 plus 12 i plus 4 times negative 1. The final simplified version is 5 plus 12 i.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169147962139\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Use the Product of Binomial Squares Pattern, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8d303193c8805487bf7985deacb82ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"188\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169145660655\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169147860309\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-781de62f38f6aaa674d1cd9cb0993c08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169147802783\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169143518464\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169143492286\">\n<div data-type=\"problem\" id=\"fs-id1169147805544\">\n<p id=\"fs-id1169147715424\">Multiply using the Binomial Squares pattern: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a70ecec1b7e5e60354595c3d7daf8236_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147840934\">\n<p id=\"fs-id1169145733710\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4792a4f71c15e586d8be2cb13b30d597_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#49;&#45;&#50;&#48;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"76\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169143295681\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145657632\">\n<div data-type=\"problem\">\n<p id=\"fs-id1169141030754\">Multiply using the Binomial Squares pattern: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d0c44d00eeb223e7ce088d09f28d8b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#43;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147808386\">\n<p id=\"fs-id1169145642404\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21a8a7821b9f9af421b6bcdcd811d811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#45;&#52;&#48;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"55\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148105017\">Since the square root of a negative number is not a real number, we cannot use the Product Property for Radicals. In order to multiply square roots of negative numbers we should first write them as complex numbers, using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0eb238af4591777552a484b63f68cd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#98;&#125;&#105;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"77\" style=\"vertical-align: -2px;\" \/> This is one place students tend to make errors, so be careful when you see multiplying with a negative square root.<\/p>\n<div data-type=\"example\" id=\"fs-id1169143266846\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1169147866217\">\n<p>Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82321d3192a2e5a853ca16f7ec8fca22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#51;&#54;&#125;&middot;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"87\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145491768\">\n<p id=\"fs-id1169147803148\">To multiply square roots of negative numbers, we first write them as complex numbers.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-737020fa9f605c4c91f9ab3f74e0d913_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#51;&#54;&#125;&middot;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#97;&#115;&#32;&#99;&#111;&#109;&#112;&#108;&#101;&#120;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#32;&#117;&#115;&#105;&#110;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#98;&#125;&#105;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&middot;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#54;&#105;&middot;&#50;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#50;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"539\" style=\"vertical-align: -48px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147870800\">\n<div data-type=\"problem\">\n<p id=\"fs-id1169147768510\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fbfdbfa955ababad4cb5d0e86f86b473_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#57;&#125;&middot;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"87\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148097432\">\n<p id=\"fs-id1169145498758\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4066673cb5834f8ce47cc6896c29963a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148250411\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169143661264\">\n<div data-type=\"problem\" id=\"fs-id1169141473006\">\n<p>Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53c11535e6c9d56fb3257a2312359456_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#51;&#54;&#125;&middot;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#56;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147833959\">\n<p id=\"fs-id1169145643813\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efb7c0d2e348d515285d291bb9f127ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165927630506\">In the next example, each binomial has a square root of a negative number. Before multiplying, each square root of a negative number must be written as a complex number.<\/p>\n<div data-type=\"example\" id=\"fs-id1169145622189\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169145606346\">\n<div data-type=\"problem\" id=\"fs-id1169148064048\">\n<p id=\"fs-id1169147979154\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b761b8baeb80b59821a84559d01918c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"194\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145663400\">\n<p id=\"fs-id1169147824279\">To multiply square roots of negative numbers, we first write them as complex numbers.<\/p>\n<p id=\"fs-id1169145520026\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-704a34c5f145252906a8b3a631aa20bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#97;&#115;&#32;&#99;&#111;&#109;&#112;&#108;&#101;&#120;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#32;&#117;&#115;&#105;&#110;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#98;&#125;&#105;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#70;&#79;&#73;&#76;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#53;&#43;&#57;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#45;&#49;&#48;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#45;&#54;&middot;&#51;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#109;&#98;&#105;&#110;&#101;&#32;&#108;&#105;&#107;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#97;&#110;&#100;&#32;&#115;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#53;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#45;&#54;&middot;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#97;&#110;&#100;&#32;&#99;&#111;&#109;&#98;&#105;&#110;&#101;&#32;&#108;&#105;&#107;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#51;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"110\" width=\"662\" style=\"vertical-align: -49px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147979332\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169143280628\">\n<div data-type=\"problem\" id=\"fs-id1169147856967\">\n<p id=\"fs-id1169147746775\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95dc0c47fb2b69d74c672f02a537b358_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"194\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147830796\">\n<p id=\"fs-id1169147776931\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9dc8c3706d7bab536e091b9499dc3350_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#45;&#50;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"100\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145775017\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145658407\">\n<div data-type=\"problem\" id=\"fs-id1169148230802\">\n<p>Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e40610fd7ef5d2aed7901b13a427aa3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"199\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169141298078\">\n<p id=\"fs-id1169143581121\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7915973acec32cd386e9451a2a8eae57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#43;&#49;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147909818\">We first looked at conjugate pairs when we studied polynomials. We said that a pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a <em data-effect=\"italics\">conjugate pair<\/em> and is of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df1b1ce95678059243781252da3af8a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1169143572992\">A <span data-type=\"term\">complex conjugate pair<\/span> is very similar. For a complex number of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e80cd255be59b21381715aaa50d79181_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\" \/> its conjugate is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28417c47389c8d21a4870d58b786836a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#45;&#98;&#105;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/> Notice they have the same first term and the same last term, but one is a sum and one is a difference.<\/p>\n<div data-type=\"note\" id=\"fs-id1169145572553\">\n<div data-type=\"title\">Complex Conjugate Pair<\/div>\n<p id=\"fs-id1169148200251\">A <strong data-effect=\"bold\">complex conjugate pair<\/strong> is of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e80cd255be59b21381715aaa50d79181_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28417c47389c8d21a4870d58b786836a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#45;&#98;&#105;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<p id=\"fs-id1169147837933\">We will multiply a complex conjugate pair in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1169147851512\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169143316403\">\n<div data-type=\"problem\" id=\"fs-id1169148080502\">\n<p id=\"fs-id1169147700609\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a82c250d03f07c86d48105ab4083e117_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143662126\">\n<p id=\"fs-id1169147877692\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8488c14f087226eafdd9df06e40890e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#70;&#79;&#73;&#76;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#43;&#54;&#105;&#45;&#54;&#105;&#45;&#52;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#109;&#98;&#105;&#110;&#101;&#32;&#108;&#105;&#107;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#97;&#110;&#100;&#32;&#115;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#97;&#110;&#100;&#32;&#99;&#111;&#109;&#98;&#105;&#110;&#101;&#32;&#108;&#105;&#107;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"83\" width=\"520\" style=\"vertical-align: -36px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147709375\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169141322580\">\n<div data-type=\"problem\" id=\"fs-id1169145607061\">\n<p id=\"fs-id1169147770009\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb94631e255eb4faacd03f3bcc8c9874_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&middot;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#43;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147949644\">\n<p id=\"fs-id1169147740020\">25<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148048174\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147868111\">\n<div data-type=\"problem\" id=\"fs-id1169147784324\">\n<p id=\"fs-id1169147744297\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95f2726e966973d7abee706b7c4c38d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#43;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&middot;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147736562\">\n<p id=\"fs-id1169147846906\">29<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147764265\">From our study of polynomials, we know the product of conjugates is always of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6ec29788f86c1d82c9c5a962e44466d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"188\" style=\"vertical-align: -4px;\" \/> The result is called a <span data-type=\"term\" class=\"no-emphasis\">difference of squares<\/span>. We can multiply a complex conjugate pair using this pattern.<\/p>\n<p id=\"fs-id1169147739518\">The last example we used FOIL. Now we will use the <span data-type=\"term\" class=\"no-emphasis\">Product of Conjugates Pattern<\/span>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147862786\" data-alt=\"The quantity a minus b in parentheses times the quantity a plus b in parentheses is written above the expression showing the product of 3 minus 2 i in parentheses and 3 plus 2 i in parentheses. In the next line a squared minus b squared is written above the expression 3 squared minus the quantity 2 i in parentheses squared. Simplifying we get 9 minus 4 i squared. This is equal to 9 minus 4 times negative 1. The final result is 13.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The quantity a minus b in parentheses times the quantity a plus b in parentheses is written above the expression showing the product of 3 minus 2 i in parentheses and 3 plus 2 i in parentheses. In the next line a squared minus b squared is written above the expression 3 squared minus the quantity 2 i in parentheses squared. Simplifying we get 9 minus 4 i squared. This is equal to 9 minus 4 times negative 1. The final result is 13.\" \/><\/span><\/p>\n<p id=\"fs-id1169147865956\">Notice this is the same result we found in <a href=\"#fs-id1169147851512\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id1169148218790\">When we multiply complex conjugates, the product of the last terms will always have an <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7fa840f261e353b4018c86a2cf033bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"13\" style=\"vertical-align: 0px;\" \/> which simplifies to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc343c70c6ffe28a6cd57dfb53f250b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1169147775175\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-229a5181281020993028cc37e81606fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"106\" width=\"119\" style=\"vertical-align: -47px;\" \/><\/div>\n<p id=\"fs-id1169148037437\">This leads us to the Product of Complex Conjugates Pattern: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2532980dab4702efcd784deeee932655_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"196\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1169147706826\">\n<div data-type=\"title\">Product of Complex Conjugates<\/div>\n<p id=\"fs-id1169145496922\">If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers, then<\/p>\n<div data-type=\"equation\" id=\"fs-id1169147848448\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2532980dab4702efcd784deeee932655_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"196\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169147867029\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148071873\">\n<div data-type=\"problem\" id=\"fs-id1169147846363\">\n<p id=\"fs-id1169145715989\">Multiply using the Product of Complex Conjugates Pattern: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3299483db6122d2c6c4742e8f3b0e4c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#43;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145667175\">\n<table id=\"fs-id1169143550228\" class=\"unnumbered unstyled can-break\" summary=\"The quantity a minus b i in parentheses times the quantity a plus b i in parentheses is written above the expression showing the product of 8 minus 2 i in parentheses and 8 plus 2 i in parentheses. In the next line a squared plus b squared is written above the expression 8 squared plus the quantity 2 squared. Simplifying we get 64 plus 4. The final result is 68.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169143573503\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Use the Product of Complex Conjugates Pattern,<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70f8ce61d5820b955500ba38b40ca864_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"201\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169143431827\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify the squares.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169147808213\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Add.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1169148250711\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148227244\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145526008\">\n<div data-type=\"problem\" id=\"fs-id1169148205847\">\n<p id=\"fs-id1169148227408\">Multiply using the Product of Complex Conjugates Pattern: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-570c3af6d090525c036d4e93895030c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#49;&#48;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#49;&#48;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147966173\">\n<p id=\"fs-id1169143520545\">109<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147776926\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148115972\">\n<div data-type=\"problem\" id=\"fs-id1169147737029\">\n<p id=\"fs-id1169147979756\">Multiply using the Product of Complex Conjugates Pattern: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e82bd72adf4af9757b788111251533a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#43;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"155\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147735634\">\n<p id=\"fs-id1169147804823\">41<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169147873364\">\n<h3 data-type=\"title\">Divide Complex Numbers<\/h3>\n<p id=\"fs-id1169143518290\">Dividing complex numbers is much like rationalizing a denominator. We want our result to be in standard form with no imaginary numbers in the denominator.<\/p>\n<div data-type=\"example\" id=\"fs-id1169147851054\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Divide Complex Numbers<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147759739\">\n<div data-type=\"problem\" id=\"fs-id1169141522111\">\n<p id=\"fs-id1169145519620\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eadb1acf6551020f8a5396dd802fccdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#43;&#51;&#105;&#125;&#123;&#51;&#45;&#52;&#105;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"35\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147730871\"><span data-type=\"media\" id=\"fs-id1169147808080\" data-alt=\"Step 1 is to write both the numerator and denominator in standard form. For this example they are both in standard form.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_006a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write both the numerator and denominator in standard form. For this example they are both in standard form.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169143662469\" data-alt=\"Step 2 is to multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 3 minus 4 i is 3 plus 4 i. The resulting expression is the quantity 4 plus 3 i in parentheses times the quantity 3 plus 4 i in parentheses divided by the product of 3 minus 4 i in parentheses and the quantity 3 plus 4 i in parentheses.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_006b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 3 minus 4 i is 3 plus 4 i. The resulting expression is the quantity 4 plus 3 i in parentheses times the quantity 3 plus 4 i in parentheses divided by the product of 3 minus 4 i in parentheses and the quantity 3 plus 4 i in parentheses.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169145729564\" data-alt=\"Step 3 is to simplify and write the result in standard form. Use the pattern the quantity a plus b i in parentheses equals a squared plus b squared in the denominator. The expression for this example then becomes the quantity 12 plus 16 i plus 9 i plus 12 i squared in parentheses divided by the sum of 9 and 16. Combining like terms we get the quantity 12 plus 25 i minus 12 in parentheses divided by 25. Simplifying we get 25 i divided by 25. Write the result in standard form. The result is i.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_006c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to simplify and write the result in standard form. Use the pattern the quantity a plus b i in parentheses equals a squared plus b squared in the denominator. The expression for this example then becomes the quantity 12 plus 16 i plus 9 i plus 12 i squared in parentheses divided by the sum of 9 and 16. Combining like terms we get the quantity 12 plus 25 i minus 12 in parentheses divided by 25. Simplifying we get 25 i divided by 25. Write the result in standard form. The result is i.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169143317947\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147880316\">\n<div data-type=\"problem\" id=\"fs-id1169147849308\">\n<p id=\"fs-id1169147846945\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d362cdcef807fec15802288961f80daf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#43;&#53;&#105;&#125;&#123;&#53;&#45;&#50;&#105;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"35\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147767699\">\n<p id=\"fs-id1169147737517\"><em data-effect=\"italics\">i<\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145643886\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147737214\">\n<div data-type=\"problem\" id=\"fs-id1169147803413\">\n<p id=\"fs-id1169145658419\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c724f723973a87a6ae82485e9bab7d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#43;&#54;&#105;&#125;&#123;&#54;&#45;&#105;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"35\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147751252\">\n<p id=\"fs-id1169145536152\"><em data-effect=\"italics\">i<\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169143578648\">We summarize the steps here.<\/p>\n<div data-type=\"note\" id=\"fs-id1169147726518\" class=\"howto\">\n<div data-type=\"title\">How to divide complex numbers.<\/div>\n<ol id=\"fs-id1169148237090\" type=\"1\" class=\"stepwise\">\n<li>Write both the numerator and denominator in standard form.<\/li>\n<li>Multiply the numerator and denominator by the complex conjugate of the denominator.<\/li>\n<li>Simplify and write the result in standard form.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169147878899\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148231374\">\n<div data-type=\"problem\" id=\"fs-id1169142122806\">\n<p id=\"fs-id1169147862468\">Divide, writing the answer in standard form: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9888900835347f6e8104e3095089674f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#125;&#123;&#53;&#43;&#50;&#105;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"35\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148071478\">\n<p id=\"fs-id1169143525632\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a543bcaf098baa16dd8289a10a3cc8a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#125;&#123;&#53;&#43;&#50;&#105;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#32;&#97;&#110;&#100;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#32;&#98;&#121;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#109;&#112;&#108;&#101;&#120;&#32;&#99;&#111;&#110;&#106;&#117;&#103;&#97;&#116;&#101;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#32;&#97;&#110;&#100;&#32;&#117;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#80;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#109;&#112;&#108;&#101;&#120;&#32;&#67;&#111;&#110;&#106;&#117;&#103;&#97;&#116;&#101;&#115;&#32;&#80;&#97;&#116;&#116;&#101;&#114;&#110;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#53;&#43;&#54;&#105;&#125;&#123;&#123;&#53;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#53;&#43;&#54;&#105;&#125;&#123;&#50;&#57;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#50;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#50;&#57;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"332\" width=\"609\" style=\"vertical-align: -161px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148053850\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145971630\">\n<div data-type=\"problem\" id=\"fs-id1169147847176\">\n<p id=\"fs-id1169142281226\">Divide, writing the answer in standard form: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-319f5967f9866afa037e34afbf9fed06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#45;&#52;&#105;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"35\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147767469\">\n<p id=\"fs-id1169148185370\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ae02f9efd0bcb1e1091881e2170550c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#55;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#125;&#123;&#49;&#55;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"61\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147862734\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147727903\">\n<div data-type=\"problem\" id=\"fs-id1169147709749\">\n<p id=\"fs-id1169147832655\">Divide, writing the answer in standard form: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d03d88a2d8fc8d9ee68a754bd811b98e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#125;&#123;&#45;&#49;&#43;&#50;&#105;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"46\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148233919\">\n<p id=\"fs-id1169145843850\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea3af639d3d6056949429a2a5496e2d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"47\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147794219\">Be careful as you find the conjugate of the denominator.<\/p>\n<div data-type=\"example\" id=\"fs-id1169148224121\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169147848442\">\n<div data-type=\"problem\" id=\"fs-id1169143748194\">\n<p id=\"fs-id1169147807705\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf98e9e5fc07f56cdd1a2e3640378c42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#43;&#51;&#105;&#125;&#123;&#52;&#105;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"35\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145607677\">\n<p id=\"fs-id1169148076300\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cebacc180b4fe476004e26bf7d7bcf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#43;&#51;&#105;&#125;&#123;&#52;&#105;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#43;&#51;&#105;&#125;&#123;&#48;&#43;&#52;&#105;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#32;&#97;&#110;&#100;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#32;&#98;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#99;&#111;&#109;&#112;&#108;&#101;&#120;&#32;&#99;&#111;&#110;&#106;&#117;&#103;&#97;&#116;&#101;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#43;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#48;&#105;&#45;&#49;&#50;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#45;&#49;&#54;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#116;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#48;&#105;&#43;&#49;&#50;&#125;&#123;&#49;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#125;&#123;&#49;&#54;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#116;&#104;&#101;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"209\" width=\"564\" style=\"vertical-align: -100px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147838114\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148070414\">\n<div data-type=\"problem\" id=\"fs-id1169147862015\">\n<p id=\"fs-id1169145575732\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b50c3cd1e4ac0b6fff4e6b8b9bd9647a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#43;&#51;&#105;&#125;&#123;&#50;&#105;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"35\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147746802\">\n<p id=\"fs-id1169147707150\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-805bdbd943b929894f02bae07f598352_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"47\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147852048\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147784202\">\n<div data-type=\"problem\" id=\"fs-id1169145640113\">\n<p id=\"fs-id1169143579172\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ba1246af7d40640f635aaead3cf1c96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#43;&#52;&#105;&#125;&#123;&#53;&#105;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"35\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148225190\">\n<p id=\"fs-id1169147946732\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52785406dc749adee534c4f1675986c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"47\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169148190703\">\n<h3 data-type=\"title\">Simplify Powers of <em data-effect=\"italics\">i<\/em><\/h3>\n<p id=\"fs-id1169147850832\">The powers of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-695d9d59bd04859c6c99e7feb11daab6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> make an interesting pattern that will help us simplify higher powers of <em data-effect=\"italics\">i<\/em>. Let\u2019s evaluate the powers of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-695d9d59bd04859c6c99e7feb11daab6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> to see the pattern.<\/p>\n<div data-type=\"equation\" id=\"fs-id1169143728354\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e5f0a091cab57eeaf94bd3540a1325c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#105;&#125;&#94;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&middot;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&middot;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&middot;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#105;&#125;&#94;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#55;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#56;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#105;&#125;&#94;&#123;&#52;&#125;&middot;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#52;&#125;&middot;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#52;&#125;&middot;&#123;&#105;&#125;&#94;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#52;&#125;&middot;&#123;&#105;&#125;&#94;&#123;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&middot;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&middot;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&middot;&#123;&#105;&#125;&#94;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&middot;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"237\" width=\"417\" style=\"vertical-align: -111px;\" \/><\/div>\n<p id=\"fs-id1169148072161\">We summarize this now.<\/p>\n<div data-type=\"equation\" id=\"fs-id1169147741472\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d95bfb30c72a4df1394d34bfdd133bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#105;&#125;&#94;&#123;&#49;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#53;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#54;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#105;&#125;&#94;&#123;&#51;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#55;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#105;&#125;&#94;&#123;&#52;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#56;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"82\" width=\"266\" style=\"vertical-align: -34px;\" \/><\/div>\n<p id=\"fs-id1169143573199\">If we continued, the pattern would keep repeating in blocks of four. We can use this pattern to help us simplify powers of <em data-effect=\"italics\">i<\/em>. Since <em data-effect=\"italics\">i<\/em><sup>4<\/sup> = 1, we rewrite each power, <em data-effect=\"italics\">i<sup>n<\/sup><\/em>, as a product using <em data-effect=\"italics\">i<\/em><sup>4<\/sup> to a power and another power of <em data-effect=\"italics\">i<\/em>.<\/p>\n<p id=\"fs-id1169145662274\">We rewrite it in the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c5a4574e8255533e486614e49422053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#105;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#113;&#125;&middot;&#123;&#105;&#125;&#94;&#123;&#114;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"94\" style=\"vertical-align: -7px;\" \/> where the exponent, <em data-effect=\"italics\">q<\/em>, is the quotient of <em data-effect=\"italics\">n<\/em> divided by 4 and the exponent, <em data-effect=\"italics\">r<\/em>, is the remainder from this division. For example, to simplify <em data-effect=\"italics\">i<\/em><sup>57<\/sup>, we divide 57 by 4 and we get 14 with a remainder of 1. In other words, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-836e0611f755b78e3a11595fc28c0201_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#55;&#61;&#52;&middot;&#49;&#52;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"103\" style=\"vertical-align: -2px;\" \/> So we write <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3fb69266e4a2cb95ae33a67ad6ec53c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#53;&#55;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#49;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#49;&#52;&#125;&middot;&#123;&#105;&#125;&#94;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"104\" style=\"vertical-align: -7px;\" \/> and then simplify from there.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1168040486190\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_008_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"example\" id=\"fs-id1169141376582\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1169145641420\">\n<p id=\"fs-id1169145641422\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d15722d7bed652b180ac3ce0d36d6082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#56;&#54;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"25\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147963111\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-660fd81e40fb8d6f03c2560c6011570e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#56;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#118;&#105;&#100;&#101;&#32;&#56;&#54;&#32;&#98;&#121;&#32;&#52;&#32;&#97;&#110;&#100;&#32;&#114;&#101;&#119;&#114;&#105;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#105;&#125;&#94;&#123;&#56;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#105;&#125;&#94;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#105;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#113;&#125;&middot;&#123;&#105;&#125;&#94;&#123;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#49;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#49;&#125;&middot;&#123;&#105;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"67\" width=\"469\" style=\"vertical-align: -29px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1168040462857\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2eb90136b412eabc4b8537c8dd89c63c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#49;&#125;&middot;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"531\" style=\"vertical-align: -15px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147766271\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145748075\">\n<div data-type=\"problem\" id=\"fs-id1169145748077\">\n<p id=\"fs-id1169142437136\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a7f4c9829c24b66882de4c7c8f454e8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#55;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"25\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148063916\">\n<p id=\"fs-id1169148063919\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbbc1af417d9937847c0c17221cb8820_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169145733988\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169145716787\">\n<div data-type=\"problem\" id=\"fs-id1169145737890\">\n<p id=\"fs-id1169145737892\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-677baa7bf53c343461ebc6dd883c5e6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#57;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"25\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147846616\">\n<p id=\"fs-id1169147960608\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169147965255\" class=\"media-2\">\n<p id=\"fs-id1169145645015\">Access these online resources for additional instruction and practice with the complex number system.<\/p>\n<ul id=\"fs-id1169143332448\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37CompNumb1\">Expressing Square Roots of Negative Numbers with i<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37CompNumb2\">Subtract and Multiply Complex Numbers<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37CompNumb3\">Dividing Complex Numbers<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37CompNumb4\">Rewriting Powers of i<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169145732671\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1169147850757\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Square Root of a Negative Number<\/strong>\n<ul id=\"fs-id1169145716315\" data-bullet-style=\"open-circle\">\n<li>If <em data-effect=\"italics\">b<\/em> is a positive real number, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20332daafa6451a81cb3899b64b0211f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#98;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -2px;\" \/>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1169147827697\" summary=\"The table has four rows and three columns. The first row is a header and the second column entry a plus b i. In the second row is b equals zero, a plus 0 i, and \u201cReal number\u201d. The third row contains b is not equal to 0, a plus b i, and \u201cImaginary number\u201d. The fourth row contains a = 0, 0 plus b i, and \u201cPure imaginary number\u201d.\">\n<tbody>\n<tr valign=\"top\">\n<td><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ffcd7d918a83e37de98a41b94384c07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-597ae6c01abdebf6816c77f3ce1fd9de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43695884e5027a485867e17fb783bfc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#97;&#43;&#48;&middot;&#105;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"37\" style=\"vertical-align: -33px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\">Real number<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1edd53778d94abb1dc1e19acff79e9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ffcd7d918a83e37de98a41b94384c07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"45\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\">Imaginary number<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-072bed0ebf929f9d7c14da365c8512a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23c8a64b4f38763727af5268c4a3f8ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#125;&#48;&#43;&#98;&#105;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#105;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"79\" width=\"45\" style=\"vertical-align: -33px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\">Pure imaginary number<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>A complex number is in <strong data-effect=\"bold\">standard form<\/strong> when written as <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bi<\/em>, where <em data-effect=\"italics\">a, b<\/em> are real numbers.\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1169143534059\" data-alt=\"The diagram has a rectangle with the labels \u201cComplex Numbers\u201d and a plus b i. A second rectangle has the labels \u201cReal Numbers\u201d, a plus b i, b = 0. A third rectangle has the labels \u201cImaginary Numbers\u201d, a plus b i, b not equal to 0. Arrows go from the Real Numbers rectangle and Imaginary Numbers rectangle and point toward the Complex Numbers rectangle.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The diagram has a rectangle with the labels \u201cComplex Numbers\u201d and a plus b i. A second rectangle has the labels \u201cReal Numbers\u201d, a plus b i, b = 0. A third rectangle has the labels \u201cImaginary Numbers\u201d, a plus b i, b not equal to 0. Arrows go from the Real Numbers rectangle and Imaginary Numbers rectangle and point toward the Complex Numbers rectangle.\" \/><\/span><\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Product of Complex Conjugates<\/strong>\n<ul id=\"fs-id1169145496103\" data-bullet-style=\"open-circle\">\n<li>If <em data-effect=\"italics\">a, b<\/em> are real numbers, then\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2532980dab4702efcd784deeee932655_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"196\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">How to Divide Complex Numbers<\/strong>\n<ol id=\"fs-id1169142400660\" type=\"1\" class=\"stepwise\">\n<li>Write both the numerator and denominator in standard form.<\/li>\n<li>Multiply the numerator and denominator by the complex conjugate of the denominator.<\/li>\n<li>Simplify and write the result in standard form.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169143332524\">\n<h3 data-type=\"title\">Section Exercises<\/h3>\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169147966479\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1169148227286\"><strong data-effect=\"bold\">Evaluate the Square Root of a Negative Number<\/strong><\/p>\n<p id=\"fs-id1169145728436\">In the following exercises, write each expression in terms of <em data-effect=\"italics\">i<\/em> and simplify if possible.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145667504\">\n<div data-type=\"problem\" id=\"fs-id1169147829166\">\n<p id=\"fs-id1169147829168\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f954dddf825fed2cc5d3b705f39a896_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c5f1c64da7be0805732ac190447b2d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-259febce4b4e7150f0dfd35be20d1c7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169147865256\">\n<p id=\"fs-id1169147878400\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16442f3dd70869ace8885d39ca3a994b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05e3ba8bc0adb3022b4db4ace719309f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34b2b5eabf54068fcd682528f6c0ff70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147797185\">\n<div data-type=\"problem\" id=\"fs-id1169147797187\">\n<p id=\"fs-id1169147841507\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8428c95cb8eee09f094a7824f1a7d5b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e5558f3692dfc100cb1b426c034b601_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0419d2946505c9713f9dcca5ed52eb90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143614499\">\n<div data-type=\"problem\" id=\"fs-id1169143614501\">\n<p id=\"fs-id1169145496257\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe16287834681030d759799a1ffc042e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b28d1b676642ac411f048683cc1c5e7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d86e626eeee76a6c5d831d159582421_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169145491586\">\n<p id=\"fs-id1169147855122\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-795907a122a059e151312b94fac7b342_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"23\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1c8a74e9393a17df938af5e82cf10e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#51;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-704f2f435c68d7ca8d8ecc214e968307_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145664846\">\n<div data-type=\"problem\" id=\"fs-id1169145644765\">\n<p id=\"fs-id1169145644767\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7dca7c0770f94a075c5454a5e22355c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff3372c1e64caa6098deddd79d24c666_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5fedcde335f7abf21237abe6a273a1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#55;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169145519944\"><strong data-effect=\"bold\">Add or Subtract Complex Numbers<\/strong> In the following exercises, add or subtract.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147961706\">\n<div data-type=\"problem\" id=\"fs-id1169147783916\">\n<p id=\"fs-id1169147783918\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e69ad0d2a5ea8fefce062d0abcec97df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#55;&#53;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1169148098416\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b061078b3c967feedac893e379bbf3ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145844083\">\n<div data-type=\"problem\" id=\"fs-id1169145844085\">\n<p id=\"fs-id1169143580639\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cc189f518095e619367f2324f96d730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#50;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#55;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147751002\">\n<div data-type=\"problem\" id=\"fs-id1169145669417\">\n<p id=\"fs-id1169145669419\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30c2ecfd1bd08351ca590f4c60471aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#53;&#48;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5988c1534ab4d9b2fb722aaa78a025a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143580475\">\n<div data-type=\"problem\" id=\"fs-id1169143580477\">\n<p id=\"fs-id1169147966380\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5713c2d59875084181204489a619162b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#55;&#50;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"105\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145505198\">\n<div data-type=\"problem\" id=\"fs-id1169145696480\">\n<p id=\"fs-id1169145696482\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72397b4976ab10920d2ee145931fc36b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#43;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145505873\">\n<p id=\"fs-id1169143688226\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65ce7a75122d946c93351e8b0a137e32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#43;&#55;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"46\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147978369\">\n<div data-type=\"problem\" id=\"fs-id1169148103770\">\n<p id=\"fs-id1169148103772\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4313950b59f45effc9bed6616e30138_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#43;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148211974\">\n<div data-type=\"problem\" id=\"fs-id1169147740283\">\n<p id=\"fs-id1169147740285\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e434a9503821fecf5f9cc8cc124fe13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#43;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147839316\">\n<p id=\"fs-id1169147839318\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd19a7b096d485214c600a27a2a6d845_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#43;&#50;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"53\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148071796\">\n<div data-type=\"problem\" id=\"fs-id1169148071799\">\n<p id=\"fs-id1169147843469\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-badeda8ed7c3f4cd654b5eade45742e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#54;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"152\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143728397\">\n<div data-type=\"problem\" id=\"fs-id1169143728399\">\n<p id=\"fs-id1169147720896\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e3ec5b0341baa944d2e8ad3a71cf84e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#54;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2046d7790e5e3eefcebbf7e930f4dd35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#43;&#50;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"58\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145661457\">\n<div data-type=\"problem\" id=\"fs-id1169145670340\">\n<p id=\"fs-id1169145670343\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f37057eb87c5ec5c22246ef13763d8f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#55;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145663778\">\n<div data-type=\"problem\" id=\"fs-id1169145505214\">\n<p id=\"fs-id1169145505217\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-840361fcc61856ef570a95d920ee2aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#43;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148229777\">\n<p id=\"fs-id1169148229779\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-160fe05bfaa449c7886131ff2018e029_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#43;&#53;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"46\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148233529\">\n<div data-type=\"problem\" id=\"fs-id1169148064110\">\n<p id=\"fs-id1169148064112\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b5831ee241351286197c1e02a8e0fad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#43;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#43;&#54;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"166\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143643797\">\n<div data-type=\"problem\" id=\"fs-id1169143643799\">\n<p id=\"fs-id1169145722644\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c63088985bed9a67c5a24d07cfbb089_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#51;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"204\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147848260\">\n<p id=\"fs-id1169147848263\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43a78eef13d24ea53c64adfc1b884287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#45;&#49;&#51;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"55\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147878950\">\n<div data-type=\"problem\" id=\"fs-id1169148229531\">\n<p id=\"fs-id1169148229533\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41d1880d7a645a1871da3dc61fce63a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#54;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"217\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147982815\">\n<div data-type=\"problem\" id=\"fs-id1169145670355\">\n<p id=\"fs-id1169145670357\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8c88b19c6f0b1274d8a08819b35484b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#53;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"240\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148207078\">\n<p id=\"fs-id1169148207080\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f04eff394cbfa64561743a49a246d692_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#45;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169141298513\">\n<div data-type=\"problem\" id=\"fs-id1169141298515\">\n<p id=\"fs-id1169145532866\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47cf7558e147770d95ed8c1ce458f635_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"231\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145737955\"><strong data-effect=\"bold\">Multiply Complex Numbers<\/strong><\/p>\n<p id=\"fs-id1169147775327\">In the following exercises, multiply.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147775330\">\n<div data-type=\"problem\" id=\"fs-id1169147775332\">\n<p id=\"fs-id1169147965722\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34eb67dae105191df555166c4041b064_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#105;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"76\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145670301\">\n<p id=\"fs-id1169145670303\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ad95e90d5fd7d81081831329027f831_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#43;&#50;&#48;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"62\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147982473\">\n<div data-type=\"problem\" id=\"fs-id1169145720264\">\n<p id=\"fs-id1169145720266\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20d3645c3dd1d03cd820d7e76d53c906_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#105;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#43;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147746296\">\n<div data-type=\"problem\" id=\"fs-id1169143534249\">\n<p id=\"fs-id1169143534251\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67ad08458c6352b9815659fa00d83434_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#105;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147863494\">\n<p id=\"fs-id1169147863496\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fabad6179aa6565410aa5c8a239ae68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#43;&#49;&#56;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"76\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147841097\">\n<div data-type=\"problem\" id=\"fs-id1169147841099\">\n<p id=\"fs-id1169147841101\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ad91f5b311bf4a7894dae3aba6a9784_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#105;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#43;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145639745\">\n<div data-type=\"problem\" id=\"fs-id1169145639748\">\n<p id=\"fs-id1169145639750\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50e621906b99c13e0d48a6471a083313_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#43;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#43;&#54;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1169141298387\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe41bdfdb9a77ed382913d82fa3206bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#56;&#43;&#43;&#57;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"81\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147739873\">\n<div data-type=\"problem\" id=\"fs-id1169147739876\">\n<p id=\"fs-id1169148233360\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b28c69bfa4805ff2555ec4172bb252e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#43;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147783892\">\n<div data-type=\"problem\" id=\"fs-id1169147783894\">\n<p id=\"fs-id1169147783896\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a934d488976edee36c746ac04c1b14cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#43;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#55;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145506798\">\n<p id=\"fs-id1169145506800\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db61566081d341dfc406d61aa2d3167d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#43;&#49;&#53;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"63\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147878970\">\n<div data-type=\"problem\" id=\"fs-id1169147878972\">\n<p id=\"fs-id1169147878974\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a14cb0adcabb5c18d6d7f4e5ea91a5bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#45;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147739869\">In the following exercises, multiply using the Product of Binomial Squares Pattern.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148077956\">\n<div data-type=\"problem\" id=\"fs-id1169148077958\">\n<p id=\"fs-id1169147963850\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e244281370556433e6afdaefcdf9dd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169142133163\">\n<p id=\"fs-id1169143520481\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07631c4691b3f55a18620a1f6a9450bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#43;&#50;&#52;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"67\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147852519\">\n<div data-type=\"problem\" id=\"fs-id1169147852521\">\n<p id=\"fs-id1169145733973\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-897d7454df050cf4e2104b869179855e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#43;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147803530\">\n<div data-type=\"problem\" id=\"fs-id1169145577064\">\n<p id=\"fs-id1169145577066\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-267e7ca6efc89bf02dd12fb4d28dbed7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147804412\">\n<p id=\"fs-id1169147804414\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1de42ccd212d3943b6cf4f04cc18a5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#45;&#49;&#50;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143573850\">\n<div data-type=\"problem\" id=\"fs-id1169143533452\">\n<p id=\"fs-id1169143533454\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea33234d46cf3951f95adb5ba8ea2af3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#45;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148123174\">In the following exercises, multiply.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148123177\">\n<div data-type=\"problem\" id=\"fs-id1169148123179\">\n<p id=\"fs-id1169147874530\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-058da86c6328f4929e9fde0b80b60be3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#53;&#125;&middot;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169141298052\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30dc93e3a7470ae35d749e6c184d7fe4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147741809\">\n<div data-type=\"problem\" id=\"fs-id1169147741811\">\n<p id=\"fs-id1169148097690\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d21706f4ebb63907a6d952830134b74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#125;&middot;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147840679\">\n<div data-type=\"problem\" id=\"fs-id1169147840681\">\n<p id=\"fs-id1169147840683\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bfb132668766bb6ebfe82723da4dd3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#57;&#125;&middot;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143442627\">\n<p id=\"fs-id1169143442630\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec1795c5af248d148a298ec6e4785280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147875254\">\n<div data-type=\"problem\" id=\"fs-id1169147875256\">\n<p id=\"fs-id1169147875258\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c0ff621f853fe522970977858c7bd70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#54;&#52;&#125;&middot;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143728425\">\n<div data-type=\"problem\" id=\"fs-id1169145605886\">\n<p id=\"fs-id1169145605888\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dda89a892c55b591881add97cc67f279_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#50;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"199\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169142416307\">\n<p id=\"fs-id1169142416309\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d608b2f2106c4e7f724d1c87158bb967_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#52;&#43;&#52;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"91\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147764582\">\n<div data-type=\"problem\" id=\"fs-id1169147764584\">\n<p id=\"fs-id1169147764587\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-610098382b7b5b2c61f0889a85dcf6e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#55;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"199\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145662602\">\n<div data-type=\"problem\" id=\"fs-id1169145662604\">\n<p id=\"fs-id1169143517978\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6bbdd9d9e44d944085417f0ca7bfc4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"190\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147965152\">\n<p id=\"fs-id1169145519425\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20082d9e2415b9287f910f386d87b86e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#48;&#45;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"91\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145601711\">\n<div data-type=\"problem\" id=\"fs-id1169145598899\">\n<p id=\"fs-id1169145598901\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6d81ed851632337d75a64daa10d48f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#53;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"199\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147768224\">\n<div data-type=\"problem\" id=\"fs-id1169147768226\">\n<p id=\"fs-id1169147768228\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57e897616e09f446e0809d096036e845_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#43;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143525861\">\n<p id=\"fs-id1169145606349\">5<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145606354\">\n<div data-type=\"problem\" id=\"fs-id1169141473728\">\n<p id=\"fs-id1169141473730\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03f23ba5798493be37118fc9fd126a13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#43;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14affd8f77ae7ccb402d0359c14de2eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#43;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145597852\">\n<p id=\"fs-id1169145597854\">53<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148225589\">\n<div data-type=\"problem\" id=\"fs-id1169148225591\">\n<p id=\"fs-id1169143586050\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f2eeaf39f3b3ed3acc947be76f74190_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#45;&#56;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#43;&#56;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148252283\">In the following exercises, multiply using the Product of Complex Conjugates Pattern.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145785940\">\n<div data-type=\"problem\" id=\"fs-id1169145785942\">\n<p id=\"fs-id1169141409123\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48c87fb8682e1062fccb3c46d769562d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#43;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147837530\">\n<p id=\"fs-id1169148125806\">50<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147771371\">\n<div data-type=\"problem\" id=\"fs-id1169147771373\">\n<p id=\"fs-id1169147771375\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2480049a94f01baf14f431940d2ed503_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#45;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#43;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145670789\">\n<div data-type=\"problem\" id=\"fs-id1169145670791\">\n<p id=\"fs-id1169148218572\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ddaa034384ff01b55e55d8e57407159b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#43;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147855189\">\n<p id=\"fs-id1169147855191\">85<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148227215\">\n<div data-type=\"problem\" id=\"fs-id1169148190446\">\n<p id=\"fs-id1169148190448\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70a1d7721e46a523827aec90e379a6f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#43;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148097665\"><strong data-effect=\"bold\">Divide Complex Numbers<\/strong><\/p>\n<p id=\"fs-id1169148097671\">In the following exercises, divide.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148125820\">\n<div data-type=\"problem\" id=\"fs-id1169148125822\">\n<p id=\"fs-id1169145622251\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c829a6b4298a4fbeca9d5ea05a6a3ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#43;&#52;&#105;&#125;&#123;&#52;&#45;&#51;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147815810\">\n<p id=\"fs-id1169147815812\"><em data-effect=\"italics\">i<\/em><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143579887\">\n<div data-type=\"problem\" id=\"fs-id1169143579889\">\n<p id=\"fs-id1169145747976\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87b3a8c5990a7e6060406895ec3ab604_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#45;&#50;&#105;&#125;&#123;&#50;&#43;&#53;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"30\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145598404\">\n<div data-type=\"problem\" id=\"fs-id1169145598406\">\n<p id=\"fs-id1169145598408\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f248cb1e992ec3ad2ce6f111d1847d4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#43;&#105;&#125;&#123;&#51;&#45;&#52;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145578886\">\n<p id=\"fs-id1169145578888\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f75115c034d09c38c27df60fd631f706_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#50;&#53;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148248849\">\n<div data-type=\"problem\" id=\"fs-id1169148248851\">\n<p id=\"fs-id1169148248853\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-376b75175e9ece2860665aeefac3c3b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#45;&#50;&#105;&#125;&#123;&#54;&#43;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"30\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145670320\">\n<div data-type=\"problem\" id=\"fs-id1169145670322\">\n<p id=\"fs-id1169148218590\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-879b02d53ab895be6784e4270e5daa9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#45;&#51;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148125778\">\n<p id=\"fs-id1169148125780\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e66b35f2b351183a4dc2ed160e5a483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#49;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#51;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"61\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148079849\">\n<div data-type=\"problem\" id=\"fs-id1169148079851\">\n<p id=\"fs-id1169148079853\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7b43717e3f2fb9741d207e344b6b506_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#52;&#45;&#53;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145716408\">\n<div data-type=\"problem\" id=\"fs-id1169145716410\">\n<p id=\"fs-id1169145716412\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65553d1da017b809a15f324b44c639e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#52;&#125;&#123;&#51;&#45;&#50;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145607991\">\n<p id=\"fs-id1169145748060\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e921366e669da7d00ada92f281a117d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#49;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#51;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"76\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145608611\">\n<div data-type=\"problem\" id=\"fs-id1169145608613\">\n<p id=\"fs-id1169145608616\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21d49ac4bce4b66a54ea6b3fc8ff9b42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#125;&#123;&#51;&#43;&#50;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"30\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143520334\">\n<div data-type=\"problem\" id=\"fs-id1169143520336\">\n<p id=\"fs-id1169141376159\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa5cc439591fa3ba4eb8d47cf846d2f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#43;&#52;&#105;&#125;&#123;&#51;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145640198\">\n<p id=\"fs-id1169145640200\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-499866de8f7572d5b0a5df6df25bbcb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"47\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1169143614240\">\n<p id=\"fs-id1169145622261\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6262f15486696b06253576bbec68463e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#43;&#51;&#105;&#125;&#123;&#55;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148248922\">\n<div data-type=\"problem\" id=\"fs-id1169148248924\">\n<p id=\"fs-id1169148248926\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3999df4bb61e5884042906aa35243ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#45;&#51;&#105;&#125;&#123;&#52;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"41\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143579814\">\n<p id=\"fs-id1169143579816\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d435d2deaa9b7d9b938e1d86d6afe16d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143385633\">\n<div data-type=\"problem\" id=\"fs-id1169143385635\">\n<p id=\"fs-id1169143385637\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1caabbaf40f078021c1c82475f609545_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#45;&#53;&#105;&#125;&#123;&#50;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"41\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169141030520\"><strong data-effect=\"bold\">Simplify Powers of <em data-effect=\"italics\">i<\/em><\/strong><\/p>\n<p id=\"fs-id1169145575830\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145575833\">\n<div data-type=\"problem\" id=\"fs-id1169145575835\">\n<p id=\"fs-id1169145575837\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-203d5c5bd66dbb2ab237d4514ae0d5aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#52;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148233571\">\n<p id=\"fs-id1169148233573\"><em data-effect=\"italics\">i<\/em><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169141036435\">\n<div data-type=\"problem\" id=\"fs-id1169141036438\">\n<p id=\"fs-id1169141036440\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6a919763a418a2d3f55d49337ec32f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#51;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145666672\">\n<div data-type=\"problem\" id=\"fs-id1169145666674\">\n<p id=\"fs-id1169145666676\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa23d9b87a80961d2fd0ec6cc5786d97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#54;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143687470\">\n<p id=\"fs-id1169143687472\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145716645\">\n<div data-type=\"problem\" id=\"fs-id1169145716647\">\n<p id=\"fs-id1169145716649\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5d71db3a82c9d128eb88b7f5b89bac3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#52;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147849782\">\n<div data-type=\"problem\" id=\"fs-id1169147849784\">\n<p id=\"fs-id1169147849786\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7b5876a05951137b5883d67f55d7934_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#49;&#50;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143637803\">\n<p id=\"fs-id1169143637805\">1<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148077997\">\n<div data-type=\"problem\" id=\"fs-id1169148077999\">\n<p id=\"fs-id1169148078001\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e8bc2f7c23c632114bceec6a5fd0f3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#49;&#54;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6ba5be0dde33a9f4d4707cde7575ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#49;&#51;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143280974\">\n<p id=\"fs-id1169143280976\"><em data-effect=\"italics\">i<\/em><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143280983\">\n<div data-type=\"problem\" id=\"fs-id1169145663864\">\n<p id=\"fs-id1169145663866\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0796aef07306fa58f775640e9763cc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#50;&#53;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1169142417355\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1169141472919\">\n<div data-type=\"problem\" id=\"fs-id1169141472921\">\n<p id=\"fs-id1169141472924\">Explain the relationship between real numbers and complex numbers.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169141472928\">\n<p id=\"fs-id1169143637743\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143637748\">\n<div data-type=\"problem\" id=\"fs-id1169143637750\">\n<p id=\"fs-id1169143637752\">Aniket multiplied as follows and he got the wrong answer. What is wrong with his reasoning?<\/p>\n<p id=\"fs-id1169143637755\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-742e9f1e7261e4bdcb320e439be5c265_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#55;&#125;&middot;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#55;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#57;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"74\" style=\"vertical-align: -22px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147878746\">\n<div data-type=\"problem\" id=\"fs-id1169147878748\">\n<p id=\"fs-id1169147878750\">Why is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0dae16787ddb7854adef86c967c784c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#54;&#52;&#125;&#61;&#56;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -3px;\" \/> but <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2d668c4bd378a6b319f3c0f7b2aed6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#45;&#54;&#52;&#125;&#61;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147878773\">\n<p id=\"fs-id1169147878775\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169141298444\">\n<div data-type=\"problem\" id=\"fs-id1169141298446\">\n<p id=\"fs-id1169141298448\">Explain how dividing complex numbers is similar to rationalizing a denominator.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169147841323\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169147841336\" data-alt=\"The table has 4 columns and 4 rows. The first row is a header row with the headers \u201cI can\u2026\u201d, \u201cConfidently\u201d, \u201cWith some help.\u201d, and \u201cNo \u2013 I don\u2019t get it!\u201d. The first column contains the phrases \u201cevaluate the square root of a negative number\u201d, \u201cadd or subtract complex numbers\u201d, \u201cmultiply complex numbers\u201d, \u201cdivide complex numbers\u201d, and \u201csimplify powers of i\u201d. The other columns are left blank so the learner can indicate their level of understanding.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The table has 4 columns and 4 rows. The first row is a header row with the headers \u201cI can\u2026\u201d, \u201cConfidently\u201d, \u201cWith some help.\u201d, and \u201cNo \u2013 I don\u2019t get it!\u201d. The first column contains the phrases \u201cevaluate the square root of a negative number\u201d, \u201cadd or subtract complex numbers\u201d, \u201cmultiply complex numbers\u201d, \u201cdivide complex numbers\u201d, and \u201csimplify powers of i\u201d. The other columns are left blank so the learner can indicate their level of understanding.\" \/><\/span><\/p>\n<p id=\"fs-id1169143581464\"><span class=\"token\">\u24d1<\/span> On a scale of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3e3c048222c38a09c036de10daa1700_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#49;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\" \/> how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p>\n<\/div>\n<\/div>\n<div class=\"review-exercises\" data-depth=\"1\" id=\"fs-id1169145519825\">\n<h3 data-type=\"title\">Chapter Review Exercises<\/h3>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169145519828\">\n<h4 data-type=\"title\"><a href=\"\/contents\/eb8b8413-de16-4b16-8555-f29918cf1207\" class=\"target-chapter\">Simplify Expressions with Roots<\/a><\/h4>\n<p id=\"fs-id1169145494084\"><strong data-effect=\"bold\">Simplify Expressions with Roots<\/strong><\/p>\n<p id=\"fs-id1169145494089\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1169145494096\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0a475f668d70cf13d563fe23a90fe34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee972e3c30a31662870fdcaf5164f877_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143533387\">\n<p id=\"fs-id1169143533389\"><span class=\"token\">\u24d0<\/span> 15 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148125835\">\n<div data-type=\"problem\" id=\"fs-id1169148125837\">\n<p id=\"fs-id1169148125840\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8530300f0961444a529a8bc356c4c38b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-259febce4b4e7150f0dfd35be20d1c7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145519880\">\n<div data-type=\"problem\" id=\"fs-id1169145519883\">\n<p id=\"fs-id1169145519885\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3ff06837ec8fca3a10b678052b4c3d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe9b91eb5b6fd4a4bcb7545fa44d8399_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2756fa02f8e7bcf9a95cc3a3f145dca4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#50;&#52;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145785915\">\n<p id=\"fs-id1169145785917\"><span class=\"token\">\u24d0<\/span> 2 <span class=\"token\">\u24d1<\/span> 3 <span class=\"token\">\u24d2<\/span> 3<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169141015057\">\n<div data-type=\"problem\" id=\"fs-id1169141015059\">\n<p id=\"fs-id1169141015061\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-112418236f1f6cc539e50f549fac9101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#45;&#53;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0eaa386750f44685e20d3af445e7d74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#45;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3e725f646d2348c928a229b15670f53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148218622\"><strong data-effect=\"bold\">Estimate and Approximate Roots<\/strong><\/p>\n<p id=\"fs-id1169148218628\">In the following exercises, estimate each root between two consecutive whole numbers.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148218631\">\n<div data-type=\"problem\" id=\"fs-id1169145716563\">\n<p id=\"fs-id1169145716565\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb80f06a557cf24280ef1dc7749b8053_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fde7c260912c897467326d290ad7651e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#56;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148125724\">\n<p id=\"fs-id1169148125726\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9cb03c7156c52fb4a1ba898a16588f23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#60;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#56;&#125;&#60;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba75f6374870f596ec008314ba77b856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#60;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#56;&#52;&#125;&#60;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169148218563\">In the following exercises, approximate each root and round to two decimal places.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148218567\">\n<div data-type=\"problem\" id=\"fs-id1169148218569\">\n<p id=\"fs-id1169147878704\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e871f68fb86c8f7769887b18985d3be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fde7c260912c897467326d290ad7651e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#56;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c9b2d9b7c5f70d979b664380541c06a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#49;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169141376105\"><strong data-effect=\"bold\">Simplify Variable Expressions with Roots<\/strong><\/p>\n<p id=\"fs-id1169143637764\">In the following exercises, simplify using absolute values as necessary.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169143637767\">\n<div data-type=\"problem\" id=\"fs-id1169143637769\">\n<p id=\"fs-id1169143637772\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39649b1944d0f0d3c46a2169265aff82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b78426a37841ad8c5dcc1117834248c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#55;&#93;&#123;&#123;&#98;&#125;&#94;&#123;&#55;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169148117671\">\n<p id=\"fs-id1169148117673\"><span class=\"token\">\u24d0<\/span><em data-effect=\"italics\">a<\/em><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e70a05d31dfb872dcdbe94cef386154_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#98;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"14\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169141376123\">\n<div data-type=\"problem\" id=\"fs-id1169141376125\">\n<p id=\"fs-id1169141376127\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d312ac72ba4b9d41f610dc71df1ae9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#97;&#125;&#94;&#123;&#49;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d310f4687b3fd4a527eeb5706f2ba453_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#119;&#125;&#94;&#123;&#50;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"42\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169141373734\">\n<div data-type=\"problem\" id=\"fs-id1169141373736\">\n<p id=\"fs-id1169141373738\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6410e98d6f34335337c5a31ea15a9dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#123;&#109;&#125;&#94;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0efc76468887d4bcb64e8b4268b8ce00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169143579829\">\n<p id=\"fs-id1169143579832\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35ec9da8c64343dba23cb9268763540d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-daac69682152d92aef78b199ba1e74fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143579862\">\n<div data-type=\"problem\" id=\"fs-id1169143579864\">\n<p id=\"fs-id1169143579866\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a80a727f06eb332782450faf30c5b2b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#49;&#123;&#109;&#125;&#94;&#123;&#50;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"71\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9618a448ec866b87ac263728c82d66d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147797054\">\n<div data-type=\"problem\" id=\"fs-id1169147797057\">\n<p id=\"fs-id1169147797059\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c71eef0773dcb41d60f0b26498d2a9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#49;&#54;&#123;&#97;&#125;&#94;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b232c746ac025fe830c56c917f28bc9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#51;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -1px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169145573794\">\n<p id=\"fs-id1169145573796\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c59d39103847ea9df0ac8f7e131ea5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"25\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0aade800cb3e98345bd41f2e2fc7a14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#98;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"24\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145748005\">\n<div data-type=\"problem\" id=\"fs-id1169145748007\">\n<p id=\"fs-id1169145748010\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eea3fbf22b60d96e6f6af9ce17a2d9a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"79\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e9f4429e1454389474644400e5ac957_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#57;&#123;&#119;&#125;&#94;&#123;&#56;&#125;&#123;&#121;&#125;&#94;&#123;&#49;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d29a4d88c4927116f1bfcd02877bae68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#56;&#123;&#97;&#125;&#94;&#123;&#53;&#49;&#125;&#123;&#98;&#125;&#94;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169143385610\">\n<h4 data-type=\"title\"><a href=\"\/contents\/dbb319b9-f421-4570-9953-ca5a39b933dc\" class=\"target-chapter\">Simplify Radical Expressions<\/a><\/h4>\n<p id=\"fs-id1169143385621\"><strong data-effect=\"bold\">Use the Product Property to Simplify Radical Expressions<\/strong><\/p>\n<p id=\"fs-id1169145977397\">In the following exercises, use the Product Property to simplify radical expressions.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145977400\">\n<div data-type=\"problem\" id=\"fs-id1169145977402\">\n<p id=\"fs-id1169145977404\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-904be03cc4596bd2b7eb61de366bd7af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145977414\">\n<p id=\"fs-id1169145977416\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d7a9343e2a2f74aa72f3278cbb70ff7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"33\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145977428\">\n<div data-type=\"problem\" id=\"fs-id1169145977430\">\n<p id=\"fs-id1169145977432\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc126a0a9ff55fd169cf1d890f7b822a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#55;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145977455\">\n<div data-type=\"problem\" id=\"fs-id1169145977458\">\n<p id=\"fs-id1169145977460\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f63e91fd18422eb46daf78068ab935e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#54;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a80473e9caa08e9c48e760bef947b962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#54;&#93;&#123;&#49;&#50;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147784223\">\n<p id=\"fs-id1169147784225\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1de1f04abc3b5355e9c0628c8e6b5b1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-092a5fa785a795bd7c695096d7615f91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#115;&#113;&#114;&#116;&#91;&#54;&#93;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147784260\">In the following exercises, simplify using absolute value signs as needed.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147784263\">\n<div data-type=\"problem\" id=\"fs-id1169147784265\">\n<p id=\"fs-id1169147784267\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-463fe7dd34b14dfe1830d76194730b3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b036aaf99bbdf8c76b84c76ac1172a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#123;&#98;&#125;&#94;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2baad2d6a77ae7e3be13c55e024c8f86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#56;&#93;&#123;&#123;&#99;&#125;&#94;&#123;&#49;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145977715\">\n<div data-type=\"problem\" id=\"fs-id1169145977717\">\n<p id=\"fs-id1169145977719\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3567bb32f28de0ac97d87c51925883f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#48;&#123;&#115;&#125;&#94;&#123;&#49;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29419fb0727128ff23bd4fe575ff6743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#57;&#54;&#123;&#97;&#125;&#94;&#123;&#55;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53bb3eecfabb2f90e8e232567de002bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#54;&#93;&#123;&#49;&#50;&#56;&#123;&#98;&#125;&#94;&#123;&#55;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -1px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169145579099\">\n<p id=\"fs-id1169145579101\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bff287474f1f3425f29e6d38f834bea2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#124;&#123;&#115;&#125;&#94;&#123;&#55;&#125;&#124;&#92;&#115;&#113;&#114;&#116;&#91;&#93;&#123;&#53;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"63\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e21edbabf49c9e5d88cb148e551f0301_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#97;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#51;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f014326841521e4aae9d1534ed0d2c7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#124;&#98;&#124;&#92;&#115;&#113;&#114;&#116;&#91;&#54;&#93;&#123;&#50;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147987538\">\n<div data-type=\"problem\" id=\"fs-id1169147987540\">\n<p id=\"fs-id1169147987542\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37398429e9a771e69324cd1113e90e20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#54;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#123;&#115;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb6dbb14bc87e608d32bd1c806c45f8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#56;&#48;&#123;&#120;&#125;&#94;&#123;&#55;&#125;&#123;&#121;&#125;&#94;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d54cbadf8081dc765eb7ea8372d93269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#56;&#48;&#123;&#120;&#125;&#94;&#123;&#56;&#125;&#123;&#121;&#125;&#94;&#123;&#57;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148063219\">\n<div data-type=\"problem\" id=\"fs-id1169148063221\">\n<p id=\"fs-id1169147725423\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f021507ab9fd854a0b93956ab083ddd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#45;&#51;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7933c09aa5d708c5b4bfaf215a3cfb4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#56;&#93;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169147725454\">\n<p id=\"fs-id1169147725457\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span> not real<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147725477\">\n<div data-type=\"problem\" id=\"fs-id1169147725479\">\n<p id=\"fs-id1169147725481\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90a5d4001c30a750c46d98f55a397fa2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4be4427171ba9873d48d96833b40fd62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#48;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169148053130\"><strong data-effect=\"bold\">Use the Quotient Property to Simplify Radical Expressions<\/strong><\/p>\n<p id=\"fs-id1169148053136\">In the following exercises, use the Quotient Property to simplify square roots.<\/p>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1169148053142\">\n<p id=\"fs-id1169148053144\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-580dd1d4ee4ef6f6f33b5dae8b547d56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#50;&#125;&#123;&#57;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"35\" style=\"vertical-align: -11px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5c424b80a4450092c7e3fbf1a69664f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#52;&#125;&#123;&#56;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"35\" style=\"vertical-align: -11px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-92742eaa5fbd10e35e69f56c73696eb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#57;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"35\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148053203\">\n<p id=\"fs-id1169148053205\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f02bad88e2a55c2a7c2d7b601e31c79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8fa03e1b526c6d07ec843385490ca4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143519248\">\n<div data-type=\"problem\" id=\"fs-id1169143519250\">\n<p id=\"fs-id1169143519253\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12fc15f2f6d9ee38d5c635f7ae4e7efa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#56;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"35\" style=\"vertical-align: -12px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f581d29a2e29a6fc8adeb42406daa64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#117;&#125;&#94;&#123;&#50;&#49;&#125;&#125;&#123;&#123;&#117;&#125;&#94;&#123;&#49;&#49;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"42\" style=\"vertical-align: -11px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af846ead7e50b1220a0e6c3ce5b30fd2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#54;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#51;&#48;&#125;&#125;&#123;&#123;&#118;&#125;&#94;&#123;&#49;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"41\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143634779\">\n<div data-type=\"problem\" id=\"fs-id1169143634782\">\n<p id=\"fs-id1169143634784\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f29cde980680097ebb985d916c86298_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#48;&#123;&#109;&#125;&#94;&#123;&#53;&#125;&#125;&#123;&#54;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"61\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143634804\">\n<p id=\"fs-id1169143634806\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-718a736d465fff22c0b55174271157e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#109;&#125;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"64\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148249775\">\n<div data-type=\"problem\" id=\"fs-id1169148249777\">\n<p id=\"fs-id1169148249779\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3fa03ec15d15b72c071ef89556419042_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#56;&#123;&#112;&#125;&#94;&#123;&#55;&#125;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"49\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eee8c97959240aac974a8b454b998ef4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#49;&#123;&#115;&#125;&#94;&#123;&#56;&#125;&#125;&#123;&#123;&#116;&#125;&#94;&#123;&#51;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"48\" style=\"vertical-align: -11px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20ddb17c3015834824b73baf2948d8e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#52;&#123;&#112;&#125;&#94;&#123;&#49;&#53;&#125;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#49;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"55\" style=\"vertical-align: -12px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143459265\">\n<div data-type=\"problem\" id=\"fs-id1169143459267\">\n<p id=\"fs-id1169143459269\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64a564699312ea877662833ceb3c8d32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#55;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#113;&#125;&#123;&#49;&#48;&#56;&#123;&#112;&#125;&#94;&#123;&#52;&#125;&#123;&#113;&#125;&#94;&#123;&#51;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"69\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d24945509d125fe885b3d6dc73fcecfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#123;&#99;&#125;&#94;&#123;&#53;&#125;&#123;&#100;&#125;&#94;&#123;&#55;&#125;&#125;&#123;&#50;&#53;&#48;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"69\" style=\"vertical-align: -11px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f596f60e13f40d5f45a2ebbcd37e651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#54;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#109;&#125;&#94;&#123;&#57;&#125;&#123;&#110;&#125;&#94;&#123;&#55;&#125;&#125;&#123;&#49;&#50;&#56;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"69\" style=\"vertical-align: -11px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169145491841\">\n<p id=\"fs-id1169145491843\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e740a0ef362a398894144d3093f8a6b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#124;&#112;&#113;&#124;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"29\" style=\"vertical-align: -9px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d02bac29875075bb67581c559629dae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#99;&#100;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#50;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"55\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fb0cfce1e1c05c639842aab07e5a958_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#124;&#109;&#110;&#124;&#92;&#115;&#113;&#114;&#116;&#91;&#54;&#93;&#123;&#50;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"49\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147834409\">\n<div data-type=\"problem\" id=\"fs-id1169147834411\">\n<p id=\"fs-id1169147834413\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3436da9f57391a0509a1a4bb7ff04c86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#48;&#123;&#113;&#125;&#94;&#123;&#53;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#113;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"43\" style=\"vertical-align: -11px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eab46db163d7b78fc49088f7a8d99571_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#45;&#54;&#50;&#53;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"46\" style=\"vertical-align: -11px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f34b8d562be1d997b006058ed91a11c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#56;&#48;&#123;&#109;&#125;&#94;&#123;&#55;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#53;&#109;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"47\" style=\"vertical-align: -11px;\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169142429740\">\n<h4 data-type=\"title\"><a href=\"\/contents\/eb676a52-0094-4ffb-95d3-c8eb0596e397\" class=\"target-chapter\">Simplify Rational Exponents<\/a><\/h4>\n<p id=\"fs-id1169142429749\"><strong data-effect=\"bold\">Simplify expressions with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-87703e837a52daa667553543cc1a4596_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"19\" style=\"vertical-align: 0px;\" \/><\/strong><\/p>\n<p id=\"fs-id1169142429768\">In the following exercises, write as a radical expression.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169142429771\">\n<div data-type=\"problem\" id=\"fs-id1169142429774\">\n<p id=\"fs-id1169142429776\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23cc4add58af74c5e73c5f6c4c409d2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"16\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c7aec388d283ea5bf41dbd4a5dc799f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#115;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"16\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f31f7ed5461c6c9f02817bf8d4fe36ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"14\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145969665\">\n<p id=\"fs-id1169145969667\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbf5d33ab6069db1568a57c158e7a880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9fce9e0fd6d8ed7f373876ae1b7f1c05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36a7c545faf443530eac9821de01aec9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"21\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145969708\">In the following exercises, write with a rational exponent.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145969712\">\n<div data-type=\"problem\" id=\"fs-id1169145969714\">\n<p id=\"fs-id1169145969716\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fd627103310ce70ee8baf78565201e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#49;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"41\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6eb8150927c3213ee14aa9495a43455_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#56;&#113;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"32\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b58bbc00cce24dd5c1337fd0add8d35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#115;&#113;&#114;&#116;&#91;&#54;&#93;&#123;&#51;&#54;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145744783\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145744786\">\n<div data-type=\"problem\" id=\"fs-id1169145744788\">\n<p id=\"fs-id1169145744790\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79b25d24cf0da05aaee281f88a023729_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#54;&#50;&#53;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17d313734d0fa254b42fb429c56f9c5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#52;&#51;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"35\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3fef56f01e713fa1d84d6453139dc051_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#50;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169145744850\">\n<p id=\"fs-id1169145744852\"><span class=\"token\">\u24d0<\/span> 5 <span class=\"token\">\u24d1<\/span> 3 <span class=\"token\">\u24d2<\/span> 2<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145924873\">\n<div data-type=\"problem\" id=\"fs-id1169145924876\">\n<p id=\"fs-id1169145924878\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-781138db18272ff19119a919d6ac381c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#48;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd9c428e585ce51d15bb66fb061b2e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#49;&#44;&#48;&#48;&#48;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e82c3c9ff5e03c48d911019404949ed0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"75\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143638122\">\n<div data-type=\"problem\" id=\"fs-id1169143638124\">\n<p id=\"fs-id1169143638126\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dccdd4ff08cc3f78a6fa730465f328a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdc005834c4eed65e666b7ef721d06a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#52;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"58\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4aefa85a89cdfc1dc4f5e553c6a57b25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#49;&#50;&#53;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -1px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169143638202\">\n<p id=\"fs-id1169143638204\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34f0e870957984f6c69249b8cf4f5813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148217684\"><strong data-effect=\"bold\">Simplify Expressions with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9c875f936d4feeb20b0213b0b02e7d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/strong><\/p>\n<p id=\"fs-id1169148217703\">In the following exercises, write with a rational exponent.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148217707\">\n<div data-type=\"problem\" id=\"fs-id1169148217709\">\n<p id=\"fs-id1169148217711\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3086b920c0449e86b551b1de5d34e0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#123;&#114;&#125;&#94;&#123;&#55;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4eeb387f49f25d2f87017ec929103b5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#50;&#112;&#113;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"63\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa499d2c8d57e777ac0f3199070372da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#109;&#125;&#123;&#55;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"71\" style=\"vertical-align: -10px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169145642785\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145642788\">\n<div data-type=\"problem\" id=\"fs-id1169145642791\">\n<p id=\"fs-id1169145642793\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-295101198ebffa95e44f28c401b5afcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#53;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfad4270396ad968bb7e9386b3012587_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#57;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"28\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b85df81ea2b8e135fe6d96a6e4af9c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169145642860\">\n<p id=\"fs-id1169145642862\"><span class=\"token\">\u24d0<\/span> 125 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3823dd2057ce967a11ddb9a2da8a389_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/> <span class=\"token\">\u24d2<\/span> 16<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143573570\">\n<div data-type=\"problem\" id=\"fs-id1169143573572\">\n<p id=\"fs-id1169143573575\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5a4e101a737e5b4b71ee03260f53078_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#54;&#52;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a797bfca7040f97bbd76392d03a735da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#54;&#52;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"36\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e34eee05b5837d06b7fa346358538e87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169143581323\"><strong data-effect=\"bold\">Use the Laws of Exponents to Simplify Expressions with Rational Exponents<\/strong><\/p>\n<p id=\"fs-id1169143581329\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169143581332\">\n<div data-type=\"problem\" id=\"fs-id1169143581334\">\n<p id=\"fs-id1169143581336\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5156c98468f9e020b8616e28add4c386_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#54;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#125;&middot;&#123;&#54;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"36\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6130fea2f514d268e4c427b4fe3c1ef9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#98;&#125;&#94;&#123;&#49;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"44\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab66f826ec56c202876d233c51da1d1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#119;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#125;&#125;&#125;&#123;&#123;&#119;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#55;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"21\" style=\"vertical-align: -13px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169148237312\">\n<p id=\"fs-id1169148237315\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1f24f4d8f12df52e638ec17c163d377_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#54;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7177812667edd7be2bf6fedb2497781a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c4c5a6780a9306cdf65041e59111cfd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#119;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"11\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148237359\">\n<div data-type=\"problem\" id=\"fs-id1169148237361\">\n<p id=\"fs-id1169148237363\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bd396724e69e6036323aabf5c209414_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#97;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#125;&middot;&#123;&#97;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#52;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"46\" style=\"vertical-align: -13px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b241b2b56ad98e1701e0d7781f8fb4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#123;&#98;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#123;&#99;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;&#125;&#123;&#99;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"91\" style=\"vertical-align: -17px;\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169148115149\">\n<h4 data-type=\"title\"><a href=\"\/contents\/a38378e8-cc28-44f4-8c29-789cc2550c6a\" class=\"target-chapter\">Add, Subtract and Multiply Radical Expressions<\/a><\/h4>\n<p id=\"fs-id1169147730989\"><strong data-effect=\"bold\">Add and Subtract Radical Expressions<\/strong><\/p>\n<p id=\"fs-id1169147730996\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147730999\">\n<div data-type=\"problem\" id=\"fs-id1169147731001\">\n<p id=\"fs-id1169147731003\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd5b12a90086074e3a0a0d17920c1989_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#45;&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"87\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d8e9b884d7cd7ffc0ce7f1b610fdcd03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#112;&#125;&#43;&#50;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45e7b5f3846249a967071f21b4deaa57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#125;&#45;&#51;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1169147731079\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d778da37035cf41dfa4fffed953663a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"33\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1dd8cc4098f36bf35a07bf71862dd3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af09c7c8d220ba8fe9e1ecbadd19920e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147977766\">\n<div data-type=\"problem\" id=\"fs-id1169147977768\">\n<p id=\"fs-id1169147977770\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff974b00f35022b40295d646bf0e9c1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#98;&#125;&#45;&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#98;&#125;&#43;&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"181\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0057a56746a7be0835ed8083acfce64f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#49;&#49;&#99;&#100;&#125;&#43;&#53;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#49;&#49;&#99;&#100;&#125;&#45;&#57;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#49;&#49;&#99;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"223\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143686190\">\n<div data-type=\"problem\" id=\"fs-id1169143686193\">\n<p id=\"fs-id1169143686195\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-921b8a7e3d261bfe60cec01f31ee018a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#56;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f16475e828aadb121490d088903863a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#52;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#49;&#50;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9cede9d40c3b7da4f3bcdb17ea7fe754_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#51;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"109\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169148075655\">\n<p id=\"fs-id1169148075657\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fcf55e07a24fef9a8092f27a130ded57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"33\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-875f9af679d71b7c1c7db80b65a084c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14fb98cd7a831d3cd87908bde262e3a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148075705\">\n<div data-type=\"problem\" id=\"fs-id1169148075708\">\n<p id=\"fs-id1169148075710\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea91fdc3bb794c01d0f588dea6475f37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#48;&#123;&#99;&#125;&#94;&#123;&#55;&#125;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#48;&#123;&#99;&#125;&#94;&#123;&#55;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e9bea0f051efaa7f3684050c891f56d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#49;&#54;&#50;&#123;&#114;&#125;&#94;&#123;&#49;&#48;&#125;&#125;&#43;&#52;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#51;&#50;&#123;&#114;&#125;&#94;&#123;&#49;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"162\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145520451\">\n<div data-type=\"problem\" id=\"fs-id1169145495463\">\n<p id=\"fs-id1169145495465\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7528cafa42c198863aa36175c94a1e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#56;&#121;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#56;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#48;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"217\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145495504\">\n<p id=\"fs-id1169145495506\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-007e6b6306e1607b0844925a986d74c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#55;&#121;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145495520\"><strong data-effect=\"bold\">Multiply Radical Expressions<\/strong><\/p>\n<p id=\"fs-id1169145495526\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145495530\">\n<div data-type=\"problem\" id=\"fs-id1169145495532\">\n<p id=\"fs-id1169145495534\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c99082e5b550fff9f0d632a731ef482_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"96\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b7c96bc3e1c1fecd8735315fb79a220_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#49;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"112\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146012528\">\n<div data-type=\"problem\" id=\"fs-id1169146012531\">\n<p id=\"fs-id1169146012533\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9766c9aab8d6488e69a849e19ea49dd2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"149\" style=\"vertical-align: -12px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14effafc696e343aa225c16522783ca7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#48;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#49;&#54;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"186\" style=\"vertical-align: -12px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169145945831\">\n<p id=\"fs-id1169145945833\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa07c9ad8d559feb2a0ae9b411d43edf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18c12ae654734eff760956f1684dc0f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#97;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145945878\"><strong data-effect=\"bold\">Use Polynomial Multiplication to Multiply Radical Expressions<\/strong><\/p>\n<p id=\"fs-id1169148125298\">In the following exercises, multiply.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148125302\">\n<div data-type=\"problem\" id=\"fs-id1169148125304\">\n<p id=\"fs-id1169148125306\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-127d9f28e415e3d2353b0a3f8f09df18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#43;&#52;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"120\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d853f7e800eecfd18e9304cbcebcde2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#57;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#49;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"120\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145969398\">\n<div data-type=\"problem\" id=\"fs-id1169145969400\">\n<p id=\"fs-id1169145969402\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7da767afc692c1a66afe91097d347107_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#52;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"157\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39f64c1677b313e2e10746fb0295c5e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#125;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#125;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"142\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169145969487\">\n<p id=\"fs-id1169145969489\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14c9a54dfd41966ccb1f9b914c85d1bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#49;&#45;&#50;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46771608807747d0f67634a33be99850_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#56;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#125;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145970420\">\n<div data-type=\"problem\" id=\"fs-id1169145970422\">\n<p id=\"fs-id1169145970424\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-760ce42f92e35e8c7730145d4561ec02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#45;&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#43;&#57;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"222\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145970489\">\n<div data-type=\"problem\" id=\"fs-id1169145491418\">\n<p id=\"fs-id1169145491420\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d46144e429fb2c859ffd59dd02da72a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"84\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-387b0e1d3be64a04e0c4542a3c770169_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"84\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169145491478\">\n<p id=\"fs-id1169145491481\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-63e0758bfe897857a062f3185e8db45a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#43;&#56;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -2px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e894166e66bba705e1c2d8e533659057_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#57;&#45;&#49;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145491521\">\n<div data-type=\"problem\" id=\"fs-id1169145491523\">\n<p id=\"fs-id1169145491525\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2c2ca4918f6cff953943a0095700475_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"157\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148063307\">\n<div data-type=\"problem\" id=\"fs-id1169148063309\">\n<p id=\"fs-id1169148063311\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6171a26ccdec82c6d55dd617e82b97b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#120;&#125;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#120;&#125;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"162\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148063355\">\n<p id=\"fs-id1169148063357\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41153cd5245c4a2925515b3ba352c28a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"71\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169143662135\">\n<h4 data-type=\"title\"><a href=\"\/contents\/94bbb796-17c2-4a4d-a202-8ead79bc0700\" class=\"target-chapter\">Divide Radical Expressions<\/a><\/h4>\n<p id=\"fs-id1169143662146\"><strong data-effect=\"bold\">Divide Square Roots<\/strong><\/p>\n<p id=\"fs-id1169143662152\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169143662155\">\n<div data-type=\"problem\" id=\"fs-id1169143662157\">\n<p id=\"fs-id1169143662159\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9672995dc5110728d10d57819ad89c70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#56;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"26\" style=\"vertical-align: -11px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f670b381f3ad12e2c89b2a29c1918c54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#56;&#49;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"28\" style=\"vertical-align: -11px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143662241\">\n<div data-type=\"problem\" id=\"fs-id1169143662243\">\n<p id=\"fs-id1169143662245\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd8d711eee02076c7d4f54648df4b23d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#50;&#48;&#109;&#123;&#110;&#125;&#94;&#123;&#45;&#53;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#53;&#123;&#109;&#125;&#94;&#123;&#45;&#55;&#125;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"69\" style=\"vertical-align: -11px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95453749f0137bd50c1e9bc84d25925c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#45;&#50;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#45;&#53;&#52;&#123;&#120;&#125;&#94;&#123;&#45;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"80\" style=\"vertical-align: -15px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169145497025\">\n<p id=\"fs-id1169145497027\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd2871360210dca9db994fc1b8737b1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#125;&#123;&#51;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7136c542043456aa6285a9772096a20d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"36\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145720654\"><strong data-effect=\"bold\">Rationalize a One Term Denominator<\/strong><\/p>\n<p id=\"fs-id1169145720660\">In the following exercises, rationalize the denominator.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145720663\">\n<div data-type=\"problem\" id=\"fs-id1169145720665\">\n<p id=\"fs-id1169145720668\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3bd935815ffd5539191aa82760d4e44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"19\" style=\"vertical-align: -11px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c7f0f07bc5f7df245dbceb0d5d2245e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"35\" style=\"vertical-align: -11px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-420ce77ccd1776d22bd92cc4177e636d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#121;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"26\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143686314\">\n<div data-type=\"problem\" id=\"fs-id1169143686316\">\n<p id=\"fs-id1169143686318\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-377ab777358e83c4bf5d97181ec89c60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#49;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"28\" style=\"vertical-align: -11px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59e4a9d7b656e4210b41c19c7e3ceac0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#53;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"35\" style=\"vertical-align: -11px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95e024317f047ab113c2a3fe4cec6746_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"36\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145714378\">\n<p id=\"fs-id1169145714380\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f066425ae4026115dee47a285e3b677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#49;&#50;&#49;&#125;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"35\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-457b2db5e3f69946c83d66b60cfa7a04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#56;&#125;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"28\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e937cf561f42c86b811666970875645d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#57;&#120;&#125;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"29\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145714442\">\n<div data-type=\"problem\" id=\"fs-id1169145714444\">\n<p id=\"fs-id1169145714446\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58dfb28249adc9e6461e0136238dc320_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"21\" style=\"vertical-align: -11px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2fc16327f08582de57d494c162c5717_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#51;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"35\" style=\"vertical-align: -11px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a63af9a15706d7fc47cbe1beec43061_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#57;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"36\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145971565\"><strong data-effect=\"bold\">Rationalize a Two Term Denominator<\/strong><\/p>\n<p id=\"fs-id1169145971571\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145971574\">\n<div data-type=\"problem\" id=\"fs-id1169145971576\">\n<p id=\"fs-id1169145971578\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2b5c47d62893ef2bda65d2ad4fc6279_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"36\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145645732\">\n<p id=\"fs-id1169145645734\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9dff04ef3dc06bc5eb84245e35861af6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145645763\">\n<div data-type=\"problem\" id=\"fs-id1169145645765\">\n<p id=\"fs-id1169145645768\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9db1c8cfa72be7feb73620988d20fee4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"49\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145645824\">\n<div data-type=\"problem\" id=\"fs-id1169145645826\">\n<p id=\"fs-id1169145645828\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc57e7a88429a0c07b353435e5106995_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"49\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169141299944\">\n<p id=\"fs-id1169141299946\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2d18ffc30cae2d43f66088c5c6d79974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#43;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#120;&#45;&#56;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"78\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169141299984\">\n<h4 data-type=\"title\"><a href=\"\/contents\/d5923faf-e056-4b0a-b96d-ee6429d48e36\" class=\"target-chapter\">Solve Radical Equations<\/a><\/h4>\n<p id=\"fs-id1169141299994\"><strong data-effect=\"bold\">Solve Radical Equations<\/strong><\/p>\n<p id=\"fs-id1169141300000\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169141300003\">\n<div data-type=\"problem\" id=\"fs-id1169141300005\">\n<p id=\"fs-id1169141300007\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ca93ac22463e77328bf6e8b861313a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#120;&#45;&#51;&#125;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143660081\">\n<div data-type=\"problem\" id=\"fs-id1169143660084\">\n<p id=\"fs-id1169143660086\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2cc9884e8397d0781980cbb20d4be7a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#120;&#43;&#49;&#125;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143660106\">\n<p id=\"fs-id1169143660108\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143660113\">\n<div data-type=\"problem\" id=\"fs-id1169143660116\">\n<p id=\"fs-id1169143660118\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96137971f03e06cb5ed1bab6338a1459_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#120;&#45;&#49;&#125;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143660156\">\n<div data-type=\"problem\" id=\"fs-id1169143660158\">\n<p id=\"fs-id1169143660160\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14c814b10f66bbe1689df9c7ec911266_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#117;&#45;&#51;&#125;&#43;&#51;&#61;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143660183\">\n<p id=\"fs-id1169142416331\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb3a77ba7b7ae0b5f67c64d485797202_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#51;&#44;&#117;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"93\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169142416352\">\n<div data-type=\"problem\" id=\"fs-id1169142416355\">\n<p id=\"fs-id1169142416357\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ed629ede1bf6b732f50db7282546d01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#120;&#43;&#53;&#125;&#45;&#50;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169142416399\">\n<div data-type=\"problem\" id=\"fs-id1169142416402\">\n<p id=\"fs-id1169142416404\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-022a04f37db41a0b5bac7fa5b7db8bb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#43;&#50;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169142416441\">\n<p id=\"fs-id1169142416443\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3dc975a98ccada6f136856736d7df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143604687\">\n<div data-type=\"problem\" id=\"fs-id1169143604689\">\n<p id=\"fs-id1169143604691\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb9e387c31bb9e8105e76cf6a91b3575_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#121;&#43;&#52;&#125;&#45;&#121;&#43;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"148\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143604733\">\n<div data-type=\"problem\" id=\"fs-id1169143604735\">\n<p id=\"fs-id1169143604737\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43ad5eb9465965536858435d01d4bc31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#114;&#43;&#49;&#125;&#45;&#56;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143604764\">\n<p id=\"fs-id1169143604766\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f57695ad3ee3297417eba5e756bb36e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169143604778\"><strong data-effect=\"bold\">Solve Radical Equations with Two Radicals<\/strong><\/p>\n<p id=\"fs-id1169143604784\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169143604787\">\n<div data-type=\"problem\" id=\"fs-id1169143604789\">\n<p id=\"fs-id1169143539917\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b58b69734d7704894d2feed331426a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#43;&#50;&#99;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#99;&#43;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"164\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143539960\">\n<div data-type=\"problem\" id=\"fs-id1169143539963\">\n<p id=\"fs-id1169143539965\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b006d5ceb98d3a9615e06f1a4a26fcfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#45;&#49;&#56;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"250\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143540015\">\n<p id=\"fs-id1169143540017\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-673b47a45398049b77add48d113519dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#56;&#44;&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145623314\">\n<div data-type=\"problem\" id=\"fs-id1169145623316\">\n<p id=\"fs-id1169145623318\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1a6447e80d10b4d9332c4af8a5e3ac2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#125;&#43;&#54;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#43;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145623349\">\n<div data-type=\"problem\" id=\"fs-id1169145623351\">\n<p id=\"fs-id1169145623353\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8c4a5c979b4dfbe19844122eecbc044_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#43;&#49;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#45;&#50;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"163\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145623382\">\n<p id=\"fs-id1169145623384\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169145623397\"><strong data-effect=\"bold\">Use Radicals in Applications<\/strong><\/p>\n<p id=\"fs-id1169145623403\">In the following exercises, solve. Round approximations to one decimal place.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145623406\">\n<div data-type=\"problem\" id=\"fs-id1169145623409\">\n<p id=\"fs-id1169145623411\"><strong data-effect=\"bold\">Landscaping<\/strong> Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-710aedf0b9cfbdd26e3c31b132a4e7a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -2px;\" \/> to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145577657\">\n<div data-type=\"problem\" id=\"fs-id1169145577659\">\n<p id=\"fs-id1169145577661\"><strong data-effect=\"bold\">Accident investigation<\/strong> An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a9a9c32372c90dc6afdb389799a1856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/> to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145577686\">\n<p id=\"fs-id1169145577688\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f5f9c7cd39f2115df10d23de0f849be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#46;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/> feet<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169145577699\">\n<h4 data-type=\"title\"><a href=\"\/contents\/d5887b1b-5ca2-40b7-99c3-43f38b1a5425\" class=\"target-chapter\">Use Radicals in Functions<\/a><\/h4>\n<p id=\"fs-id1169145577709\"><strong data-effect=\"bold\">Evaluate a Radical Function<\/strong><\/p>\n<p id=\"fs-id1169145577715\">In the following exercises, evaluate each function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145577718\">\n<div data-type=\"problem\" id=\"fs-id1169145577721\">\n<p id=\"fs-id1169145577723\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9a317b4d77475614a3dfa9395e555ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#120;&#43;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49fcaf9fe488cacd08d464d30c0c480c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6cfbd3ae14c544c46b05ffbdd1f16f77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147816061\">\n<div data-type=\"problem\" id=\"fs-id1169147816063\">\n<p id=\"fs-id1169147816065\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ccee924d000f67b01b5ed82ddbb1b00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#120;&#45;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f044ff5a23d3030f01e94c59bda68dda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89130f10e39971c048f08cf68fddf955_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169145535783\">\n<p id=\"fs-id1169145535785\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12a31481e68c5282ddc4e072954e545c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"96\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d122fbb11c08f0ba1417b5abc5e90fcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"72\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145535842\">\n<div data-type=\"problem\" id=\"fs-id1169141037377\">\n<p id=\"fs-id1169141037379\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2bccd351e306a78c1a1f779fc07ee988_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"129\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-290021335f8d9db8fe0ca90975fe52e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4901b87d7d3f8b91b3e665f92895170c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143442804\">\n<div data-type=\"problem\" id=\"fs-id1169143442806\">\n<p id=\"fs-id1169143442808\">For the function<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46437a40ee52e2a21f4880e7ec488a03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#52;&#45;&#52;&#120;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/> find<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ef2b6c65c15b3dab9cdab30a64cd96f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4ec920e80233e6f592d8618e7d699e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1169143442881\">\n<p id=\"fs-id1169143442883\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1de932db937dbdef508a418ecd01d313_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d27b627a63246d461d0580b9c5920af2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169143580727\"><strong data-effect=\"bold\">Find the Domain of a Radical Function<\/strong><\/p>\n<p id=\"fs-id1169143580734\">In the following exercises, find the domain of the function and write the domain in interval notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169143580738\">\n<div data-type=\"problem\" id=\"fs-id1169143580740\">\n<p id=\"fs-id1169143580742\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4707f8dfae246d679f39ac16247da6a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#45;&#51;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143580796\">\n<div data-type=\"problem\" id=\"fs-id1169143580798\">\n<p id=\"fs-id1169143580801\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-813bd0c2074dc829ba6731acf359a4f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#43;&#51;&#125;&#123;&#120;&#45;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"112\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169141522401\">\n<p id=\"fs-id1169141522403\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3103559f9a65200f03c9dd13b132df4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169141522422\">\n<div data-type=\"problem\" id=\"fs-id1169141522424\">\n<p id=\"fs-id1169141522426\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d828dbe77584e8c29ca29896a20010d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169141522484\">\n<div data-type=\"problem\" id=\"fs-id1169142133190\">\n<p id=\"fs-id1169142133192\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd0b0589aa2182714489a2e2a9bf4c3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#49;&#48;&#45;&#55;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169142133223\">\n<p id=\"fs-id1169142133225\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4cd17611a81ea49a13c7af790816b810_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#48;&#125;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169142133249\"><strong data-effect=\"bold\">Graph Radical Functions<\/strong><\/p>\n<p id=\"fs-id1169142133255\">In the following exercises, <span class=\"token\">\u24d0<\/span> find the domain of the function <span class=\"token\">\u24d1<\/span> graph the function <span class=\"token\">\u24d2<\/span> use the graph to determine the range.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169142133269\">\n<div data-type=\"problem\" id=\"fs-id1169142133271\">\n<p id=\"fs-id1169142133273\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-093a372d7bccc123f572867cd3cbefc2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147765384\">\n<div data-type=\"problem\" id=\"fs-id1169147765386\">\n<p id=\"fs-id1169147765389\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7209f1e1666a26f16b85e4aa6aba34c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"94\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147765412\">\n<p id=\"fs-id1169147765414\"><span class=\"token\">\u24d0<\/span> domain: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2640870d7533edf8778f6e64ee0db9c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#48;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"44\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169143576361\" data-alt=\"The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 0) and goes through the points (1, 2) and (4, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_302_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 0) and goes through the points (1, 2) and (4, 4).\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> range: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2640870d7533edf8778f6e64ee0db9c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#48;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"44\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143576399\">\n<div data-type=\"problem\" id=\"fs-id1169143576401\">\n<p id=\"fs-id1169143576404\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad383e42eaafe233a64a67f906f809a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143295372\">\n<div data-type=\"problem\" id=\"fs-id1169143295374\">\n<p id=\"fs-id1169143295376\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-576abc85a87a50d8cea6c44df2b08954_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143295404\">\n<p id=\"fs-id1169143295406\"><span class=\"token\">\u24d0<\/span> domain: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90eaa2f0f55290692476556ec134a082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169143295437\" data-alt=\"The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a center point at (0, 3) and goes through the points (negative 1, 2) and (1, 4).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_304_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a center point at (0, 3) and goes through the points (negative 1, 2) and (1, 4).\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> range: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90eaa2f0f55290692476556ec134a082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169143658297\">\n<h4 data-type=\"title\"><a href=\"\/contents\/916a2094-3b51-4f1a-803d-95909a359123\" class=\"target-chapter\">Use the Complex Number System<\/a><\/h4>\n<p id=\"fs-id1169143658308\"><strong data-effect=\"bold\">Evaluate the Square Root of a Negative Number<\/strong><\/p>\n<p id=\"fs-id1169143658314\">In the following exercises, write each expression in terms of <em data-effect=\"italics\">i<\/em> and simplify if possible.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169143658322\">\n<div data-type=\"problem\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe16287834681030d759799a1ffc042e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b28d1b676642ac411f048683cc1c5e7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d86e626eeee76a6c5d831d159582421_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169148225776\"><strong data-effect=\"bold\">Add or Subtract Complex Numbers<\/strong><\/p>\n<p id=\"fs-id1168037345963\">In the following exercises, add or subtract.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148225784\">\n<div data-type=\"problem\" id=\"fs-id1169148225786\">\n<p id=\"fs-id1169148225788\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30c2ecfd1bd08351ca590f4c60471aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#53;&#48;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148225804\">\n<p id=\"fs-id1169148225806\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5988c1534ab4d9b2fb722aaa78a025a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148225820\">\n<div data-type=\"problem\" id=\"fs-id1169148225822\">\n<p id=\"fs-id1169148225824\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e434a9503821fecf5f9cc8cc124fe13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#43;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148225876\">\n<div data-type=\"problem\" id=\"fs-id1169143600008\">\n<p id=\"fs-id1169143600010\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-840361fcc61856ef570a95d920ee2aa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#43;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#52;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143600046\">\n<p id=\"fs-id1169143600048\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-160fe05bfaa449c7886131ff2018e029_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#43;&#53;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"46\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143600063\">\n<div data-type=\"problem\" id=\"fs-id1169143600065\">\n<p id=\"fs-id1169143600067\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8c88b19c6f0b1274d8a08819b35484b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#53;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"240\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169143600125\"><strong data-effect=\"bold\">Multiply Complex Numbers<\/strong><\/p>\n<p id=\"fs-id1169147910490\">In the following exercises, multiply.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147910493\">\n<div data-type=\"problem\" id=\"fs-id1169147910495\">\n<p id=\"fs-id1169147910497\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b28c69bfa4805ff2555ec4172bb252e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#53;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#43;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147910532\">\n<p id=\"fs-id1169147910535\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ea171506895cc121c04898627f09275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#51;&#43;&#49;&#52;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"63\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147910550\">\n<div data-type=\"problem\" id=\"fs-id1169147910552\">\n<p id=\"fs-id1169147910554\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67ad08458c6352b9815659fa00d83434_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#105;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147910595\">\n<div data-type=\"problem\" id=\"fs-id1169147910598\">\n<p id=\"fs-id1169147910600\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d21706f4ebb63907a6d952830134b74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#125;&middot;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147948334\">\n<p id=\"fs-id1169147948336\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147948344\">\n<div data-type=\"problem\" id=\"fs-id1169147948346\">\n<p id=\"fs-id1169147948348\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-610098382b7b5b2c61f0889a85dcf6e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#55;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"199\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147948404\">In the following exercises, multiply using the Product of Binomial Squares Pattern.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147948407\">\n<div data-type=\"problem\" id=\"fs-id1169147948409\">\n<p id=\"fs-id1169147948411\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-267e7ca6efc89bf02dd12fb4d28dbed7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147948436\">\n<p id=\"fs-id1169147948438\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1de42ccd212d3943b6cf4f04cc18a5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#45;&#49;&#50;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148206456\">In the following exercises, multiply using the Product of Complex Conjugates Pattern.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148206459\">\n<div data-type=\"problem\" id=\"fs-id1169148206461\">\n<p id=\"fs-id1169148206463\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ddaa034384ff01b55e55d8e57407159b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#45;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#43;&#50;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148206509\"><strong data-effect=\"bold\">Divide Complex Numbers<\/strong><\/p>\n<p id=\"fs-id1169148206515\">In the following exercises, divide.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148206519\">\n<div data-type=\"problem\" id=\"fs-id1169148206521\">\n<p id=\"fs-id1169148206523\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f248cb1e992ec3ad2ce6f111d1847d4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#43;&#105;&#125;&#123;&#51;&#45;&#52;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148206546\">\n<p id=\"fs-id1169148206548\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f75115c034d09c38c27df60fd631f706_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#50;&#53;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148071897\">\n<div data-type=\"problem\" id=\"fs-id1169148071899\">\n<p id=\"fs-id1169148071901\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65553d1da017b809a15f324b44c639e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#52;&#125;&#123;&#51;&#45;&#50;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148071949\"><strong data-effect=\"bold\">Simplify Powers of <em data-effect=\"italics\">i<\/em><\/strong><\/p>\n<p id=\"fs-id1169148071958\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148071962\">\n<div data-type=\"problem\" id=\"fs-id1169148071964\">\n<p id=\"fs-id1169148071966\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5d71db3a82c9d128eb88b7f5b89bac3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#52;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148071978\">\n<p id=\"fs-id1169148071980\">1<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148071985\">\n<div data-type=\"problem\" id=\"fs-id1169148071987\">\n<p id=\"fs-id1169148071989\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0796aef07306fa58f775640e9763cc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#50;&#53;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"practice-test\" data-depth=\"1\" id=\"fs-id1169145548148\">\n<h3 data-type=\"title\">Practice Test<\/h3>\n<p id=\"fs-id1169145548155\">In the following exercises, simplify using absolute values as necessary.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169145548158\">\n<div data-type=\"problem\" id=\"fs-id1169145548160\">\n<p id=\"fs-id1169145548162\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62916fc046615849829ce19701a4a0c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#49;&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#57;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145548180\">\n<p id=\"fs-id1169145548182\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ad8655a65bb986a62c563c119edd22c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145548196\">\n<div data-type=\"problem\" id=\"fs-id1169145548198\">\n<p id=\"fs-id1169145548200\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b18d66b235bb730c2c491916af346fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#57;&#123;&#120;&#125;&#94;&#123;&#56;&#125;&#123;&#121;&#125;&#94;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"79\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145548241\">\n<div data-type=\"problem\" id=\"fs-id1169145548244\">\n<p id=\"fs-id1169145548246\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-611687b0ab147ac1151ac33db161fda6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#55;&#50;&#123;&#120;&#125;&#94;&#123;&#56;&#125;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143600773\">\n<p id=\"fs-id1169143600775\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7537a03f0c45a26b8b5f1de71e4f56b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#125;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"93\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143600808\">\n<div data-type=\"problem\" id=\"fs-id1169143600810\">\n<p id=\"fs-id1169143600812\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ece2679f4f5e2c993449953dcc57ab98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#53;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#125;&#123;&#49;&#56;&#48;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"71\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169147948649\">In the following exercises, simplify. Assume all variables are positive.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169147948652\">\n<div data-type=\"problem\" id=\"fs-id1169147948655\">\n<p id=\"fs-id1169147948657\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b4eff45fe4d80210b4cd74090c3b61c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#49;&#54;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"45\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e97dc66dc6f8adafd1eca7f2f50cf94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#52;&#57;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147948701\">\n<p id=\"fs-id1169147948703\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83dc4a903812856a369855508d0b2637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9db6d9268510b1ef156b293d0a9d9947_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#52;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147948731\">\n<div data-type=\"problem\" id=\"fs-id1169147948733\">\n<p id=\"fs-id1169147948735\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d86e626eeee76a6c5d831d159582421_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147948761\">\n<div data-type=\"problem\" id=\"fs-id1169147948763\">\n<p id=\"fs-id1169147948765\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71370c930bd1d459cae3e4e26c540002_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&middot;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#125;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"47\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145734818\">\n<p id=\"fs-id1169145734820\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78271ce0fe2cad76988913b00e2edf1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145734836\">\n<div data-type=\"problem\" id=\"fs-id1169145734838\">\n<p id=\"fs-id1169145734841\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bd13bac4e7033a8a8779982137998f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#123;&#121;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#125;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#51;&#125;&#125;&#123;&#121;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"87\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147775930\">\n<div data-type=\"problem\" id=\"fs-id1169147775932\">\n<p id=\"fs-id1169147775934\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48a71369a98b87f618954a6701e88c73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#56;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#53;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147775961\">\n<p id=\"fs-id1169147775963\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e2ad073cdb1e791f0452dae9ab3442b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147775983\">\n<div data-type=\"problem\" id=\"fs-id1169147775985\">\n<p id=\"fs-id1169147775987\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b891713177fa4dbb9ada998d0448923_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#55;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#52;&#120;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"204\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145519492\">\n<div data-type=\"problem\" id=\"fs-id1169145519494\">\n<p id=\"fs-id1169145519496\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-700e35af234609b3dbb621204c88244e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#125;&middot;&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145519527\">\n<p id=\"fs-id1169145519529\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19ab4b46246f53245c5c1487d200b50b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145519546\">\n<div data-type=\"problem\" id=\"fs-id1169145519549\">\n<p id=\"fs-id1169145519551\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff8ec4273d165c88afbf2a72495ebcf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#49;&#54;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"120\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143519724\">\n<div data-type=\"problem\" id=\"fs-id1169143519726\">\n<p id=\"fs-id1169143519729\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d07b463e18cdca94448bed7a4fda309e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"157\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143519766\">\n<p id=\"fs-id1169143519768\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15bc51ea6f8e046d160b28aeffae31be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#45;&#55;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143519784\">\n<div data-type=\"problem\" id=\"fs-id1169143519786\">\n<p id=\"fs-id1169143519788\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93e1abfc972f97eb95eafe7c76e44bc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#49;&#50;&#56;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"35\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169143519826\">\n<div data-type=\"problem\" id=\"fs-id1169148122896\">\n<p id=\"fs-id1169148122898\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1e5ff3a04fbc250ce7457e904ef15ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#53;&#120;&#123;&#121;&#125;&#94;&#123;&#45;&#52;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#53;&#123;&#120;&#125;&#94;&#123;&#45;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"67\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148122935\">\n<p id=\"fs-id1169148122938\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6182a6d8b6607585ee307d6d84f4f3e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#125;&#123;&#51;&#123;&#121;&#125;&#94;&#123;&#55;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"22\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148122962\">\n<div data-type=\"problem\" id=\"fs-id1169148122964\">\n<p id=\"fs-id1169148122966\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24c7b94eee17b2f5945881e85db1a940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"21\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148123000\">\n<div data-type=\"problem\" id=\"fs-id1169148123003\">\n<p id=\"fs-id1169148123005\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7915fb5a509fc6fee838045c7d775e4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"36\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145760791\">\n<p id=\"fs-id1169145760793\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2095cccf4d6424f1e24c30fdb9b761c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"80\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145760816\">\n<div data-type=\"problem\" id=\"fs-id1169145760818\">\n<p id=\"fs-id1169145760820\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f99cc9d48304bebe4fdf977bdc4b1ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#52;&#125;&middot;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145760847\">\n<div data-type=\"problem\" id=\"fs-id1169145760849\">\n<p id=\"fs-id1169145760851\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9f56cd55796ef93e5f2a4a6a27ea894_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#105;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#45;&#51;&#105;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169145760876\">\n<p id=\"fs-id1169145760878\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-63cccec136a84c1e4c8e9c4734bf28ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#43;&#56;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"67\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169145760893\">\n<div data-type=\"problem\" id=\"fs-id1169145760895\">\n<p id=\"fs-id1169145760897\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02a2bacfb7fb3ac8ce5780a2c84907d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#43;&#105;&#125;&#123;&#51;&#45;&#50;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148107267\">\n<div data-type=\"problem\" id=\"fs-id1169148107269\">\n<p id=\"fs-id1169148107271\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6198aa364ad6178c5073133349d83d52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#105;&#125;&#94;&#123;&#49;&#55;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148107283\">\n<p id=\"fs-id1169148107285\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbbc1af417d9937847c0c17221cb8820_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148107296\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148107299\">\n<div data-type=\"problem\" id=\"fs-id1169148107301\">\n<p id=\"fs-id1169148107303\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-452eb231694daf75210a1890951a2363_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#43;&#53;&#125;&#43;&#56;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148107335\">\n<div data-type=\"problem\" id=\"fs-id1169148107337\">\n<p id=\"fs-id1169148107339\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d2e7becaf247ff86deeeb3b0ce8a7a21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#43;&#53;&#125;&#43;&#49;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148105697\">\n<p id=\"fs-id1169148105699\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2145acc2878ed61214887e120f2485b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148105712\">\n<div data-type=\"problem\" id=\"fs-id1169148105714\">\n<p id=\"fs-id1169148105716\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34302f27421e2520b22aca69d48fa7b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#50;&#51;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"250\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148105789\">In the following exercise, <span class=\"token\">\u24d0<\/span> find the domain of the function <span class=\"token\">\u24d1<\/span> graph the function <span class=\"token\">\u24d2<\/span> use the graph to determine the range.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148105802\">\n<div data-type=\"problem\" id=\"fs-id1169143517571\">\n<p id=\"fs-id1169143517574\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fcb4ec6f1873e3117735aceaad94e92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169143517599\">\n<p id=\"fs-id1169143517601\"><span class=\"token\">\u24d0<\/span> domain: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58cba6b887275185a1fe142174f2bd42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#45;&#50;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"57\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1169143517631\" data-alt=\"The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 6. The y-axis runs from 0 to 8. The function has a starting point at (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_08_305_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 6. The y-axis runs from 0 to 8. The function has a starting point at (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> range: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2640870d7533edf8778f6e64ee0db9c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#48;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"44\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1169143517674\">\n<dt>complex conjugate pair<\/dt>\n<dd id=\"fs-id1169143517677\">A complex conjugate pair is of the form <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bi<\/em>, <em data-effect=\"italics\">a<\/em> \u2013 <em data-effect=\"italics\">bi<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1169148236939\">\n<dt>complex number<\/dt>\n<dd id=\"fs-id1169148236942\">A complex number is of the form <em data-effect=\"italics\">a<\/em> + <em data-effect=\"italics\">bi<\/em>, where <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers. We call <em data-effect=\"italics\">a<\/em> the real part and <em data-effect=\"italics\">b<\/em> the imaginary part.<\/dd>\n<\/dl>\n<dl id=\"fs-id1169148236978\">\n<dt>complex number system<\/dt>\n<dd id=\"fs-id1169148236981\">The complex number system is made up of both the real numbers and the imaginary numbers.<\/dd>\n<\/dl>\n<dl id=\"fs-id1169148236986\">\n<dt>imaginary unit<\/dt>\n<dd id=\"fs-id1169148236989\">The imaginary unit <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-695d9d59bd04859c6c99e7feb11daab6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> is the number whose square is \u20131. <em data-effect=\"italics\">i<\/em><sup>2<\/sup> = \u20131 or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea25b85ef5770d31220170be5af546f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"71\" style=\"vertical-align: -3px;\" \/><\/dd>\n<\/dl>\n<dl id=\"fs-id1169148237018\">\n<dt>standard form<\/dt>\n<dd id=\"fs-id1169148237021\">A complex number is in standard form when written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e80cd255be59b21381715aaa50d79181_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#43;&#98;&#105;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\" \/> where <em data-effect=\"italics\">a, b<\/em> are real numbers.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":9,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3676","chapter","type-chapter","status-publish","hentry"],"part":3472,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3676","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3676\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/3472"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3676\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3676"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3676"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3676"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3676"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}