{"id":3583,"date":"2018-12-11T13:57:40","date_gmt":"2018-12-11T18:57:40","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/divide-radical-expressions\/"},"modified":"2018-12-11T13:57:40","modified_gmt":"2018-12-11T18:57:40","slug":"divide-radical-expressions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/divide-radical-expressions\/","title":{"raw":"Divide Radical Expressions","rendered":"Divide Radical Expressions"},"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>Divide radical expressions<\/li><li>Rationalize a one term denominator<\/li><li>Rationalize a two term denominator<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1169146738116\" class=\"be-prepared\"><p id=\"fs-id1169134309346\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1169147085484\" type=\"1\"><li>Simplify: \\(\\frac{30}{48}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836620030\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: \\({x}^{2}\u00b7{x}^{4}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/3fa6a6c5-9a36-4dee-aea1-0166229f52fb#fs-id1167835512989\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Multiply: \\(\\left(7+3x\\right)\\left(7-3x\\right).\\)<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-id1169144516870\"><h3 data-type=\"title\">Divide Radical Expressions<\/h3><p id=\"fs-id1169148869846\">We have used the <span data-type=\"term\" class=\"no-emphasis\">Quotient Property of Radical Expressions<\/span> to simplify roots of fractions. We will need to use this property \u2018in reverse\u2019 to simplify a fraction with radicals.<\/p><p id=\"fs-id1169146630664\">We give the Quotient Property of Radical Expressions again for easy reference. Remember, we assume all variables are greater than or equal to zero so that no absolute value bars re needed.<\/p><div data-type=\"note\" id=\"fs-id1169148933282\"><div data-type=\"title\">Quotient Property of Radical Expressions<\/div><p id=\"fs-id1169146638883\">If \\(\\sqrt[n]{a}\\) and \\(\\sqrt[n]{b}\\) are real numbers, \\(b\\ne 0,\\) and for any integer \\(n\\ge 2\\) then,<\/p><div data-type=\"equation\" id=\"fs-id1169148993746\" class=\"unnumbered\" data-label=\"\">\\(\\sqrt[n]{\\frac{a}{b}}=\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}}\\phantom{\\rule{1em}{0ex}}\\text{and}\\phantom{\\rule{1em}{0ex}}\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}}=\\sqrt[n]{\\frac{a}{b}}\\)<\/div><\/div><p id=\"fs-id1169148859881\">We will use the Quotient Property of Radical Expressions when the fraction we start with is the quotient of two radicals, and neither radicand is a perfect power of the index. When we write the fraction in a single radical, we may find common factors in the numerator and denominator.<\/p><div data-type=\"example\" id=\"fs-id1169147136256\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148992709\"><div data-type=\"problem\" id=\"fs-id1169146669434\"><p id=\"fs-id1169148837638\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{\\sqrt{72{x}^{3}}}{\\sqrt{162x}}\\) <span class=\"token\">\u24d1<\/span> \\(\\frac{\\sqrt[3]{32{x}^{2}}}{\\sqrt[3]{4{x}^{5}}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149094103\"><p id=\"fs-id1169149370840\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\sqrt{72{x}^{3}}}{\\sqrt{162x}}\\hfill \\\\ \\begin{array}{c}\\text{Rewrite using the quotient property,}\\hfill \\\\ \\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}}=\\sqrt[n]{\\frac{a}{b}}.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{\\frac{72{x}^{3}}{162x}}\\hfill \\\\ \\text{Remove common factors.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{\\frac{\\overline{)18}\u00b74\u00b7{x}^{2}\u00b7\\overline{)x}}{\\overline{)18}\u00b79\u00b7\\overline{)x}}}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{\\frac{4{x}^{2}}{9}}\\hfill \\\\ \\text{Simplify the radical.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{2x}{3}\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\sqrt[3]{32{x}^{2}}}{\\sqrt[3]{4{x}^{5}}}\\hfill \\\\ \\begin{array}{c}\\text{Rewrite using the quotient property,}\\hfill \\\\ \\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}}=\\sqrt[n]{\\frac{a}{b}}.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt[3]{\\frac{32{x}^{2}}{4{x}^{5}}}\\hfill \\\\ \\text{Simplify the fraction under the radical.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt[3]{\\frac{8}{{x}^{3}}}\\hfill \\\\ \\text{Simplify the radical.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{2}{x}\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169146668779\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169146978086\"><div data-type=\"problem\" id=\"fs-id1169146743962\"><p id=\"fs-id1169149024013\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{\\sqrt{50{s}^{3}}}{\\sqrt{128s}}\\) <span class=\"token\">\u24d1<\/span> \\(\\frac{\\sqrt[3]{56{a}^{}}}{\\sqrt[3]{7{a}^{4}}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146626214\"><p id=\"fs-id1169140091518\"><span class=\"token\">\u24d0<\/span>\\(\\frac{5s}{8}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{2}{a}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149003808\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169149222208\"><div data-type=\"problem\" id=\"fs-id1169149350056\"><p id=\"fs-id1169148911048\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{\\sqrt{75{q}^{5}}}{\\sqrt{108q}}\\) <span class=\"token\">\u24d1<\/span> \\(\\frac{\\sqrt[3]{72{b}^{2}}}{\\sqrt[3]{9{b}^{5}}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148997993\"><p id=\"fs-id1169147136095\"><span class=\"token\">\u24d0<\/span>\\(\\frac{5{q}^{2}}{6}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{2}{b}\\)<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1169149373645\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169149012671\"><div data-type=\"problem\" id=\"fs-id1169148889871\"><p id=\"fs-id1169147027763\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{\\sqrt{147a{b}^{8}}}{\\sqrt{3{a}^{3}{b}^{4}}}\\) <span class=\"token\">\u24d1<\/span> \\(\\frac{\\sqrt[3]{-250{m}^{}{n}^{-2}}}{\\sqrt[3]{2{m}^{-2}{n}^{4}}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148990099\"><p id=\"fs-id1169144373998\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\sqrt{147a{b}^{8}}}{\\sqrt{3{a}^{3}{b}^{4}}}\\hfill \\\\ \\text{Rewrite using the quotient property.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{\\frac{147a{b}^{8}}{3{a}^{3}{b}^{4}}}\\hfill \\\\ \\text{Remove common factors in the fraction.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{\\frac{49{b}^{4}}{{a}^{2}}}\\hfill \\\\ \\text{Simplify the radical.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{7{b}^{2}}{a}\\hfill \\end{array}\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\sqrt[3]{-250{m}^{}{n}^{-2}}}{\\sqrt[3]{2{m}^{-2}{n}^{4}}}\\hfill \\\\ \\text{Rewrite using the quotient property.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt[3]{\\frac{-250{m}^{}{n}^{-2}}{2{m}^{-2}{n}^{4}}}\\hfill \\\\ \\text{Simplify the fraction under the radical.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt[3]{\\frac{-125{m}^{3}}{{n}^{6}}}\\hfill \\\\ \\text{Simplify the radical.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}-\\frac{5m}{{n}^{2}}\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149329581\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169146607917\"><div data-type=\"problem\" id=\"fs-id1169149367228\"><p id=\"fs-id1169146646547\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{\\sqrt{162{x}^{10}{y}^{2}}}{\\sqrt{2{x}^{6}{y}^{6}}}\\) <span class=\"token\">\u24d1<\/span> \\(\\frac{\\sqrt[3]{-128{x}^{2}{y}^{-1}}}{\\sqrt[3]{2{x}^{-1}{y}^{2}}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148863029\"><p id=\"fs-id1169148884863\"><span class=\"token\">\u24d0<\/span>\\(\\frac{9{x}^{2}}{{y}^{2}}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{-4x}{y}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149105200\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147115617\"><div data-type=\"problem\" id=\"fs-id1169146815722\"><p id=\"fs-id1169144553970\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{\\sqrt{300{m}^{3}{n}^{7}}}{\\sqrt{3{m}^{5}n}}\\) <span class=\"token\">\u24d1<\/span> \\(\\frac{\\sqrt[3]{-81p{q}^{-1}}}{\\sqrt[3]{3{p}^{-2}{q}^{5}}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149016702\"><p id=\"fs-id1169149349179\"><span class=\"token\">\u24d0<\/span>\\(\\frac{10{n}^{3}}{m}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{-3p}{{q}^{2}}\\)<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1169149316122\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169149330242\"><div data-type=\"problem\" id=\"fs-id1169149344431\"><p id=\"fs-id1169144728872\">Simplify: \\(\\frac{\\sqrt{54{x}^{5}{y}^{3}}}{\\sqrt{3{x}^{2}y}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146620795\"><p id=\"fs-id1169146648111\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\sqrt{54{x}^{5}{y}^{3}}}{\\sqrt{3{x}^{2}y}}\\hfill \\\\ \\text{Rewrite using the quotient property.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{\\frac{54{x}^{5}{y}^{3}}{3{x}^{2}y}}\\hfill \\\\ \\text{Remove common factors in the fraction.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{18{x}^{3}{y}^{2}}\\hfill \\\\ \\begin{array}{c}\\text{Rewrite the radicand as a product}\\hfill \\\\ \\text{using the largest perfect square factor.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{9{x}^{2}{y}^{2}\\cdot 2x}\\hfill \\\\ \\begin{array}{c}\\text{Rewrite the radical as the product of two}\\hfill \\\\ \\text{radicals.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\sqrt{9{x}^{2}{y}^{2}}\\cdot \\sqrt{2x}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}3xy\\sqrt{2x}\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148996331\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169146645855\"><div data-type=\"problem\" id=\"fs-id1169146654752\"><p id=\"fs-id1169149033237\">Simplify: \\(\\frac{\\sqrt{64{x}^{4}{y}^{5}}}{\\sqrt{2x{y}^{3}}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148963012\"><p id=\"fs-id1169149123517\">\\(4xy\\sqrt{2x}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169144523044\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148957840\"><div data-type=\"problem\" id=\"fs-id1169148859738\"><p id=\"fs-id1169144560010\">Simplify: \\(\\frac{\\sqrt{96{a}^{5}{b}^{4}}}{\\sqrt{2{a}^{3}b}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149347376\"><p id=\"fs-id1169148876020\">\\(4ab\\sqrt{3b}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149140677\"><h3 data-type=\"title\">Rationalize a One Term Denominator<\/h3><p id=\"fs-id1169148909612\">Before the calculator became a tool of everyday life, approximating the value of a fraction with a radical in the denominator was a very cumbersome process!<\/p><p id=\"fs-id1169149294686\">For this reason, a process called <span data-type=\"term\">rationalizing the denominator<\/span> was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator.<\/p><p id=\"fs-id1169146655337\">This process is still used today, and is useful in other areas of mathematics, too.<\/p><div data-type=\"note\" id=\"fs-id1169146620764\"><div data-type=\"title\">Rationalizing the Denominator<\/div><p><strong data-effect=\"bold\">Rationalizing the denominator<\/strong> is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.<\/p><\/div><p id=\"fs-id1169149297227\">Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a radical.<\/p><p id=\"fs-id1169148939113\">Similarly, a <span data-type=\"term\" class=\"no-emphasis\">radical expression<\/span> is not considered simplified if the radicand contains a fraction.<\/p><div data-type=\"note\" id=\"fs-id1169149172275\"><div data-type=\"title\">Simplified Radical Expressions<\/div><p id=\"fs-id1169149136595\">A radical expression is considered simplified if there are<\/p><ul id=\"fs-id1169149326602\" data-bullet-style=\"bullet\"><li>no factors in the radicand have perfect powers of the index<\/li><li>no fractions in the radicand<\/li><li>no radicals in the denominator of a fraction<\/li><\/ul><\/div><p id=\"fs-id1169149367820\">To rationalize a denominator with a square root, we use the property that \\({\\left(\\sqrt{a}\\right)}^{2}=a.\\) If we square an irrational square root, we get a rational number.<\/p><p id=\"fs-id1169149114181\">We will use this property to rationalize the denominator in the next example.<\/p><div data-type=\"example\" id=\"fs-id1169146612955\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169149087434\"><div data-type=\"problem\" id=\"fs-id1169148998388\"><p id=\"fs-id1169149029418\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{4}{\\sqrt{3}}\\) <span class=\"token\">\u24d1<\/span> \\(\\sqrt{\\frac{3}{20}}\\) <span class=\"token\">\u24d2<\/span> \\(\\frac{3}{\\sqrt{6x}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148869202\"><p id=\"fs-id1169148858445\">To rationalize a denominator with one term, we can multiply a square root by itself. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.<\/p><p id=\"fs-id1169149017598\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1169149346904\" class=\"unnumbered unstyled can-break\" summary=\"To rationalize the denominator of 4 divided by square root 3 we multiply both the numerator and denominator by square root 3. The result is the 4 times square root 3 divided by the quantity square root 3 times square root 3 in parentheses. Simplifying we get 4 times square root 3 divided by 3.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146838836\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply both the numerator and denominator by \\(\\sqrt{3}.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149312218\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149214546\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169148881651\"><span class=\"token\">\u24d1<\/span> We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.<\/p><table id=\"fs-id1169146732575\" class=\"unnumbered unstyled can-break\" summary=\"With the quantity 3 divided by 20 in parentheses we first notive that the fraction is not a perfect square, so we rewrite using the quotient property to get square root 3 divided by square root 20. Simplifying the denominator we get square root 3 divided by the quantity 2 times square root 5 in parentheses. To rationalize the denominator we multiply the numerator and denominator by square root 5. This is written as square root 3 times square root 5 divided by the quantity 2 square root 5 times square root 5 in parentheses. Simplifying we get square root 15 divided by the quantity 2 times 5 in parentheses. Simplifying furthere we get square root 15 divided by 10.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149217996\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The fraction is not a perfect square, so rewrite using the<div data-type=\"newline\"><br><\/div>Quotient Property.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149116163\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149292163\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by \\(\\sqrt{5}.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148890540\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149001128\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_002e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147085588\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_002f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169149346365\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1169149023603\" class=\"unnumbered unstyled can-break\" summary=\"To rationalize the denominator of 3 divided by square root of the quantity 6 x in parentheses we multiply both the numerator and denominator by square root of the quantity 6 x in parentheses. This is written out as 3 times square root of the quantity 6 x in parentheses divided by the quantity square root 6 x times square root 6 x in parentheses. The result is 3 times square root of the quantity 6 x in parentheses divided by the quantity 6 x in parentheses. Simplifying we square root of the quantity 6 x in parentheses divided by the quantity 2 x.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146645267\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by \\(\\sqrt{6x}.\\)\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148973792\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149302939\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148984763\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149029816\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169144567202\"><div data-type=\"problem\" id=\"fs-id1169148932927\"><p id=\"fs-id1169148967517\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{5}{\\sqrt{3}}\\) <span class=\"token\">\u24d1<\/span> \\(\\sqrt{\\frac{3}{32}}\\) <span class=\"token\">\u24d2<\/span> \\(\\frac{2}{\\sqrt{2x}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169147086760\"><p id=\"fs-id1169144745469\"><span class=\"token\">\u24d0<\/span>\\(\\frac{5\\sqrt{3}}{3}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt{6}}{8}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{\\sqrt{2x}}{x}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149095729\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169149156285\"><div data-type=\"problem\" id=\"fs-id1169149219746\"><p id=\"fs-id1169148912377\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{6}{\\sqrt{5}}\\) <span class=\"token\">\u24d1<\/span> \\(\\sqrt{\\frac{7}{18}}\\) <span class=\"token\">\u24d2<\/span> \\(\\frac{5}{\\sqrt{5x}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149036848\"><p id=\"fs-id1169149099890\"><span class=\"token\">\u24d0<\/span>\\(\\frac{6\\sqrt{5}}{5}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt{14}}{6}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{\\sqrt{5x}}{x}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169148883936\">When we rationalized a square root, we multiplied the numerator and denominator by a square root that would give us a perfect square under the radical in the denominator. When we took the square root, the denominator no longer had a radical.<\/p><p id=\"fs-id1169149026920\">We will follow a similar process to rationalize higher roots. To rationalize a denominator with a higher index radical, we multiply the numerator and denominator by a radical that would give us a radicand that is a perfect power of the index. When we simplify the new radical, the denominator will no longer have a radical.<\/p><p id=\"fs-id1169149100260\">For example,<\/p><span data-type=\"media\" id=\"fs-id1169148843962\" data-alt=\"Two examples of rationalizing denominators are shown. The first example is 1 divided by cube root 2. A note is made that the radicand in the denominator is 1 power of 2 and that we need 2 more to get a perfect cube. We multiply numerator and denominator by the cube root of the quantity 2 squared. The result is cube root 4 divided by cube root of quantity 2 cubed. This simplifies to cube root 4 divided by 2. The second example is 1 divided by fourth root 5. A note is made that the radicand in the denominator is 1 power of 5 and that we need 3 more to get a perfect fourth. We multiply numerator and denominator by the fourth root of the quantity 5 cubed. The result is fourth root of 125 divided by fourth root of quantity 5 to the fourth. This simplifies to fourth root 125 divided by 5.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Two examples of rationalizing denominators are shown. The first example is 1 divided by cube root 2. A note is made that the radicand in the denominator is 1 power of 2 and that we need 2 more to get a perfect cube. We multiply numerator and denominator by the cube root of the quantity 2 squared. The result is cube root 4 divided by cube root of quantity 2 cubed. This simplifies to cube root 4 divided by 2. The second example is 1 divided by fourth root 5. A note is made that the radicand in the denominator is 1 power of 5 and that we need 3 more to get a perfect fourth. We multiply numerator and denominator by the fourth root of the quantity 5 cubed. The result is fourth root of 125 divided by fourth root of quantity 5 to the fourth. This simplifies to fourth root 125 divided by 5.\"><\/span><p id=\"fs-id1169146626332\">We will use this technique in the next examples.<\/p><div data-type=\"example\" id=\"fs-id1169149178429\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169146631585\"><div data-type=\"problem\" id=\"fs-id1169147088334\"><p id=\"fs-id1169148972794\">Simplify <span class=\"token\">\u24d0<\/span> \\(\\frac{1}{\\sqrt[3]{6}}\\) <span class=\"token\">\u24d1<\/span> \\(\\sqrt[3]{\\frac{7}{24}}\\) <span class=\"token\">\u24d2<\/span> \\(\\frac{3}{\\sqrt[3]{4x}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146612334\"><p id=\"fs-id1169144380220\">To rationalize a denominator with a cube root, we can multiply by a cube root that will give us a perfect cube in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.<\/p><p id=\"fs-id1169148884074\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1169149313679\" class=\"unnumbered unstyled\" summary=\"The example is 1 divided by cube root 6. The radicand in the denominator is 1 factor of 6. Multiplying both the numerator and denominator by cube root of quantity 6 squared gives us 2 more factors of 6. The result is cube root of the quantity 6 squared in parentheses divided by cube root of quantity 6 cubed. Notice the radicand in the denominator has 3 powers of 6. This simplifies to cube root 36 divided by 6.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149358452\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The radical in the denominator has one factor of 6.<div data-type=\"newline\"><br><\/div>Multiply both the numerator and denominator by \\(\\sqrt[3]{{6}^{2}},\\)<div data-type=\"newline\"><br><\/div>which gives us 2 more factors of 6.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149007127\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply. Notice the radicand in the denominator<div data-type=\"newline\"><br><\/div>has 3 powers of 6.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149114418\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the cube root in the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148995146\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169148821019\"><span class=\"token\">\u24d1<\/span> We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.<\/p><table id=\"fs-id1169148820343\" class=\"unnumbered unstyled can-break\" summary=\"The example is cube root of the quantity 7 divided by 24. The fraction is not a perfect cube so rewrite using the quotient property. The new expression is cube root 7 divided by cube root 24. Simplifying the denominator gives cube root 7 divided by the quantity 2 cube root 3. Multiply the numerator and denominator by cube root quantity 3 squared. This will give us 3 factors of 3. This is written as cube root 7 times cube root quantity 3 squared in parentheses divided by the quantity 2 cube root 3 cube root 3 squared in parentheses. Simplifying we get cube root 63 divided by the quantity 2 cube root quantity 3 cubed in parentheses. Remember that cube root quantity 3 cubed equals 3. This gives cube root 63 divided by quantity 2 times 3. Simplifying once more we get cube root 63 divided by 6.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144421220\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The fraction is not a perfect cube, so<div data-type=\"newline\"><br><\/div>rewrite using the Quotient Property.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144601623\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148825672\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator\u2003\u2003\u2003\u2003\u2003\u2003\u2003<div data-type=\"newline\"><br><\/div>by \\(\\sqrt[3]{{3}^{2}}.\\) This will give us 3 factors of 3.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148918570\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Remember, \\(\\sqrt[3]{{3}^{3}}=3.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144715806\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149030330\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169146917946\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1169144517151\" class=\"unnumbered unstyled\" summary=\"The example is 3 divided by cube root of the quantity 4 x. Rewrite the radical to show the factors. The new expression is 3 divided by cube root of the quantity 2 squared times x in parentheses. Multiply the numerator and denominator by cube root quantity 2 x squared in parentheses. This will give us 3 factors of 2 and 3 factors of x. This is written as 3 times cube root quantity 2 x squared in parentheses divided by the quantity cube root quantity 2 x squared times cube root quantity 2 x squared in parentheses . Simplifying we get 3 times cube root quantity 2 x squared in parentheses divided by cube root of the quantity 2 cubed times x cubed in parentheses. Simplifying the radical in the denominator we get 3 cube root quantity 2 x squared in parentheses divided by the quantity 2 x.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144876452\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite the radicand to show the factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149110142\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by \\(\\sqrt[3]{2\u00b7{x}^{2}}.\\)<div data-type=\"newline\"><br><\/div>This will get us 3 factors of 2 and 3 factors of <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169138971669\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146815185\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the radical in the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148937504\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_007e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148968862\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148837048\"><div data-type=\"problem\" id=\"fs-id1169147028800\"><p id=\"fs-id1169146737825\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{1}{\\sqrt[3]{7}}\\) <span class=\"token\">\u24d1<\/span> \\(\\sqrt[3]{\\frac{5}{12}}\\) <span class=\"token\">\u24d2<\/span> \\(\\frac{5}{\\sqrt[3]{9y}}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1169146719598\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt[3]{49}}{7}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{90}}{6}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{5\\sqrt[3]{3{y}^{2}}}{3y}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148821337\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169144768704\"><div data-type=\"problem\" id=\"fs-id1169149102495\"><p id=\"fs-id1169144606258\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{1}{\\sqrt[3]{2}}\\) <span class=\"token\">\u24d1<\/span> \\(\\sqrt[3]{\\frac{3}{20}}\\) <span class=\"token\">\u24d2<\/span> \\(\\frac{2}{\\sqrt[3]{25n}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149093039\"><p id=\"fs-id1169148879748\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt[3]{4}}{2}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{150}}{10}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{2\\sqrt[3]{5{n}^{2}}}{5n}\\)<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1169148944647\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169149149549\"><div data-type=\"problem\"><p id=\"fs-id1169149309461\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{1}{\\sqrt[4]{2}}\\) <span class=\"token\">\u24d1<\/span> \\(\\sqrt[4]{\\frac{5}{64}}\\) <span class=\"token\">\u24d2<\/span> \\(\\frac{2}{\\sqrt[4]{8x}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149037326\"><p id=\"fs-id1169149102350\">To rationalize a denominator with a fourth root, we can multiply by a fourth root that will give us a perfect fourth power in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.<\/p><p id=\"fs-id1169148934751\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1169149304217\" class=\"unnumbered unstyled can-break\" summary=\"The example is 1 divided by fourth root 2. The radicand in the denominator has 1 factor of 2. Multiply both the numerator and denominator by fourth root quantity 2 cubed, which gives 3 more factors of 2. This is written as 1 times fourth root quantity 2 cubed divided by the quantity fourth root quantity 2 cubed times fourth root quantity 2 cubed in parentheses. Multiplying we get fourth root 8 divided by fourth root quantity 2 to the fourth. Notice the radicand in the denominator has 4 powers of 2. Simplifying the fourth root in the denominator results in fourth root 8 divided by 2.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148869643\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The radical in the denominator has one factor of 2.<div data-type=\"newline\"><br><\/div>Multiply both the numerator and denominator by \\(\\sqrt[4]{{2}^{3}},\\)\u2003\u2003\u2003<div data-type=\"newline\"><br><\/div>which gives us 3 more factors of 2.<\/td><td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146652418\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply. Notice the radicand in the denominator<div data-type=\"newline\"><br><\/div>has 4 powers of 2.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148837591\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the fourth root in the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149109247\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169148990220\"><span class=\"token\">\u24d1<\/span> We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.<\/p><table id=\"fs-id1169146669491\" class=\"unnumbered unstyled can-break\" summary=\"The example is fourth root of the quantity 5 divided by 64. The fraction is not a perfect cube so rewrite using the quotient property. The new expression is fourth root 5 divided by fourth root quantity 2 to the sixth. Simplifying the denominator gives fourth root 5 divided by the quantity 2 fourth root quantity 2 squared. Multiply the numerator and denominator by fourth root quantity 2 squared. This will give us 4 factors of 2. This is written as fourth root 5 times fourth root quantity 2 squared in parentheses divided by the quantity 2 fourth root 2 squared times fourth root 2 squared in parentheses. Simplifying we get fourth root 20 divided by the quantity 2 fourth root quantity 2 to the fourth in parentheses. Remember that fourth root quantity 2 to the fourth equals 2. This gives fourth root 20 divided by quantity 2 times 2. Simplifying once more we get fourth root 20 divided by 4.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148943899\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The fraction is not a perfect fourth power, so rewrite<div data-type=\"newline\"><br><\/div>using the Quotient Property.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144366838\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite the radicand in the denominator to show the factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149370876\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144885732\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by \\(\\sqrt[4]{{2}^{2}}.\\)<div data-type=\"newline\"><br><\/div>This will give us 4 factors of 2.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148838178\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146742186\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Remember, \\(\\sqrt[4]{{2}^{4}}=2.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144400096\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144365491\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169149112743\"><span class=\"token\">\u24d2<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1169149329077\" class=\"unnumbered unstyled can-break\" summary=\"The example is 2 divided by fourth root of the quantity 8 x. Rewrite the radical to show the factors. The new expression is 2 divided by fourth root of the quantity 2 cubed times x in parentheses. Multiply the numerator and denominator by fourth root quantity 2 x cubed in parentheses. This will give us 4 factors of 2 and 4 factors of x. This is written as 2 times fourth root quantity 2 x cubed in parentheses divided by the quantity fourth root quantity 2 x cubed times fourth root quantity 2 x cubed in parentheses . Simplifying we get 2 times fourth root quantity 2 x cubed in parentheses divided by fourth root of the quantity 2 to the fourth times x to the fourth in parentheses. Simplifying the radical in the denominator we get 2 fourth root quantity 2 x cubed in parentheses divided by the quantity 2 x. Simplifying the fraction results in fourth root quantity 2 x cubed in parentheses divided by x.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149016796\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite the radicand to show the factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149064900\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by \\(\\sqrt[4]{2\u00b7{x}^{3}}.\\)\u2003\u2003\u2003<div data-type=\"newline\"><br><\/div>This will get us 4 factors of 2 and 4 factors of <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147028339\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144451038\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the radical in the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149005582\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the fraction.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148971366\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149116201\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169149006904\"><div data-type=\"problem\" id=\"fs-id1169144563844\"><p id=\"fs-id1169149344287\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{1}{\\sqrt[4]{3}}\\) <span class=\"token\">\u24d1<\/span> \\(\\sqrt[4]{\\frac{3}{64}}\\) <span class=\"token\">\u24d2<\/span> \\(\\frac{3}{\\sqrt[4]{125x}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146699399\"><p id=\"fs-id1169149294726\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt[4]{27}}{3}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[4]{12}}{4}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{3\\sqrt[4]{5{x}^{3}}}{5x}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169144744512\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148890606\"><div data-type=\"problem\" id=\"fs-id1169149369284\"><p id=\"fs-id1169148837658\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{1}{\\sqrt[4]{5}}\\) <span class=\"token\">\u24d1<\/span> \\(\\sqrt[4]{\\frac{7}{128}}\\) <span class=\"token\">\u24d2<\/span> \\(\\frac{4}{\\sqrt[4]{4x}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149025360\"><p id=\"fs-id1169147086622\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt[4]{125}}{5}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[4]{224}}{8}\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span>\\(\\frac{\\sqrt[4]{64{x}^{3}}}{x}\\)<\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169148935104\"><h3 data-type=\"title\">Rationalize a Two Term Denominator<\/h3><p id=\"fs-id1169149064766\">When the denominator of a fraction is a sum or difference with square roots, we use the <span data-type=\"term\" class=\"no-emphasis\">Product of Conjugates Pattern<\/span> to <span data-type=\"term\" class=\"no-emphasis\">rationalize the denominator<\/span>.<\/p><div data-type=\"equation\" id=\"fs-id1169148879894\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccc}\\hfill \\left(a-b\\right)\\left(a+b\\right)\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\left(2-\\sqrt{5}\\right)\\left(2+\\sqrt{5}\\right)\\hfill \\\\ \\hfill {a}^{2}-{b}^{2}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{2}^{2}-{\\left(\\sqrt{5}\\right)}^{2}\\hfill \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}4-5\\hfill \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}-1\\hfill \\end{array}\\)<\/div><p id=\"fs-id1169147087680\">When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.<\/p><div data-type=\"example\" id=\"fs-id1169146812808\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148994436\"><div data-type=\"problem\" id=\"fs-id1169146848618\"><p id=\"fs-id1169149285866\">Simplify: \\(\\frac{5}{2-\\sqrt{3}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148918408\"><table id=\"fs-id1169149155739\" class=\"unnumbered unstyled can-break\" summary=\"To rationalize the denominator in 5 divided by the quantity 2 minus square root 3 in parentheses we multiply the numerator and denominator by the conjugate of the denominator. This is written as 5 times the quantity 2 plus square root 3 in parentheses divided by the product of the quantity 2 minus square root 3 in parentheses with the quantity 2 plus square root 3 in parentheses. Muliplying the conjugates in the denominator results in 5 times the quantity 2 plus square root 3 in parentheses divided by the difference of 2 squared and the quantity square root 3 squared. Simplifying the denominator gives 5 times the quantity 2 plus square root 3 in parentheses divided by the quantity 4 minus 3 in parentheses. This simplifies to 5 times the quantity 2 plus square root 3 in parentheses.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148971687\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by the<div data-type=\"newline\"><br><\/div>conjugate of the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149302426\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the conjugates in the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148868647\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149213386\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_011d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149001659\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_011e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148994799\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_011f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148875850\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169146607241\"><div data-type=\"problem\" id=\"fs-id1169149004978\"><p id=\"fs-id1169144566191\">Simplify: \\(\\frac{3}{1-\\sqrt{5}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149223924\"><p id=\"fs-id1169148930197\">\\(-\\frac{3\\left(1+\\sqrt{5}\\right)}{4}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149291553\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148866919\"><div data-type=\"problem\" id=\"fs-id1169149308919\"><p id=\"fs-id1169148959569\">Simplify: \\(\\frac{2}{4-\\sqrt{6}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146623122\"><p id=\"fs-id1169147089653\">\\(\\frac{4+\\sqrt{6}}{5}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169149347057\">Notice we did not distribute the 5 in the answer of the last example. By leaving the result factored we can see if there are any factors that may be common to both the numerator and denominator.<\/p><div data-type=\"example\" id=\"fs-id1169149346571\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169146637455\"><div data-type=\"problem\" id=\"fs-id1169149032875\"><p id=\"fs-id1169146750688\">Simplify: \\(\\frac{\\sqrt{3}}{\\sqrt{u}-\\sqrt{6}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148883988\"><table id=\"fs-id1169144893528\" class=\"unnumbered unstyled can-break\" summary=\"To rationalize the denominator in square root 3 divided by the quantity square root u minus square root 6 in parentheses we multiply the numerator and denominator by the conjugate of the denominator. This is written as square root 3 times the quantity square root u plus square root 6 in parentheses divided by the product of the quantity square root u minus square root 6 in parentheses with the quantity square root u plus square root 6 in parentheses. Muliplying the conjugates in the denominator results in square root 3 times the quantity square root u plus square root 6 in parentheses divided by the difference of square root u squared and square root 6 squared. Simplifying the denominator gives square root 3 times the quantity square root u plus square root 6 in parentheses divided by the quantity u minus 6 in parentheses. Simplifying the numerator results in the difference of square root quantity 3 u in parentheses and 3 square root 2 divided by the quantity u minus 6.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149219717\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_012a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by the<div data-type=\"newline\"><br><\/div>conjugate of the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144419340\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the conjugates in the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146947408\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149315736\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_012d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169144381649\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169149114706\"><div data-type=\"problem\" id=\"fs-id1169148974522\"><p id=\"fs-id1169148868702\">Simplify: \\(\\frac{\\sqrt{5}}{\\sqrt{x}+\\sqrt{2}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144873576\"><p id=\"fs-id1169149178345\">\\(\\frac{\\sqrt{5}\\left(\\sqrt{x}-\\sqrt{2}\\right)}{x-2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149000052\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148867898\"><div data-type=\"problem\" id=\"fs-id1169148964598\"><p id=\"fs-id1169149031341\">Simplify: \\(\\frac{\\sqrt{10}}{\\sqrt{y}-\\sqrt{3}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149197086\"><p id=\"fs-id1169149037952\">\\(\\frac{\\sqrt{10}\\left(\\sqrt{y}+\\sqrt{3}\\right)}{y-3}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165926512140\">Be careful of the signs when multiplying. The numerator and denominator look very similar when you multiply by the conjugate.<\/p><div data-type=\"example\" id=\"fs-id1169149345703\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169149108831\"><div data-type=\"problem\" id=\"fs-id1169148992913\"><p id=\"fs-id1169146644820\">Simplify: \\(\\frac{\\sqrt{x}+\\sqrt{7}}{\\sqrt{x}-\\sqrt{7}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144560259\"><table id=\"fs-id1169146606154\" class=\"unnumbered unstyled can-break\" summary=\"To rationalize the denominator in the quantity square root x plus square root 7 in parentheses divided by the quantity square root x minus square root 7 in parentheses we multiply the numerator and denominator by the conjugate of the denominator. This is written as the quantity square root x plus square root 7 in parentheses times the quantity square root x plus square root 7 in parentheses divided by the product of the quantity square root x minus square root 7 in parentheses with the quantity square root x plus square root 7 in parentheses. Muliplying the conjugates in the denominator results in the quantity square root x plus square root 7 in parentheses times the quantity square root x plus square root 7 in parentheses divided by the difference of square root x squared and square root 7 squared. Simplifying the denominator gives the quantity square root x plus square root 7 in parentheses squared divided by the quantity x minus 7 in parentheses.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144484645\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by the<div data-type=\"newline\"><br><\/div>conjugate of the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149306398\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the conjugates in the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149015016\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149213947\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169149314052\">We do not square the numerator. Leaving it in factored form, we can see there are no common factors to remove from the numerator and denominator.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169144522987\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147054334\"><div data-type=\"problem\" id=\"fs-id1169144484785\"><p id=\"fs-id1169148890263\">Simplify: \\(\\frac{\\sqrt{p}+\\sqrt{2}}{\\sqrt{p}-\\sqrt{2}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148991499\"><p id=\"fs-id1169144422081\">\\({\\frac{\\left(\\sqrt{p}+\\sqrt{2}\\right)}{p-2}}^{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148971696\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148968242\"><div data-type=\"problem\" id=\"fs-id1169148938406\"><p id=\"fs-id1169144683706\">Simplify: \\(\\frac{\\sqrt{q}-\\sqrt{10}}{\\sqrt{q}+\\sqrt{10}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149204814\"><p id=\"fs-id1169146620540\">\\({\\frac{\\left(\\sqrt{q}-\\sqrt{10}\\right)}{q-10}}^{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169144769214\" class=\"media-2\"><p id=\"fs-id1169148997642\">Access these online resources for additional instruction and practice with dividing radical expressions.<\/p><ul id=\"fs-id1169148992311\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37RatDenom1\">Rationalize the Denominator<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37RatDenom2\">Dividing Radical Expressions and Rationalizing the Denominator<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37RatDenom3\">Simplifying a Radical Expression with a Conjugate<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37RatDenom4\">Rationalize the Denominator of a Radical Expression<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169148984716\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1169146739168\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Quotient Property of Radical Expressions<\/strong><ul id=\"fs-id1169144685624\" data-bullet-style=\"bullet\"><li>If \\(\\sqrt[n]{a}\\) and \\(\\sqrt[n]{b}\\) are real numbers, \\(b\\ne 0,\\) and for any integer \\(n\\ge 2\\) then,<div data-type=\"newline\"><br><\/div> \\(\\sqrt[n]{\\frac{a}{b}}=\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}}\\) and \\(\\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}}=\\sqrt[n]{\\frac{a}{b}}\\)<\/li><\/ul><\/li><li><strong data-effect=\"bold\">Simplified Radical Expressions<\/strong><ul id=\"fs-id1169144567375\" data-bullet-style=\"bullet\"><li>A radical expression is considered simplified if there are: <ul id=\"fs-id1169148880634\" data-bullet-style=\"bullet\"><li>no factors in the radicand that have perfect powers of the index<\/li><li>no fractions in the radicand<\/li><li>no radicals in the denominator of a fraction<\/li><\/ul><\/li><\/ul><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169139949101\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169149157492\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1169144451023\"><strong data-effect=\"bold\">Divide Square Roots<\/strong><\/p><p id=\"fs-id1169149312167\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1169148869456\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169146666523\"><p id=\"fs-id1169149193736\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{128}}{\\sqrt{72}}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{128}}{\\sqrt[3]{54}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144601274\"><p id=\"fs-id1169148964069\"><span class=\"token\">\u24d0<\/span>\\(\\frac{4}{3}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{4}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148951118\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169144874488\"><p id=\"fs-id1169146644476\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{48}}{\\sqrt{75}}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{81}}{\\sqrt[3]{24}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149196629\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169146744588\"><p id=\"fs-id1169144768763\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{200{m}^{5}}}{\\sqrt{98m}}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{54{y}^{2}}}{\\sqrt[3]{2{y}^{5}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149240580\"><p id=\"fs-id1169146621806\"><span class=\"token\">\u24d0<\/span>\\(\\frac{10{m}^{2}}{7}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{3}{y}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149287043\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169146731710\"><p id=\"fs-id1169149016261\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{108{n}^{7}}}{\\sqrt{243{n}^{3}}}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{54{y}^{}}}{\\sqrt[3]{16{y}^{4}}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169146740915\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169149113858\"><p id=\"fs-id1169144873851\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{75{r}^{3}}}{\\sqrt{108{r}^{7}}}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{24{x}^{7}}}{\\sqrt[3]{81{x}^{4}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144550544\"><p id=\"fs-id1169144686560\"><span class=\"token\">\u24d0<\/span>\\(\\frac{5}{6{r}^{2}}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{2x}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148970327\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169149065505\"><p id=\"fs-id1169149294580\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{196{q}^{}}}{\\sqrt{484{q}^{5}}}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{16{m}^{4}}}{\\sqrt[3]{54{m}^{}}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149328074\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169146664810\"><p id=\"fs-id1169148955565\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{108{p}^{5}{q}^{2}}}{\\sqrt{3{p}^{3}{q}^{6}}}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{-16{a}^{4}{b}^{-2}}}{\\sqrt[3]{2{a}^{-2}{b}^{}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146663944\"><p id=\"fs-id1169148956592\"><span class=\"token\">\u24d0<\/span>\\(\\frac{6p}{{q}^{2}}\\)<span class=\"token\">\u24d1<\/span>\\(-\\frac{2{a}^{2}}{b}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149339448\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169149339450\"><p id=\"fs-id1169148956040\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{98r{s}^{10}}}{\\sqrt{2{r}^{3}{s}^{4}}}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{-375{y}^{4}{z}^{-2}}}{\\sqrt[3]{3{y}^{-2}{z}^{4}}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144565587\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169144565590\"><p id=\"fs-id1169149042640\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{320m{n}^{-5}}}{\\sqrt{45{m}^{-7}{n}^{3}}}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{16{x}^{4}{y}^{-2}}}{\\sqrt[3]{-54{x}^{-2}{y}^{4}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149328417\"><p id=\"fs-id1169148995445\"><span class=\"token\">\u24d0<\/span>\\(\\frac{8{m}^{4}}{3{n}^{4}}\\)<span class=\"token\">\u24d1<\/span>\\(-\\frac{2{x}^{2}}{3{y}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148880013\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169148992184\"><p id=\"fs-id1169148992187\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt{810{c}^{-3}{d}^{7}}}{\\sqrt{1000{c}^{}{d}^{-1}}}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{24{a}^{7}{b}^{-1}}}{\\sqrt[3]{-81{a}^{-2}{b}^{2}}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144614783\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169144614785\"><p id=\"fs-id1169149013905\">\\(\\frac{\\sqrt{56{x}^{5}{y}^{4}}}{\\sqrt{2x{y}^{3}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146643067\"><p id=\"fs-id1169149214011\">\\(4{x}^{4}\\sqrt{7y}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169146667820\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169146667822\"><p id=\"fs-id1169149065808\">\\(\\frac{\\sqrt{72{a}^{3}{b}^{6}}}{\\sqrt{3a{b}^{3}}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149109868\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169149109870\"><p id=\"fs-id1169148957157\">\\(\\frac{\\sqrt[3]{48{a}^{3}{b}^{6}}}{\\sqrt[3]{3{a}^{-1}{b}^{3}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149350593\"><p id=\"fs-id1169149118426\">\\(2ab\\sqrt[3]{2a}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149280925\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1169149280927\"><p id=\"fs-id1169148993218\">\\(\\frac{\\sqrt[3]{162{x}^{-3}{y}^{6}}}{\\sqrt[3]{2{x}^{3}{y}^{-2}}}\\)<\/p><\/div><\/div><p id=\"fs-id1169148969800\"><strong data-effect=\"bold\">Rationalize a One Term Denominator<\/strong><\/p><p id=\"fs-id1169146841907\">In the following exercises, rationalize the denominator.<\/p><div data-type=\"exercise\" id=\"fs-id1169149012708\"><div data-type=\"problem\" id=\"fs-id1169149012710\"><p id=\"fs-id1169144550856\"><span class=\"token\">\u24d0<\/span>\\(\\frac{10}{\\sqrt{6}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt{\\frac{4}{27}}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{10}{\\sqrt{5x}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149348903\"><p id=\"fs-id1169149153399\"><span class=\"token\">\u24d0<\/span>\\(\\frac{5\\sqrt{6}}{3}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{2\\sqrt{3}}{9}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{2\\sqrt{5x}}{x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149316475\"><div data-type=\"problem\" id=\"fs-id1169149361122\"><p id=\"fs-id1169149361124\"><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-id1169148996426\"><div data-type=\"problem\" id=\"fs-id1169148996428\"><p id=\"fs-id1169149340430\"><span class=\"token\">\u24d0<\/span>\\(\\frac{6}{\\sqrt{7}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt{\\frac{8}{45}}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{12}{\\sqrt{3p}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144605586\"><p id=\"fs-id1169144605588\"><span class=\"token\">\u24d0<\/span>\\(\\frac{6\\sqrt{7}}{7}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{2\\sqrt{10}}{15}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{4\\sqrt{3p}}{p}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144523288\"><div data-type=\"problem\" id=\"fs-id1169149004586\"><p id=\"fs-id1169149004588\"><span class=\"token\">\u24d0<\/span>\\(\\frac{4}{\\sqrt{5}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt{\\frac{27}{80}}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{18}{\\sqrt{6q}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144796674\"><div data-type=\"problem\" id=\"fs-id1169144796676\"><p id=\"fs-id1169149156704\"><span class=\"token\">\u24d0<\/span>\\(\\frac{1}{\\sqrt[3]{5}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{\\frac{5}{24}}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{4}{\\sqrt[3]{36a}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144381822\"><p id=\"fs-id1169144381824\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt[3]{25}}{5}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[3]{45}}{6}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{2\\sqrt[3]{6{a}^{2}}}{3a}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149303500\"><div data-type=\"problem\" id=\"fs-id1169149303502\"><p id=\"fs-id1169146738463\"><span class=\"token\">\u24d0<\/span>\\(\\frac{1}{\\sqrt[3]{3}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{\\frac{5}{32}}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{7}{\\sqrt[3]{49b}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149367796\"><div data-type=\"problem\" id=\"fs-id1169149367798\"><p id=\"fs-id1169149367801\"><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-id1169149224185\"><p id=\"fs-id1169149003529\"><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-id1169149210154\"><div data-type=\"problem\" id=\"fs-id1169149210156\"><p id=\"fs-id1169146659953\"><span class=\"token\">\u24d0<\/span>\\(\\frac{1}{\\sqrt[3]{13}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[3]{\\frac{3}{128}}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{3}{\\sqrt[3]{6{y}^{2}}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144835169\"><div data-type=\"problem\" id=\"fs-id1169144835172\"><p id=\"fs-id1169148969699\"><span class=\"token\">\u24d0<\/span>\\(\\frac{1}{\\sqrt[4]{7}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[4]{\\frac{5}{32}}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{4}{\\sqrt[4]{4{x}^{2}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149351259\"><p id=\"fs-id1169144422013\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt[4]{343}}{7}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[4]{40}}{4}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{2\\sqrt[4]{4{x}^{2}}}{x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149376982\"><div data-type=\"problem\" id=\"fs-id1169149376984\"><p id=\"fs-id1169149376986\"><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><div data-type=\"exercise\" id=\"fs-id1169148962629\"><div data-type=\"problem\" id=\"fs-id1169148962631\"><p id=\"fs-id1169148969883\"><span class=\"token\">\u24d0<\/span>\\(\\frac{1}{\\sqrt[4]{9}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[4]{\\frac{25}{128}}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{6}{\\sqrt[4]{27a}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144567993\"><p id=\"fs-id1169144567995\"><span class=\"token\">\u24d0<\/span>\\(\\frac{\\sqrt[4]{9}}{3}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{\\sqrt[4]{50}}{4}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{2\\sqrt[4]{3{a}^{2}}}{a}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149011945\"><div data-type=\"problem\" id=\"fs-id1169144686266\"><p id=\"fs-id1169144686269\"><span class=\"token\">\u24d0<\/span>\\(\\frac{1}{\\sqrt[4]{8}}\\)<span class=\"token\">\u24d1<\/span>\\(\\sqrt[4]{\\frac{27}{128}}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{16}{\\sqrt[4]{64{b}^{2}}}\\)<\/p><\/div><\/div><p id=\"fs-id1169144784948\"><strong data-effect=\"bold\">Rationalize a Two Term Denominator<\/strong><\/p><p id=\"fs-id1169149189450\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1169149290498\"><div data-type=\"problem\" id=\"fs-id1169149290500\"><p id=\"fs-id1169149290502\">\\(\\frac{8}{1-\\sqrt{5}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149183950\"><p id=\"fs-id1169149183952\">\\(-2\\left(1+\\sqrt{5}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149066158\"><div data-type=\"problem\" id=\"fs-id1169149066160\"><p id=\"fs-id1169149373773\">\\(\\frac{7}{2-\\sqrt{6}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169146632382\"><div data-type=\"problem\" id=\"fs-id1169146632384\"><p id=\"fs-id1169144374242\">\\(\\frac{6}{3-\\sqrt{7}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144757266\"><p id=\"fs-id1169148995749\">\\(3\\left(3+\\sqrt{7}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149308966\"><div data-type=\"problem\" id=\"fs-id1169140115605\"><p id=\"fs-id1169140115607\">\\(\\frac{5}{4-\\sqrt{11}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148870248\"><div data-type=\"problem\" id=\"fs-id1169149006551\"><p id=\"fs-id1169149006553\">\\(\\frac{\\sqrt{3}}{\\sqrt{m}-\\sqrt{5}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149347193\"><p id=\"fs-id1169144605142\">\\(\\frac{\\sqrt{3}\\left(\\sqrt{m}+\\sqrt{5}\\right)}{m-5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149312792\"><div data-type=\"problem\" id=\"fs-id1169149312794\"><p id=\"fs-id1169148939800\">\\(\\frac{\\sqrt{5}}{\\sqrt{n}-\\sqrt{7}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144359929\"><div data-type=\"problem\" id=\"fs-id1169149315892\"><p id=\"fs-id1169149315894\">\\(\\frac{\\sqrt{2}}{\\sqrt{x}-\\sqrt{6}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149370046\"><p id=\"fs-id1169146625890\">\\(\\frac{\\sqrt{2}\\left(\\sqrt{x}+\\sqrt{6}\\right)}{x-6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149100564\"><div data-type=\"problem\" id=\"fs-id1169149100566\"><p id=\"fs-id1169148871397\">\\(\\frac{\\sqrt{7}}{\\sqrt{y}+\\sqrt{3}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149193810\"><div data-type=\"problem\" id=\"fs-id1169146613048\"><p id=\"fs-id1169146613050\">\\(\\frac{\\sqrt{r}+\\sqrt{5}}{\\sqrt{r}-\\sqrt{5}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144560494\"><p id=\"fs-id1169144560496\">\\({\\frac{\\left(\\sqrt{r}+\\sqrt{5}\\right)}{r-5}}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148989470\"><div data-type=\"problem\" id=\"fs-id1169148989472\"><p id=\"fs-id1169149039557\">\\(\\frac{\\sqrt{s}-\\sqrt{6}}{\\sqrt{s}+\\sqrt{6}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149351357\"><div data-type=\"problem\" id=\"fs-id1169149330488\"><p id=\"fs-id1169149330490\">\\(\\frac{\\sqrt{x}+\\sqrt{8}}{\\sqrt{x}-\\sqrt{8}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144730314\"><p id=\"fs-id1169144730317\">\\({\\frac{\\left(\\sqrt{x}+2\\sqrt{2}\\right)}{x-8}}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169146745049\"><div data-type=\"problem\" id=\"fs-id1169147106841\"><p id=\"fs-id1169147106843\">\\(\\frac{\\sqrt{m}-\\sqrt{3}}{\\sqrt{m}+\\sqrt{3}}\\)<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1169149001285\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1169148933998\"><div data-type=\"problem\" id=\"fs-id1169148934000\"><p id=\"fs-id1169149030522\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> Simplify \\(\\sqrt{\\frac{27}{3}}\\) and explain all your steps.<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Simplify \\(\\sqrt{\\frac{27}{5}}\\) and explain all your steps.<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span> Why are the two methods of simplifying square roots different?<\/div><div data-type=\"solution\" id=\"fs-id1169149121884\"><p id=\"fs-id1169149121886\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148947253\"><div data-type=\"problem\" id=\"fs-id1169148947255\"><p id=\"fs-id1169149310618\">Explain what is meant by the word rationalize in the phrase, \u201crationalize a denominator.\u201d<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149315954\"><div data-type=\"problem\" id=\"fs-id1169149315956\"><p id=\"fs-id1169149315958\">Explain why multiplying \\(\\sqrt{2x}-3\\) by its conjugate results in an expression with no radicals.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144567005\"><p id=\"fs-id1169144567007\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149183626\"><div data-type=\"problem\" id=\"fs-id1169147085094\"><p id=\"fs-id1169147085096\">Explain why multiplying \\(\\frac{7}{\\sqrt[3]{x}}\\) by \\(\\frac{\\sqrt[3]{x}}{\\sqrt[3]{x}}\\) does not rationalize the denominator.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169149026970\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1169149026976\"><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-id1169144601314\" data-alt=\"This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cConfidently\u201d, the third is \u201cWith some help\u201d, and the fourth is \u201cNo, I don\u2019t get it\u201d. Under the first column are the phrases \u201cdivide radical expressions.\u201d, \u201crationalize a one term denominator\u201d, and \u201crationalize a two term denominator\u201d. The other columns are left blank so that the learner may indicate their mastery level for each topic.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cConfidently\u201d, the third is \u201cWith some help\u201d, and the fourth is \u201cNo, I don\u2019t get it\u201d. Under the first column are the phrases \u201cdivide radical expressions.\u201d, \u201crationalize a one term denominator\u201d, and \u201crationalize a two term denominator\u201d. The other columns are left blank so that the learner may indicate their mastery level for each topic.\"><\/span><p id=\"fs-id1169149113217\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1169149039257\"><dt>rationalizing the denominator<\/dt><dd id=\"fs-id1169149039262\">Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.<\/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>Divide radical expressions<\/li>\n<li>Rationalize a one term denominator<\/li>\n<li>Rationalize a two term denominator<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169146738116\" class=\"be-prepared\">\n<p id=\"fs-id1169134309346\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1169147085484\" type=\"1\">\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98fd62475d96dfb73aaa052701eaeb3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#125;&#123;&#52;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"20\" style=\"vertical-align: -6px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/425620d9-51dd-45e5-8a21-953998a4a77f#fs-id1167836620030\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31a2167c193fe59c4ba0d6f5e88eb18e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&middot;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"40\" style=\"vertical-align: 0px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/3fa6a6c5-9a36-4dee-aea1-0166229f52fb#fs-id1167835512989\" 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-40a763cb69470634ee34fbe65b70ea3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#43;&#51;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#51;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"136\" 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-id1167836392219\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169144516870\">\n<h3 data-type=\"title\">Divide Radical Expressions<\/h3>\n<p id=\"fs-id1169148869846\">We have used the <span data-type=\"term\" class=\"no-emphasis\">Quotient Property of Radical Expressions<\/span> to simplify roots of fractions. We will need to use this property \u2018in reverse\u2019 to simplify a fraction with radicals.<\/p>\n<p id=\"fs-id1169146630664\">We give the Quotient Property of Radical Expressions again for easy reference. Remember, we assume all variables are greater than or equal to zero so that no absolute value bars re needed.<\/p>\n<div data-type=\"note\" id=\"fs-id1169148933282\">\n<div data-type=\"title\">Quotient Property of Radical Expressions<\/div>\n<p id=\"fs-id1169146638883\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c6b7326bec5fc086e8335bb7e382f30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-489e8ff9827ba6988dfa32390ed30650_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"21\" style=\"vertical-align: -2px;\" \/> are real numbers, <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;\" \/> and for any integer <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5482ac64c45006c6548c713888a5d34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\" \/> then,<\/p>\n<div data-type=\"equation\" id=\"fs-id1169148993746\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79cae890d0aa45e31e370a8a06f7c8ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#97;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#98;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#97;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#98;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"224\" style=\"vertical-align: -11px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169148859881\">We will use the Quotient Property of Radical Expressions when the fraction we start with is the quotient of two radicals, and neither radicand is a perfect power of the index. When we write the fraction in a single radical, we may find common factors in the numerator and denominator.<\/p>\n<div data-type=\"example\" id=\"fs-id1169147136256\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148992709\">\n<div data-type=\"problem\" id=\"fs-id1169146669434\">\n<p id=\"fs-id1169148837638\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cab6b5c2a5494fc64699d0345a6ba795_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#50;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"40\" 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-acc58733efb7774f1407912c48035d86_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;&#51;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"48\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149094103\">\n<p id=\"fs-id1169149370840\"><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-f5753ea8d973bb21aef905c4fb99a379_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;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#50;&#120;&#125;&#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;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#117;&#115;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#111;&#116;&#105;&#101;&#110;&#116;&#32;&#112;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#44;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#97;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#98;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#125;&#46;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#49;&#54;&#50;&#120;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#109;&#111;&#118;&#101;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#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;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#49;&#56;&#125;&middot;&#52;&middot;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&middot;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#120;&#125;&#125;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#49;&#56;&#125;&middot;&#57;&middot;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#120;&#125;&#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;&#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;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#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;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#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;&#50;&#120;&#125;&#123;&#51;&#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=\"179\" width=\"495\" style=\"vertical-align: -85px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><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-f9bbe633b0487125a8cffd9ec25cc748_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;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#125;&#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;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#117;&#115;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#111;&#116;&#105;&#101;&#110;&#116;&#32;&#112;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#44;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#97;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#98;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#125;&#46;&#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;&#91;&#51;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#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;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#32;&#117;&#110;&#100;&#101;&#114;&#32;&#116;&#104;&#101;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#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;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#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;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#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;&#50;&#125;&#123;&#120;&#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=\"137\" width=\"475\" style=\"vertical-align: -64px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169146668779\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169146978086\">\n<div data-type=\"problem\" id=\"fs-id1169146743962\">\n<p id=\"fs-id1169149024013\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f57a1f0ca8dcb22f2c73b033bdf6db9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#48;&#123;&#115;&#125;&#94;&#123;&#51;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#56;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"39\" 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-bb5074ee23596dd2e1ab7eb589e4b1e1_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;&#53;&#54;&#123;&#97;&#125;&#94;&#123;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#55;&#123;&#97;&#125;&#94;&#123;&#52;&#125;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"41\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146626214\">\n<p id=\"fs-id1169140091518\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5031234bb4872439850896a44ac75a75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#115;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" 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-1538e092d111f6ad81a34bc0370bd1b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"8\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149003808\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149222208\">\n<div data-type=\"problem\" id=\"fs-id1169149350056\">\n<p id=\"fs-id1169148911048\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b30dc846393065a768ae34d1755c0d28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#53;&#123;&#113;&#125;&#94;&#123;&#53;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#56;&#113;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"43\" 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-0a504cab039a1d78690b5533d7880aa5_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;&#55;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#57;&#123;&#98;&#125;&#94;&#123;&#53;&#125;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"46\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148997993\">\n<p id=\"fs-id1169147136095\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc14b096fcedf441fdf018d92f2b44bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"21\" 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-79e998ca29e33ea404f8bb18d1bfff7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169149373645\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149012671\">\n<div data-type=\"problem\" id=\"fs-id1169148889871\">\n<p id=\"fs-id1169147027763\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a79451b19a450103cd4c08bc79f346cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#55;&#97;&#123;&#98;&#125;&#94;&#123;&#56;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#123;&#98;&#125;&#94;&#123;&#52;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"53\" 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-480235e76b6c3eea2c22144b90f076f6_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;&#50;&#53;&#48;&#123;&#109;&#125;&#94;&#123;&#125;&#123;&#110;&#125;&#94;&#123;&#45;&#50;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#123;&#109;&#125;&#94;&#123;&#45;&#50;&#125;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"88\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148990099\">\n<p id=\"fs-id1169144373998\"><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-777a5d5e61a1ac23046087992856aa87_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;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#55;&#97;&#123;&#98;&#125;&#94;&#123;&#56;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#123;&#98;&#125;&#94;&#123;&#52;&#125;&#125;&#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;&#117;&#115;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#111;&#116;&#105;&#101;&#110;&#116;&#32;&#112;&#114;&#111;&#112;&#101;&#114;&#116;&#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;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#55;&#97;&#123;&#98;&#125;&#94;&#123;&#56;&#125;&#125;&#123;&#51;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#123;&#98;&#125;&#94;&#123;&#52;&#125;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#109;&#111;&#118;&#101;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#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;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#57;&#123;&#98;&#125;&#94;&#123;&#52;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#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;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#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;&#55;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#97;&#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=\"119\" width=\"493\" style=\"vertical-align: -55px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><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-b12c43d537e73e9077c0e61f6a0c8347_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;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#45;&#50;&#53;&#48;&#123;&#109;&#125;&#94;&#123;&#125;&#123;&#110;&#125;&#94;&#123;&#45;&#50;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#123;&#109;&#125;&#94;&#123;&#45;&#50;&#125;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#125;&#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;&#117;&#115;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#111;&#116;&#105;&#101;&#110;&#116;&#32;&#112;&#114;&#111;&#112;&#101;&#114;&#116;&#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;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#53;&#48;&#123;&#109;&#125;&#94;&#123;&#125;&#123;&#110;&#125;&#94;&#123;&#45;&#50;&#125;&#125;&#123;&#50;&#123;&#109;&#125;&#94;&#123;&#45;&#50;&#125;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#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;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#32;&#117;&#110;&#100;&#101;&#114;&#32;&#116;&#104;&#101;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#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;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#50;&#53;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#54;&#125;&#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;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#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;&#53;&#109;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#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=\"117\" width=\"515\" style=\"vertical-align: -54px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149329581\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169146607917\">\n<div data-type=\"problem\" id=\"fs-id1169149367228\">\n<p id=\"fs-id1169146646547\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a355594b64f83b56e6690230123b60dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#54;&#50;&#123;&#120;&#125;&#94;&#123;&#49;&#48;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#54;&#125;&#123;&#121;&#125;&#94;&#123;&#54;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"71\" style=\"vertical-align: -15px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b16a97dba5b67ca0c8ace000bc1be614_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;&#49;&#50;&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#45;&#49;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#45;&#49;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"92\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148863029\">\n<p id=\"fs-id1169148884863\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72caf9ce0614e2c1646316007387ac93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"22\" style=\"vertical-align: -10px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bc1f0a09d3d714a5617c5e40313cb3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#52;&#120;&#125;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149105200\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147115617\">\n<div data-type=\"problem\" id=\"fs-id1169146815722\">\n<p id=\"fs-id1169144553970\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47f4ab2d1d68f3eeb88264739216a31a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#48;&#48;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#123;&#110;&#125;&#94;&#123;&#55;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#123;&#109;&#125;&#94;&#123;&#53;&#125;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"67\" 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-22c37dbbb3c7ba6981560284f894f967_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;&#56;&#49;&#112;&#123;&#113;&#125;&#94;&#123;&#45;&#49;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#123;&#112;&#125;&#94;&#123;&#45;&#50;&#125;&#123;&#113;&#125;&#94;&#123;&#53;&#125;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"77\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149016702\">\n<p id=\"fs-id1169149349179\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9923cf5c57a6ba7dda0c85d89955387b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"29\" 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-c46f7751e588db3c968f905a1a6abd07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#51;&#112;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"25\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169149316122\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149330242\">\n<div data-type=\"problem\" id=\"fs-id1169149344431\">\n<p id=\"fs-id1169144728872\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-42cf42bb4a01cd01f7e0ef3daa31b279_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#52;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"63\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146620795\">\n<p id=\"fs-id1169146648111\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d150b0b47cdb9856877ef26d46d5e481_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;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#52;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#125;&#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;&#117;&#115;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#111;&#116;&#105;&#101;&#110;&#116;&#32;&#112;&#114;&#111;&#112;&#101;&#114;&#116;&#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;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#52;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#109;&#111;&#118;&#101;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#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;&#115;&#113;&#114;&#116;&#123;&#49;&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#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;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#110;&#100;&#32;&#97;&#115;&#32;&#97;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#117;&#115;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#32;&#108;&#97;&#114;&#103;&#101;&#115;&#116;&#32;&#112;&#101;&#114;&#102;&#101;&#99;&#116;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#102;&#97;&#99;&#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;&#115;&#113;&#114;&#116;&#123;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#120;&#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;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#32;&#97;&#115;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#111;&#102;&#32;&#116;&#119;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#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;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#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;&#51;&#120;&#121;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#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=\"203\" width=\"565\" style=\"vertical-align: -96px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148996331\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169146645855\">\n<div data-type=\"problem\" id=\"fs-id1169146654752\">\n<p id=\"fs-id1169149033237\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfbc9a6f9d8c63649d162ba516e21476_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#52;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#53;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"63\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148963012\">\n<p id=\"fs-id1169149123517\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-811255205b524f09ace99fea29a5ed6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#121;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"62\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144523044\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148957840\">\n<div data-type=\"problem\" id=\"fs-id1169148859738\">\n<p id=\"fs-id1169144560010\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18ad3401029b5ca157baa65305eeb2c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#54;&#123;&#97;&#125;&#94;&#123;&#53;&#125;&#123;&#98;&#125;&#94;&#123;&#52;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#98;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"58\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149347376\">\n<p id=\"fs-id1169148876020\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e324a00205e5b2d8632454eae5b1522_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#97;&#98;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"58\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149140677\">\n<h3 data-type=\"title\">Rationalize a One Term Denominator<\/h3>\n<p id=\"fs-id1169148909612\">Before the calculator became a tool of everyday life, approximating the value of a fraction with a radical in the denominator was a very cumbersome process!<\/p>\n<p id=\"fs-id1169149294686\">For this reason, a process called <span data-type=\"term\">rationalizing the denominator<\/span> was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator.<\/p>\n<p id=\"fs-id1169146655337\">This process is still used today, and is useful in other areas of mathematics, too.<\/p>\n<div data-type=\"note\" id=\"fs-id1169146620764\">\n<div data-type=\"title\">Rationalizing the Denominator<\/div>\n<p><strong data-effect=\"bold\">Rationalizing the denominator<\/strong> is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.<\/p>\n<\/div>\n<p id=\"fs-id1169149297227\">Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a radical.<\/p>\n<p id=\"fs-id1169148939113\">Similarly, a <span data-type=\"term\" class=\"no-emphasis\">radical expression<\/span> is not considered simplified if the radicand contains a fraction.<\/p>\n<div data-type=\"note\" id=\"fs-id1169149172275\">\n<div data-type=\"title\">Simplified Radical Expressions<\/div>\n<p id=\"fs-id1169149136595\">A radical expression is considered simplified if there are<\/p>\n<ul id=\"fs-id1169149326602\" data-bullet-style=\"bullet\">\n<li>no factors in the radicand have perfect powers of the index<\/li>\n<li>no fractions in the radicand<\/li>\n<li>no radicals in the denominator of a fraction<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1169149367820\">To rationalize a denominator with a square root, we use the property that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-832dfd9c4ffcaa657d4337721d47fe6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"81\" style=\"vertical-align: -4px;\" \/> If we square an irrational square root, we get a rational number.<\/p>\n<p id=\"fs-id1169149114181\">We will use this property to rationalize the denominator in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1169146612955\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149087434\">\n<div data-type=\"problem\" id=\"fs-id1169148998388\">\n<p id=\"fs-id1169149029418\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e5f411f7cd463db5ac1ca7263245e1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#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-eca13c52f38daae7728dbc902b412107_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#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-1ada2cc2e0e01d3a46ddc567ac1e52ad_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;&#123;&#54;&#120;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"32\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148869202\">\n<p id=\"fs-id1169148858445\">To rationalize a denominator with one term, we can multiply a square root by itself. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.<\/p>\n<p id=\"fs-id1169149017598\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1169149346904\" class=\"unnumbered unstyled can-break\" summary=\"To rationalize the denominator of 4 divided by square root 3 we multiply both the numerator and denominator by square root 3. The result is the 4 times square root 3 divided by the quantity square root 3 times square root 3 in parentheses. Simplifying we get 4 times square root 3 divided by 3.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146838836\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply both the numerator and denominator by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aac479c0f329133abffc4313452bca95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -2px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149312218\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149214546\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169148881651\"><span class=\"token\">\u24d1<\/span> We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.<\/p>\n<table id=\"fs-id1169146732575\" class=\"unnumbered unstyled can-break\" summary=\"With the quantity 3 divided by 20 in parentheses we first notive that the fraction is not a perfect square, so we rewrite using the quotient property to get square root 3 divided by square root 20. Simplifying the denominator we get square root 3 divided by the quantity 2 times square root 5 in parentheses. To rationalize the denominator we multiply the numerator and denominator by square root 5. This is written as square root 3 times square root 5 divided by the quantity 2 square root 5 times square root 5 in parentheses. Simplifying we get square root 15 divided by the quantity 2 times 5 in parentheses. Simplifying furthere we get square root 15 divided by 10.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149217996\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The fraction is not a perfect square, so rewrite using the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Quotient Property.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149116163\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149292163\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by <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;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148890540\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149001128\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_002e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147085588\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_002f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169149346365\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1169149023603\" class=\"unnumbered unstyled can-break\" summary=\"To rationalize the denominator of 3 divided by square root of the quantity 6 x in parentheses we multiply both the numerator and denominator by square root of the quantity 6 x in parentheses. This is written out as 3 times square root of the quantity 6 x in parentheses divided by the quantity square root 6 x times square root 6 x in parentheses. The result is 3 times square root of the quantity 6 x in parentheses divided by the quantity 6 x in parentheses. Simplifying we square root of the quantity 6 x in parentheses divided by the quantity 2 x.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146645267\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-245423f463c480596cdb1f92c2bdd53b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -2px;\" \/>\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148973792\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149302939\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148984763\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_003d_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-id1169149029816\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169144567202\">\n<div data-type=\"problem\" id=\"fs-id1169148932927\">\n<p id=\"fs-id1169148967517\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-201a67903518b1af4b03fdfe895a8bde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#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-a4b0dea232a5e8f91990023c45112e62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#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-e6bed7a54502ac7c8fe32999f08c24d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"32\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169147086760\">\n<p id=\"fs-id1169144745469\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97c5aaa9541c61547de4e51b057c8427_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" 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-d485848bd799c754079e76687b20ed05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"19\" 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-e5907927b0fcfed4018edafd9415ccf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#125;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"27\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149095729\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149156285\">\n<div data-type=\"problem\" id=\"fs-id1169149219746\">\n<p id=\"fs-id1169148912377\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a1db363220a97b31d9e16819b5e4b26_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;&#123;&#53;&#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-215a1b04dddbc3fb5257f8c98a6e0274_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;&#49;&#56;&#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-5b4b5ff4b4821789a62b1fef75eed132_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#120;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"32\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149036848\">\n<p id=\"fs-id1169149099890\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4fc19c9767e35ca1155b8a8a3f2e31f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" 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-7af1c1b220d4505f2df11679160ff6ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#52;&#125;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" 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-f914f2f87910a11b6b6ef49db2cdf123_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#120;&#125;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"27\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148883936\">When we rationalized a square root, we multiplied the numerator and denominator by a square root that would give us a perfect square under the radical in the denominator. When we took the square root, the denominator no longer had a radical.<\/p>\n<p id=\"fs-id1169149026920\">We will follow a similar process to rationalize higher roots. To rationalize a denominator with a higher index radical, we multiply the numerator and denominator by a radical that would give us a radicand that is a perfect power of the index. When we simplify the new radical, the denominator will no longer have a radical.<\/p>\n<p id=\"fs-id1169149100260\">For example,<\/p>\n<p><span data-type=\"media\" id=\"fs-id1169148843962\" data-alt=\"Two examples of rationalizing denominators are shown. The first example is 1 divided by cube root 2. A note is made that the radicand in the denominator is 1 power of 2 and that we need 2 more to get a perfect cube. We multiply numerator and denominator by the cube root of the quantity 2 squared. The result is cube root 4 divided by cube root of quantity 2 cubed. This simplifies to cube root 4 divided by 2. The second example is 1 divided by fourth root 5. A note is made that the radicand in the denominator is 1 power of 5 and that we need 3 more to get a perfect fourth. We multiply numerator and denominator by the fourth root of the quantity 5 cubed. The result is fourth root of 125 divided by fourth root of quantity 5 to the fourth. This simplifies to fourth root 125 divided by 5.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Two examples of rationalizing denominators are shown. The first example is 1 divided by cube root 2. A note is made that the radicand in the denominator is 1 power of 2 and that we need 2 more to get a perfect cube. We multiply numerator and denominator by the cube root of the quantity 2 squared. The result is cube root 4 divided by cube root of quantity 2 cubed. This simplifies to cube root 4 divided by 2. The second example is 1 divided by fourth root 5. A note is made that the radicand in the denominator is 1 power of 5 and that we need 3 more to get a perfect fourth. We multiply numerator and denominator by the fourth root of the quantity 5 cubed. The result is fourth root of 125 divided by fourth root of quantity 5 to the fourth. This simplifies to fourth root 125 divided by 5.\" \/><\/span><\/p>\n<p id=\"fs-id1169146626332\">We will use this technique in the next examples.<\/p>\n<div data-type=\"example\" id=\"fs-id1169149178429\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169146631585\">\n<div data-type=\"problem\" id=\"fs-id1169147088334\">\n<p id=\"fs-id1169148972794\">Simplify <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90027f78667b5610b9c91686e8b5e814_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;&#54;&#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-6446c7a99fd9f88e459ad04cbb57354e_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;&#50;&#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-a0450d7bc3c67fdda2e4c6b05a52660e_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;&#52;&#120;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"34\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146612334\">\n<p id=\"fs-id1169144380220\">To rationalize a denominator with a cube root, we can multiply by a cube root that will give us a perfect cube in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.<\/p>\n<p id=\"fs-id1169148884074\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1169149313679\" class=\"unnumbered unstyled\" summary=\"The example is 1 divided by cube root 6. The radicand in the denominator is 1 factor of 6. Multiplying both the numerator and denominator by cube root of quantity 6 squared gives us 2 more factors of 6. The result is cube root of the quantity 6 squared in parentheses divided by cube root of quantity 6 cubed. Notice the radicand in the denominator has 3 powers of 6. This simplifies to cube root 36 divided by 6.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149358452\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The radical in the denominator has one factor of 6.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Multiply both the numerator and denominator by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf841298230a7fc70b8f864c8c709689_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#123;&#54;&#125;&#94;&#123;&#50;&#125;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"34\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>which gives us 2 more factors of 6.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149007127\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply. Notice the radicand in the denominator<\/p>\n<div data-type=\"newline\"><\/div>\n<p>has 3 powers of 6.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149114418\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the cube root in the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148995146\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169148821019\"><span class=\"token\">\u24d1<\/span> We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.<\/p>\n<table id=\"fs-id1169148820343\" class=\"unnumbered unstyled can-break\" summary=\"The example is cube root of the quantity 7 divided by 24. The fraction is not a perfect cube so rewrite using the quotient property. The new expression is cube root 7 divided by cube root 24. Simplifying the denominator gives cube root 7 divided by the quantity 2 cube root 3. Multiply the numerator and denominator by cube root quantity 3 squared. This will give us 3 factors of 3. This is written as cube root 7 times cube root quantity 3 squared in parentheses divided by the quantity 2 cube root 3 cube root 3 squared in parentheses. Simplifying we get cube root 63 divided by the quantity 2 cube root quantity 3 cubed in parentheses. Remember that cube root quantity 3 cubed equals 3. This gives cube root 63 divided by quantity 2 times 3. Simplifying once more we get cube root 63 divided by 6.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144421220\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The fraction is not a perfect cube, so<\/p>\n<div data-type=\"newline\"><\/div>\n<p>rewrite using the Quotient Property.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144601623\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148825672\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator\u2003\u2003\u2003\u2003\u2003\u2003\u2003<\/p>\n<div data-type=\"newline\"><\/div>\n<p>by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9a99b707a7bc8e2ab6c0c5f0f06e31b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -1px;\" \/> This will give us 3 factors of 3.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148918570\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Remember, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fd96bd7af68651cfe496f6cf93bcc56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#123;&#51;&#125;&#94;&#123;&#51;&#125;&#125;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144715806\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149030330\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_006g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169146917946\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1169144517151\" class=\"unnumbered unstyled\" summary=\"The example is 3 divided by cube root of the quantity 4 x. Rewrite the radical to show the factors. The new expression is 3 divided by cube root of the quantity 2 squared times x in parentheses. Multiply the numerator and denominator by cube root quantity 2 x squared in parentheses. This will give us 3 factors of 2 and 3 factors of x. This is written as 3 times cube root quantity 2 x squared in parentheses divided by the quantity cube root quantity 2 x squared times cube root quantity 2 x squared in parentheses . Simplifying we get 3 times cube root quantity 2 x squared in parentheses divided by cube root of the quantity 2 cubed times x cubed in parentheses. Simplifying the radical in the denominator we get 3 cube root quantity 2 x squared in parentheses divided by the quantity 2 x.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144876452\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite the radicand to show the factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149110142\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4f68752c390960552e6f8bfc8198513_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&middot;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>This will get us 3 factors of 2 and 3 factors of <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169138971669\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146815185\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the radical in the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148937504\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_007e_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-id1169148968862\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148837048\">\n<div data-type=\"problem\" id=\"fs-id1169147028800\">\n<p id=\"fs-id1169146737825\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80728308e40dd0904683773ddb04e5a0_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;&#55;&#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-f2fee185e6d7a3df78025eb74f57c433_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;&#53;&#125;&#123;&#49;&#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-3af817e7fc3018353296db81af3e1d12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#57;&#121;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"34\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1169146719598\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81f7002adf947b3d771e924954827c1d_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;&#52;&#57;&#125;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"28\" 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-b1160e593a02165af96eae9658b2bae1_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;&#48;&#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-ca678c04b598e043e2d264e5fcad7443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#51;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"45\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148821337\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169144768704\">\n<div data-type=\"problem\" id=\"fs-id1169149102495\">\n<p id=\"fs-id1169144606258\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9f4b50992e61289766fcac3b3d7bb00_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;&#50;&#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-c6492a4e8282f74395cbe8fb36af015c_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;&#51;&#125;&#123;&#50;&#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-7fd089d52680f5bbd6ef6a9068712a55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#53;&#110;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"42\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149093039\">\n<p id=\"fs-id1169148879748\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb4850a3fee8031ec009b32038f776d5_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;&#52;&#125;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"21\" 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-0873a9d575fd33a791650ef97eaef6cc_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;&#53;&#48;&#125;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"35\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-559543e7f6ed384c6d071494df81dfec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#53;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1169148944647\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149149549\">\n<div data-type=\"problem\">\n<p id=\"fs-id1169149309461\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f93e9bf5dd2449c1548c57adab4523f_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;&#50;&#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-8b8e49a9e1779f2d8c4a6b887bd03bc0_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;&#53;&#125;&#123;&#54;&#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-c26dfc98112332f8a666b2b2759c6885_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#56;&#120;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"34\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149037326\">\n<p id=\"fs-id1169149102350\">To rationalize a denominator with a fourth root, we can multiply by a fourth root that will give us a perfect fourth power in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.<\/p>\n<p id=\"fs-id1169148934751\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1169149304217\" class=\"unnumbered unstyled can-break\" summary=\"The example is 1 divided by fourth root 2. The radicand in the denominator has 1 factor of 2. Multiply both the numerator and denominator by fourth root quantity 2 cubed, which gives 3 more factors of 2. This is written as 1 times fourth root quantity 2 cubed divided by the quantity fourth root quantity 2 cubed times fourth root quantity 2 cubed in parentheses. Multiplying we get fourth root 8 divided by fourth root quantity 2 to the fourth. Notice the radicand in the denominator has 4 powers of 2. Simplifying the fourth root in the denominator results in fourth root 8 divided by 2.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148869643\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The radical in the denominator has one factor of 2.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Multiply both the numerator and denominator by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e9bc3ea1f1bd3650a3691ed7f20b34a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#123;&#50;&#125;&#94;&#123;&#51;&#125;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"34\" style=\"vertical-align: -4px;\" \/>\u2003\u2003\u2003<\/p>\n<div data-type=\"newline\"><\/div>\n<p>which gives us 3 more factors of 2.<\/td>\n<td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146652418\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply. Notice the radicand in the denominator<\/p>\n<div data-type=\"newline\"><\/div>\n<p>has 4 powers of 2.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148837591\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the fourth root in the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149109247\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169148990220\"><span class=\"token\">\u24d1<\/span> We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.<\/p>\n<table id=\"fs-id1169146669491\" class=\"unnumbered unstyled can-break\" summary=\"The example is fourth root of the quantity 5 divided by 64. The fraction is not a perfect cube so rewrite using the quotient property. The new expression is fourth root 5 divided by fourth root quantity 2 to the sixth. Simplifying the denominator gives fourth root 5 divided by the quantity 2 fourth root quantity 2 squared. Multiply the numerator and denominator by fourth root quantity 2 squared. This will give us 4 factors of 2. This is written as fourth root 5 times fourth root quantity 2 squared in parentheses divided by the quantity 2 fourth root 2 squared times fourth root 2 squared in parentheses. Simplifying we get fourth root 20 divided by the quantity 2 fourth root quantity 2 to the fourth in parentheses. Remember that fourth root quantity 2 to the fourth equals 2. This gives fourth root 20 divided by quantity 2 times 2. Simplifying once more we get fourth root 20 divided by 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148943899\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The fraction is not a perfect fourth power, so rewrite<\/p>\n<div data-type=\"newline\"><\/div>\n<p>using the Quotient Property.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144366838\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite the radicand in the denominator to show the factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149370876\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144885732\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1fb4e9f7dd239cbe79509c5495b32d04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>This will give us 4 factors of 2.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148838178\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146742186\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Remember, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c08e21f6417b6774a86c250d7b4e266d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#123;&#50;&#125;&#94;&#123;&#52;&#125;&#125;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144400096\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144365491\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_009h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169149112743\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1169149329077\" class=\"unnumbered unstyled can-break\" summary=\"The example is 2 divided by fourth root of the quantity 8 x. Rewrite the radical to show the factors. The new expression is 2 divided by fourth root of the quantity 2 cubed times x in parentheses. Multiply the numerator and denominator by fourth root quantity 2 x cubed in parentheses. This will give us 4 factors of 2 and 4 factors of x. This is written as 2 times fourth root quantity 2 x cubed in parentheses divided by the quantity fourth root quantity 2 x cubed times fourth root quantity 2 x cubed in parentheses . Simplifying we get 2 times fourth root quantity 2 x cubed in parentheses divided by fourth root of the quantity 2 to the fourth times x to the fourth in parentheses. Simplifying the radical in the denominator we get 2 fourth root quantity 2 x cubed in parentheses divided by the quantity 2 x. Simplifying the fraction results in fourth root quantity 2 x cubed in parentheses divided by x.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149016796\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite the radicand to show the factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149064900\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f0507a5d96563301fcba1aaf99b74a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#50;&middot;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -1px;\" \/>\u2003\u2003\u2003<\/p>\n<div data-type=\"newline\"><\/div>\n<p>This will get us 4 factors of 2 and 4 factors of <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147028339\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144451038\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the radical in the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149005582\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the fraction.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148971366\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_010f_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-id1169149116201\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149006904\">\n<div data-type=\"problem\" id=\"fs-id1169144563844\">\n<p id=\"fs-id1169149344287\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de0e1a03893cad2eca4db45230ec237e_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;&#51;&#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-6296897f71bf8737be319db48e135e13_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;&#51;&#125;&#123;&#54;&#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-204d14457317ec0af47f8135b17c9438_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;&#52;&#93;&#123;&#49;&#50;&#53;&#120;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"48\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146699399\">\n<p id=\"fs-id1169149294726\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3e7de1e28a9d38fc5672a10f26f4381_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;&#50;&#55;&#125;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"28\" 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-9d10d7cbb58ec02cdb8d8bd4f856042d_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;&#49;&#50;&#125;&#125;&#123;&#52;&#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-1018c574da00aaf0eca5052540a18d0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#53;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#125;&#125;&#123;&#53;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144744512\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148890606\">\n<div data-type=\"problem\" id=\"fs-id1169149369284\">\n<p id=\"fs-id1169148837658\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-791033c6a51e61b6e5bfd0d5fc706f99_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;&#53;&#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-fe505ee17a233118443e83efe381fd3f_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;&#55;&#125;&#123;&#49;&#50;&#56;&#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-05389fc338fdecd7218ab9dd73986774_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#52;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"29\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149025360\">\n<p id=\"fs-id1169147086622\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca7c42aa6f42286c9b9f245d64889720_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;&#49;&#50;&#53;&#125;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"35\" 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-e53b15de90fcf23256c1a8ad946d5688_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;&#50;&#50;&#52;&#125;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"35\" 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-f9112f6aa866bd26ddd55e3d2dec76d0_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;&#54;&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#125;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169148935104\">\n<h3 data-type=\"title\">Rationalize a Two Term Denominator<\/h3>\n<p id=\"fs-id1169149064766\">When the denominator of a fraction is a sum or difference with square roots, we use the <span data-type=\"term\" class=\"no-emphasis\">Product of Conjugates Pattern<\/span> to <span data-type=\"term\" class=\"no-emphasis\">rationalize the denominator<\/span>.<\/p>\n<div data-type=\"equation\" id=\"fs-id1169148879894\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-accb4f0febbe0f743c2bbecd51de5703_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;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#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;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#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;&#98;&#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;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#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;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#45;&#53;&#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;&#52;&#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=\"86\" width=\"371\" style=\"vertical-align: -36px;\" \/><\/div>\n<p id=\"fs-id1169147087680\">When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.<\/p>\n<div data-type=\"example\" id=\"fs-id1169146812808\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148994436\">\n<div data-type=\"problem\" id=\"fs-id1169146848618\">\n<p id=\"fs-id1169149285866\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0504c3a56fc1a0ba69f75c28e4751782_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"42\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148918408\">\n<table id=\"fs-id1169149155739\" class=\"unnumbered unstyled can-break\" summary=\"To rationalize the denominator in 5 divided by the quantity 2 minus square root 3 in parentheses we multiply the numerator and denominator by the conjugate of the denominator. This is written as 5 times the quantity 2 plus square root 3 in parentheses divided by the product of the quantity 2 minus square root 3 in parentheses with the quantity 2 plus square root 3 in parentheses. Muliplying the conjugates in the denominator results in 5 times the quantity 2 plus square root 3 in parentheses divided by the difference of 2 squared and the quantity square root 3 squared. Simplifying the denominator gives 5 times the quantity 2 plus square root 3 in parentheses divided by the quantity 4 minus 3 in parentheses. This simplifies to 5 times the quantity 2 plus square root 3 in parentheses.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148971687\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>conjugate of the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149302426\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the conjugates in the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148868647\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149213386\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_011d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149001659\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_011e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148994799\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_011f_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-id1169148875850\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169146607241\">\n<div data-type=\"problem\" id=\"fs-id1169149004978\">\n<p id=\"fs-id1169144566191\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-946c433552d52f1cddf056aa5181a247_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"42\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149223924\">\n<p id=\"fs-id1169148930197\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c42880504dac59b7a585d854af14c64e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149291553\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148866919\">\n<div data-type=\"problem\" id=\"fs-id1169149308919\">\n<p id=\"fs-id1169148959569\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37d68fb87b7777a159b1ac23657adbbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#52;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"42\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146623122\">\n<p id=\"fs-id1169147089653\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa2a248bf6ec087ccab35f72486e7ae9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"36\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149347057\">Notice we did not distribute the 5 in the answer of the last example. By leaving the result factored we can see if there are any factors that may be common to both the numerator and denominator.<\/p>\n<div data-type=\"example\" id=\"fs-id1169149346571\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169146637455\">\n<div data-type=\"problem\" id=\"fs-id1169149032875\">\n<p id=\"fs-id1169146750688\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fefa975e253c47b9ebbe8add7ff6fc31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#117;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"54\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148883988\">\n<table id=\"fs-id1169144893528\" class=\"unnumbered unstyled can-break\" summary=\"To rationalize the denominator in square root 3 divided by the quantity square root u minus square root 6 in parentheses we multiply the numerator and denominator by the conjugate of the denominator. This is written as square root 3 times the quantity square root u plus square root 6 in parentheses divided by the product of the quantity square root u minus square root 6 in parentheses with the quantity square root u plus square root 6 in parentheses. Muliplying the conjugates in the denominator results in square root 3 times the quantity square root u plus square root 6 in parentheses divided by the difference of square root u squared and square root 6 squared. Simplifying the denominator gives square root 3 times the quantity square root u plus square root 6 in parentheses divided by the quantity u minus 6 in parentheses. Simplifying the numerator results in the difference of square root quantity 3 u in parentheses and 3 square root 2 divided by the quantity u minus 6.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149219717\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_012a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>conjugate of the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144419340\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the conjugates in the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146947408\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149315736\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_012d_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-id1169144381649\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149114706\">\n<div data-type=\"problem\" id=\"fs-id1169148974522\">\n<p id=\"fs-id1169148868702\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ce38c3b3517be34f76f3bb4a29a0a60_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;&#120;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"54\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144873576\">\n<p id=\"fs-id1169149178345\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-129267eaaaabed5313c4d1b4ffde739a_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;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#120;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"81\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149000052\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148867898\">\n<div data-type=\"problem\" id=\"fs-id1169148964598\">\n<p id=\"fs-id1169149031341\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1db4ebc2f74e1dca3b70f2c1ee80e430_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#121;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"54\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149197086\">\n<p id=\"fs-id1169149037952\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69fa4d11d9046eab92dae440eacd2e53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#121;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#121;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"87\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165926512140\">Be careful of the signs when multiplying. The numerator and denominator look very similar when you multiply by the conjugate.<\/p>\n<div data-type=\"example\" id=\"fs-id1169149345703\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149108831\">\n<div data-type=\"problem\" id=\"fs-id1169148992913\">\n<p id=\"fs-id1169146644820\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b8a0e985bd88fe3a6a80cbfbff55e6c_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;&#55;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"54\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144560259\">\n<table id=\"fs-id1169146606154\" class=\"unnumbered unstyled can-break\" summary=\"To rationalize the denominator in the quantity square root x plus square root 7 in parentheses divided by the quantity square root x minus square root 7 in parentheses we multiply the numerator and denominator by the conjugate of the denominator. This is written as the quantity square root x plus square root 7 in parentheses times the quantity square root x plus square root 7 in parentheses divided by the product of the quantity square root x minus square root 7 in parentheses with the quantity square root x plus square root 7 in parentheses. Muliplying the conjugates in the denominator results in the quantity square root x plus square root 7 in parentheses times the quantity square root x plus square root 7 in parentheses divided by the difference of square root x squared and square root 7 squared. Simplifying the denominator gives the quantity square root x plus square root 7 in parentheses squared divided by the quantity x minus 7 in parentheses.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144484645\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerator and denominator by the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>conjugate of the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149306398\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the conjugates in the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149015016\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify the denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149213947\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169149314052\">We do not square the numerator. Leaving it in factored form, we can see there are no common factors to remove from the numerator and denominator.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144522987\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147054334\">\n<div data-type=\"problem\" id=\"fs-id1169144484785\">\n<p id=\"fs-id1169148890263\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2fc80f27314fc761d83889f83594981_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#112;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#112;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"53\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148991499\">\n<p id=\"fs-id1169144422081\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7c12558aacf9056c8bb4c86366beaff7_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;&#112;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#112;&#45;&#50;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"70\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148971696\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148968242\">\n<div data-type=\"problem\" id=\"fs-id1169148938406\">\n<p id=\"fs-id1169144683706\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd6b805f9aee5ad8f645ec368c4b0b42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#113;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#113;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"55\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149204814\">\n<p id=\"fs-id1169146620540\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b90d8fa99ed0e60dfe48292e792a5f8_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;&#113;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#113;&#45;&#49;&#48;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"77\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144769214\" class=\"media-2\">\n<p id=\"fs-id1169148997642\">Access these online resources for additional instruction and practice with dividing radical expressions.<\/p>\n<ul id=\"fs-id1169148992311\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37RatDenom1\">Rationalize the Denominator<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37RatDenom2\">Dividing Radical Expressions and Rationalizing the Denominator<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37RatDenom3\">Simplifying a Radical Expression with a Conjugate<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37RatDenom4\">Rationalize the Denominator of a Radical Expression<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169148984716\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1169146739168\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Quotient Property of Radical Expressions<\/strong>\n<ul id=\"fs-id1169144685624\" data-bullet-style=\"bullet\">\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c6b7326bec5fc086e8335bb7e382f30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-489e8ff9827ba6988dfa32390ed30650_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"21\" style=\"vertical-align: -2px;\" \/> are real numbers, <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;\" \/> and for any integer <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5482ac64c45006c6548c713888a5d34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#103;&#101;&#32;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"42\" style=\"vertical-align: -3px;\" \/> 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-993437a43b9b8fe9ee825a6bf5b41a7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#97;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#98;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"76\" style=\"vertical-align: -11px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0adc894935a48ac2937673ac0a02d09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#97;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#98;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"80\" style=\"vertical-align: -11px;\" \/><\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Simplified Radical Expressions<\/strong>\n<ul id=\"fs-id1169144567375\" data-bullet-style=\"bullet\">\n<li>A radical expression is considered simplified if there are:\n<ul id=\"fs-id1169148880634\" data-bullet-style=\"bullet\">\n<li>no factors in the radicand that have perfect powers of the index<\/li>\n<li>no fractions in the radicand<\/li>\n<li>no radicals in the denominator of a fraction<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169139949101\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169149157492\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1169144451023\"><strong data-effect=\"bold\">Divide Square Roots<\/strong><\/p>\n<p id=\"fs-id1169149312167\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169148869456\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169146666523\">\n<p id=\"fs-id1169149193736\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a76092dea82d9402f2f20e8b2fe70be3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#56;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"33\" 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-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 data-type=\"solution\" id=\"fs-id1169144601274\">\n<p id=\"fs-id1169148964069\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d50443e9b625e846a8f38e891bba33d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#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-0d50443e9b625e846a8f38e891bba33d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#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-id1169148951118\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169144874488\">\n<p id=\"fs-id1169146644476\"><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;\" \/><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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149196629\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169146744588\">\n<p id=\"fs-id1169144768763\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae36cfc16cf6479403ffa014112decc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#48;&#48;&#123;&#109;&#125;&#94;&#123;&#53;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#56;&#109;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"52\" 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-5fd8facd215fc8ccf628fe764d4b6ed5_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;&#53;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#123;&#121;&#125;&#94;&#123;&#53;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"45\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149240580\">\n<p id=\"fs-id1169146621806\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c4ca02dbb54260b1282b0e543a56fd32_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;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"33\" 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-66e31551db10647b0a630e067487830e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"8\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149287043\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169146731710\">\n<p id=\"fs-id1169149016261\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-577b5d4954e3727621f7a5fd8e16fd5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#56;&#123;&#110;&#125;&#94;&#123;&#55;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#51;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"48\" 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-41cd09dd07f097b4b2cbb1b3b2db366d_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;&#53;&#52;&#123;&#121;&#125;&#94;&#123;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"45\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146740915\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169149113858\">\n<p id=\"fs-id1169144873851\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8aa168d7618f046e5590b438e60efce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#53;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#56;&#123;&#114;&#125;&#94;&#123;&#55;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"46\" 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-1e8151cfdf335a9c66d30ea44ac08f1d_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;&#52;&#123;&#120;&#125;&#94;&#123;&#55;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#56;&#49;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"43\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144550544\">\n<p id=\"fs-id1169144686560\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-344893e197af068b008c2da37208e3a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"21\" 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-60b66c089abbc195db6cc97bab15c674_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#120;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"15\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148970327\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169149065505\">\n<p id=\"fs-id1169149294580\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1533cbedc02957bb64793a4d1aee32c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#57;&#54;&#123;&#113;&#125;&#94;&#123;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#56;&#52;&#123;&#113;&#125;&#94;&#123;&#53;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"49\" style=\"vertical-align: -15px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29a015ff666db0562c1c562925b02e1c_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;&#109;&#125;&#94;&#123;&#52;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#52;&#123;&#109;&#125;&#94;&#123;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"47\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149328074\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169146664810\">\n<p id=\"fs-id1169148955565\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0849b6364156487368acd1ee03a6ab8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#56;&#123;&#112;&#125;&#94;&#123;&#53;&#125;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#123;&#113;&#125;&#94;&#123;&#54;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"63\" style=\"vertical-align: -15px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a49610eba93f25c0f4021b749a34a2b7_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;&#49;&#54;&#123;&#97;&#125;&#94;&#123;&#52;&#125;&#123;&#98;&#125;&#94;&#123;&#45;&#50;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#123;&#97;&#125;&#94;&#123;&#45;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"75\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146663944\">\n<p id=\"fs-id1169148956592\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd5fc890206f10d6042e3d151fa222a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#112;&#125;&#123;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"14\" style=\"vertical-align: -10px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1fb44f377612fc41ee7c76af613e45d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"36\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149339448\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169149339450\">\n<p id=\"fs-id1169148956040\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b001cfeeef97169233553c6ac31726ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#56;&#114;&#123;&#115;&#125;&#94;&#123;&#49;&#48;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#123;&#115;&#125;&#94;&#123;&#52;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"51\" 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-7c8bee9bd1686b2e432ec6e94ba6d245_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;&#51;&#55;&#53;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#123;&#122;&#125;&#94;&#123;&#45;&#50;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#123;&#121;&#125;&#94;&#123;&#45;&#50;&#125;&#123;&#122;&#125;&#94;&#123;&#52;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"86\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144565587\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169144565590\">\n<p id=\"fs-id1169149042640\"><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;\" \/><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;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149328417\">\n<p id=\"fs-id1169148995445\"><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-3c27b41251b0d105b3d5e99ad4aa5064_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"37\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148880013\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169148992184\">\n<p id=\"fs-id1169148992187\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a713e2c3e41f12c62640c165f90832b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#49;&#48;&#123;&#99;&#125;&#94;&#123;&#45;&#51;&#125;&#123;&#100;&#125;&#94;&#123;&#55;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#48;&#48;&#123;&#99;&#125;&#94;&#123;&#125;&#123;&#100;&#125;&#94;&#123;&#45;&#49;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"69\" 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-27efca66cfbe81456bfec57c539c4e5a_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;&#52;&#123;&#97;&#125;&#94;&#123;&#55;&#125;&#123;&#98;&#125;&#94;&#123;&#45;&#49;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#45;&#56;&#49;&#123;&#97;&#125;&#94;&#123;&#45;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"75\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144614783\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169144614785\">\n<p id=\"fs-id1169149013905\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-498a84e0167d9154b7d896a509a66be9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#54;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"58\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146643067\">\n<p id=\"fs-id1169149214011\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67cecd17cdc9a8c13db14e5bb72bf763_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146667820\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169146667822\">\n<p id=\"fs-id1169149065808\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1984f9efd05ebc68109d663ccdfadb85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#50;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#123;&#98;&#125;&#94;&#123;&#54;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#97;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"52\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149109868\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169149109870\">\n<p id=\"fs-id1169148957157\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-636fa5eeb9d67a7ce76afed95bdf56a3_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;&#52;&#56;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#123;&#98;&#125;&#94;&#123;&#54;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#123;&#97;&#125;&#94;&#123;&#45;&#49;&#125;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"57\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149350593\">\n<p id=\"fs-id1169149118426\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b73531798c7dafed04d5e406bce6b844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#97;&#98;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149280925\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1169149280927\">\n<p id=\"fs-id1169148993218\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0d2cfc57ef217bf43c05b729bedee83_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;&#50;&#123;&#120;&#125;&#94;&#123;&#45;&#51;&#125;&#123;&#121;&#125;&#94;&#123;&#54;&#125;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#123;&#121;&#125;&#94;&#123;&#45;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"76\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148969800\"><strong data-effect=\"bold\">Rationalize a One Term Denominator<\/strong><\/p>\n<p id=\"fs-id1169146841907\">In the following exercises, rationalize the denominator.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149012708\">\n<div data-type=\"problem\" id=\"fs-id1169149012710\">\n<p id=\"fs-id1169144550856\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4515678d2da24bfc96a31c9628d4e724_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#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-f3bfb4a0f79d1ac4621d0169b0b64ca9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#50;&#55;&#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-b448c19be7a34b562c663fccc34fe663_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"27\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149348903\">\n<p id=\"fs-id1169149153399\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc9c6a85e4abeaa029adc78d24babc4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" 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-d73d02dd9916dfeb901b81ce7c349b47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" 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-7652747110fd60be61d618a5260e7218_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#120;&#125;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"33\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149316475\">\n<div data-type=\"problem\" id=\"fs-id1169149361122\">\n<p id=\"fs-id1169149361124\"><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-id1169148996426\">\n<div data-type=\"problem\" id=\"fs-id1169148996428\">\n<p id=\"fs-id1169149340430\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d9d567994ef075bf7d94acab0c020c9_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;&#123;&#55;&#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-709aa09cdfa34798f6332d3b13ba90c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#52;&#53;&#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-5f1ea455b6afb1fc910c0ce2a79ddcfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#112;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"26\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144605586\">\n<p id=\"fs-id1169144605588\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26160643f3ce2ee28da2e42b0a007f9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"26\" 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-8faafff4457d8f55c3be0df266668340_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#125;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"33\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4a9001827b45657658ba84c56c7ec09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#112;&#125;&#125;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"33\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144523288\">\n<div data-type=\"problem\" id=\"fs-id1169149004586\">\n<p id=\"fs-id1169149004588\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31bbe9d96ec1f2d169ba3d1e620d5fe7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#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-034550d3d1f099033c2382603bce45ad_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;&#125;&#123;&#56;&#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-757bde5042afe9f40c0533e686fab963_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#113;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"25\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144796674\">\n<div data-type=\"problem\" id=\"fs-id1169144796676\">\n<p id=\"fs-id1169149156704\"><span class=\"token\">\u24d0<\/span><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;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a38ef76bc8f752fc3861a0a584446dbb_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;&#53;&#125;&#123;&#50;&#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-f5887afe77d4d7d8fe0c9535bed43da4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#54;&#97;&#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-id1169144381822\">\n<p id=\"fs-id1169144381824\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07760e00b6af95928c7383edee0f85e7_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;&#53;&#125;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"28\" 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-fde8fa9513409ad27efa7f010c355af4_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;&#52;&#53;&#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-f0c862a66f7e94e5f29e9039f1d8ef09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#51;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149303500\">\n<div data-type=\"problem\" id=\"fs-id1169149303502\">\n<p id=\"fs-id1169146738463\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-995cbe931379f8029de9a7f8f600e9a4_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;&#51;&#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-bfaefd2dfec104cb4d1c849e4ac29067_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;&#53;&#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-1c7d6895a26b5d989bdf3f50dcc614ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#57;&#98;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"34\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149367796\">\n<div data-type=\"problem\" id=\"fs-id1169149367798\">\n<p id=\"fs-id1169149367801\"><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-id1169149224185\">\n<p id=\"fs-id1169149003529\"><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-id1169149210154\">\n<div data-type=\"problem\" id=\"fs-id1169149210156\">\n<p id=\"fs-id1169146659953\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14c99c5e373231ed00ff35b1570aab7d_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;&#51;&#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-fbe72c72ed1de6d751c4331dbcc975cc_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;&#51;&#125;&#123;&#49;&#50;&#56;&#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-264b7cee5722db8fdb8d2ca207b227d9_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;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"38\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144835169\">\n<div data-type=\"problem\" id=\"fs-id1169144835172\">\n<p id=\"fs-id1169148969699\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da2074a1c0cad4d60c1fe21275c2f599_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;&#55;&#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-6b1387c4984e1b284cca71fe3f06a736_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;&#53;&#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-b679074065de036c71fd3e5d76b8d958_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#52;&#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-id1169149351259\">\n<p id=\"fs-id1169144422013\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-119d5e49ee18ba0078c096892a242d52_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;&#51;&#52;&#51;&#125;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"35\" 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-d3ede5e97dd7043c42458ed7061bb19d_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;&#52;&#48;&#125;&#125;&#123;&#52;&#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-d20b3718fd3069ac473583663a21b0a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149376982\">\n<div data-type=\"problem\" id=\"fs-id1169149376984\">\n<p id=\"fs-id1169149376986\"><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<div data-type=\"exercise\" id=\"fs-id1169148962629\">\n<div data-type=\"problem\" id=\"fs-id1169148962631\">\n<p id=\"fs-id1169148969883\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae290f85de030961151410ba85be9a8b_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;&#57;&#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-6e83334c10e6a2a5520a068fafb65fb6_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;&#50;&#53;&#125;&#123;&#49;&#50;&#56;&#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-f61cba5fca78d29d715b78b858925730_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;&#50;&#55;&#97;&#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-id1169144567993\">\n<p id=\"fs-id1169144567995\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f50630356aac2cd6dcb69c21e1f28de8_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;&#57;&#125;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"21\" 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-93131af4ca27f69d2b5f1fbca4fdbdf0_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;&#53;&#48;&#125;&#125;&#123;&#52;&#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-4f92449908bad72cf55b2dbf980e5c38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#51;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149011945\">\n<div data-type=\"problem\" id=\"fs-id1169144686266\">\n<p id=\"fs-id1169144686269\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec07073017a6f31a29de1affc594b1a1_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;&#56;&#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-fd759ef01953d082193e56a83a2671d2_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;&#50;&#55;&#125;&#123;&#49;&#50;&#56;&#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-5c6ddb0d277b24f862a31ef866d7c9b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#54;&#52;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"41\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169144784948\"><strong data-effect=\"bold\">Rationalize a Two Term Denominator<\/strong><\/p>\n<p id=\"fs-id1169149189450\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149290498\">\n<div data-type=\"problem\" id=\"fs-id1169149290500\">\n<p id=\"fs-id1169149290502\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8769c6c09b6820913473126a2e5f2450_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#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-id1169149183950\">\n<p id=\"fs-id1169149183952\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fbd4328729617d8ad4edab578221f935_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"93\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149066158\">\n<div data-type=\"problem\" id=\"fs-id1169149066160\">\n<p id=\"fs-id1169149373773\"><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>\n<div data-type=\"exercise\" id=\"fs-id1169146632382\">\n<div data-type=\"problem\" id=\"fs-id1169146632384\">\n<p id=\"fs-id1169144374242\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5705b12afbd4e0240793d167fcc62b0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#51;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#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-id1169144757266\">\n<p id=\"fs-id1169148995749\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-931493b95c45617bf97a5d7ed8987764_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#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-id1169149308966\">\n<div data-type=\"problem\" id=\"fs-id1169140115605\">\n<p id=\"fs-id1169140115607\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a88ff770357de69bec763abff015a8dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"43\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148870248\">\n<div data-type=\"problem\" id=\"fs-id1169149006551\">\n<p id=\"fs-id1169149006553\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-72ec6710d4c6c93101e142fb29a27cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"53\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149347193\">\n<p id=\"fs-id1169144605142\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2097bc3f8ce17bba1baf291a3feca038_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#109;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"85\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149312792\">\n<div data-type=\"problem\" id=\"fs-id1169149312794\">\n<p id=\"fs-id1169148939800\"><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-id1169144359929\">\n<div data-type=\"problem\" id=\"fs-id1169149315892\">\n<p id=\"fs-id1169149315894\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-796e60ea67c3586210400db58ca0de2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"49\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149370046\">\n<p id=\"fs-id1169146625890\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70f5088be6f2ae1ffed15a084c7e893b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#120;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"81\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149100564\">\n<div data-type=\"problem\" id=\"fs-id1169149100566\">\n<p id=\"fs-id1169148871397\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acd450d4543b1085e04ddf308e3cc381_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#121;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"48\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149193810\">\n<div data-type=\"problem\" id=\"fs-id1169146613048\">\n<p id=\"fs-id1169146613050\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aca3431c41826e734e11afe452a4dec7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"48\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144560494\">\n<p id=\"fs-id1169144560496\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44333a465c50f6433959c60622b78d6d_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;&#114;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#114;&#45;&#53;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"70\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148989470\">\n<div data-type=\"problem\" id=\"fs-id1169148989472\">\n<p id=\"fs-id1169149039557\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bc838451356f8c4f4d818adb25585f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#115;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#115;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"47\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149351357\">\n<div data-type=\"problem\" id=\"fs-id1169149330488\">\n<p id=\"fs-id1169149330490\"><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-id1169144730314\">\n<p id=\"fs-id1169144730317\"><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 data-type=\"exercise\" id=\"fs-id1169146745049\">\n<div data-type=\"problem\" id=\"fs-id1169147106841\">\n<p id=\"fs-id1169147106843\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a09a56e6235f1f8a03add3adc925b168_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"53\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1169149001285\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1169148933998\">\n<div data-type=\"problem\" id=\"fs-id1169148934000\">\n<p id=\"fs-id1169149030522\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> Simplify <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-498ca022aa59906c21510e9caad0f05e_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;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"35\" style=\"vertical-align: -11px;\" \/> and explain all your steps.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Simplify <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75b6a6ea985b1795c33188bff1665040_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;&#125;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"35\" style=\"vertical-align: -11px;\" \/> and explain all your steps.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> Why are the two methods of simplifying square roots different?<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149121884\">\n<p id=\"fs-id1169149121886\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148947253\">\n<div data-type=\"problem\" id=\"fs-id1169148947255\">\n<p id=\"fs-id1169149310618\">Explain what is meant by the word rationalize in the phrase, \u201crationalize a denominator.\u201d<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149315954\">\n<div data-type=\"problem\" id=\"fs-id1169149315956\">\n<p id=\"fs-id1169149315958\">Explain why multiplying <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d3ac4dbbc110537f921e01c1247c608_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -2px;\" \/> by its conjugate results in an expression with no radicals.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144567005\">\n<p id=\"fs-id1169144567007\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149183626\">\n<div data-type=\"problem\" id=\"fs-id1169147085094\">\n<p id=\"fs-id1169147085096\">Explain why multiplying <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55141c6689e12e704a986bae663fa420_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"22\" style=\"vertical-align: -11px;\" \/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81b1bb7012d4d07b45a9120cf70504af_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;&#120;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"22\" style=\"vertical-align: -11px;\" \/> does not rationalize the denominator.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169149026970\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1169149026976\"><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-id1169144601314\" data-alt=\"This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cConfidently\u201d, the third is \u201cWith some help\u201d, and the fourth is \u201cNo, I don\u2019t get it\u201d. Under the first column are the phrases \u201cdivide radical expressions.\u201d, \u201crationalize a one term denominator\u201d, and \u201crationalize a two term denominator\u201d. The other columns are left blank so that the learner may indicate their mastery level for each topic.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_05_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is \u201cI can\u2026\u201d, the second is \u201cConfidently\u201d, the third is \u201cWith some help\u201d, and the fourth is \u201cNo, I don\u2019t get it\u201d. Under the first column are the phrases \u201cdivide radical expressions.\u201d, \u201crationalize a one term denominator\u201d, and \u201crationalize a two term denominator\u201d. The other columns are left blank so that the learner may indicate their mastery level for each topic.\" \/><\/span><\/p>\n<p id=\"fs-id1169149113217\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1169149039257\">\n<dt>rationalizing the denominator<\/dt>\n<dd id=\"fs-id1169149039262\">Rationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3583","chapter","type-chapter","status-publish","hentry"],"part":3472,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3583","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\/3583\/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\/3583\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3583"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3583"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3583"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3583"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}