{"id":3633,"date":"2018-12-11T13:57:47","date_gmt":"2018-12-11T18:57:47","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-radical-equations\/"},"modified":"2018-12-11T13:57:47","modified_gmt":"2018-12-11T18:57:47","slug":"solve-radical-equations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-radical-equations\/","title":{"raw":"Solve Radical Equations","rendered":"Solve Radical Equations"},"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>Solve radical equations<\/li><li>Solve radical equations with two radicals<\/li><li>Use radicals in applications<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1169149032213\" class=\"be-prepared\"><p id=\"fs-id1169144829761\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1169146738034\" type=\"1\"><li>Simplify: \\({\\left(y-3\\right)}^{2}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836660220\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve: \\(2x-5=0.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836432956\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve \\({n}^{2}-6n+8=0.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/da8478b4-93bc-4919-81a1-5e3267050e7e#fs-id1167836625705\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169146610807\"><h3 data-type=\"title\">Solve Radical Equations<\/h3><p id=\"fs-id1169148909008\">In this section we will solve equations that have a variable in the radicand of a radical expression. An equation of this type is called a <span data-type=\"term\">radical equation<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1169149330689\"><div data-type=\"title\">Radical Equation<\/div><p id=\"fs-id1169148989002\">An equation in which a variable is in the radicand of a radical expression is called a <strong data-effect=\"bold\">radical equation<\/strong>.<\/p><\/div><p id=\"fs-id1169146646052\">As usual, when solving these equations, what we do to one side of an equation we must do to the other side as well. Once we isolate the radical, our strategy will be to raise both sides of the equation to the power of the index. This will eliminate the radical.<\/p><p id=\"fs-id1169149319453\">Solving radical equations containing an even index by raising both sides to the power of the index may introduce an algebraic solution that would not be a solution to the original radical equation. Again, we call this an extraneous solution as we did when we solved rational equations.<\/p><p id=\"fs-id1169149011290\">In the next example, we will see how to solve a radical equation. Our strategy is based on raising a radical with index <em data-effect=\"italics\">n<\/em> to the <em data-effect=\"italics\">n<\/em><sup>th<\/sup> power. This will eliminate the radical.<\/p><div data-type=\"equation\" id=\"fs-id1169146621794\" class=\"unnumbered\" data-label=\"\">\\(\\text{For}\\phantom{\\rule{0.2em}{0ex}}a\\ge 0,\\phantom{\\rule{0.2em}{0ex}}{\\left(\\sqrt[n]{a}\\right)}^{n}=a.\\)<\/div><div data-type=\"example\" id=\"fs-id1169149296250\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a Radical Equation<\/div><div data-type=\"exercise\" id=\"fs-id1169148959770\"><div data-type=\"problem\" id=\"fs-id1169149340153\"><p id=\"fs-id1169147107160\">Solve: \\(\\sqrt{5n-4}-9=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144382249\"><span data-type=\"media\" id=\"fs-id1169149094355\" data-alt=\"Step 1 is to isolate the radical on one side of the equation. To isolate the radical add 9 to both sides. The resulting equation is square root of the quantity 5 n minus 4 in parentheses minus 9 plus 9 equals 0 plus 9. This simplifies to square root of the quantity 5 n minus 4 in parentheses equals 9.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_001a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to isolate the radical on one side of the equation. To isolate the radical add 9 to both sides. The resulting equation is square root of the quantity 5 n minus 4 in parentheses minus 9 plus 9 equals 0 plus 9. This simplifies to square root of the quantity 5 n minus 4 in parentheses equals 9.\"><\/span><span data-type=\"media\" id=\"fs-id1169146741252\" data-alt=\"Step 2 is to raise both sides of the equation to the power of the index. Since the index of a square root is 2, we square both sides. Remember that the square of the square root of \u201ca\u201d is equal to \u201ca\u201d. The equation that results is the square of the square root of the quantity 5 n minus 4 in parentheses equals 9 squared. This simplifies to 5 n minus 4 equals 81.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_001b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to raise both sides of the equation to the power of the index. Since the index of a square root is 2, we square both sides. Remember that the square of the square root of \u201ca\u201d is equal to \u201ca\u201d. The equation that results is the square of the square root of the quantity 5 n minus 4 in parentheses equals 9 squared. This simplifies to 5 n minus 4 equals 81.\"><\/span><span data-type=\"media\" id=\"fs-id1169149169210\" data-alt=\"Step 3 is to solve the new equation. We get 5 n equals 85 and then n equals 17.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_001c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to solve the new equation. We get 5 n equals 85 and then n equals 17.\"><\/span><span data-type=\"media\" id=\"fs-id1169148971297\" data-alt=\"Step 4 is to check the answer in the original equation. Does the square root of the quantity 5 times 17 minus 4 in parentheses minus 9 equal zero? Simplifying the left side we get square root of the quantity 85 minus 4 in parentheses minus 9 and then square root of 81 minus 9 and then 9 minus 9 which does equal 0. This verifies that the solution is n equals 17.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_001d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to check the answer in the original equation. Does the square root of the quantity 5 times 17 minus 4 in parentheses minus 9 equal zero? Simplifying the left side we get square root of the quantity 85 minus 4 in parentheses minus 9 and then square root of 81 minus 9 and then 9 minus 9 which does equal 0. This verifies that the solution is n equals 17.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169144560187\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147133968\"><div data-type=\"problem\" id=\"fs-id1169146632210\"><p id=\"fs-id1169148858103\">Solve: \\(\\sqrt{3m+2}-5=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148965261\"><p id=\"fs-id1169149223370\">\\(m=\\frac{23}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148939407\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169149144348\"><div data-type=\"problem\" id=\"fs-id1169146648761\"><p id=\"fs-id1169146645178\">Solve: \\(\\sqrt{10z+1}-2=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149196136\"><p id=\"fs-id1169144383780\">\\(z=\\frac{3}{10}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149319438\" class=\"howto\"><div data-type=\"title\">Solve a radical equation with one radical.<\/div><ol id=\"fs-id1169149302699\" type=\"1\" class=\"stepwise\"><li>Isolate the radical on one side of the equation.<\/li><li>Raise both sides of the equation to the power of the index.<\/li><li>Solve the new equation.<\/li><li>Check the answer in the original equation.<\/li><\/ol><\/div><p>When we use a radical sign, it indicates the principal or positive root. If an equation has a radical with an even index equal to a negative number, that equation will have no solution.<\/p><div data-type=\"example\" id=\"fs-id1169144604426\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169146719065\"><div data-type=\"problem\" id=\"fs-id1169149004291\"><p id=\"fs-id1169149103407\">Solve: \\(\\sqrt{9k-2}+1=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149196386\"><table id=\"fs-id1169148968245\" class=\"unnumbered unstyled\" summary=\"To isolate the radical, subtract 1 from both sides. The resulting equation is square root of the quantity 9 k minus 2 in parentheses plus 1 minus 1 equals 0 minus 1. Simplifying this we get square root of the quantity 9 k minus 2 in parentheses equals negative 1. Since the square root of a real number is always positive there is no solution to the equation.\" 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-id1169149194537\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To isolate the radical, subtract 1 to both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149358284\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_002b_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-id1169144383049\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169149094818\">Because the square root is equal to a negative number, the equation has no solution.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169144523599\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169146799266\"><div data-type=\"problem\" id=\"fs-id1169144359277\"><p id=\"fs-id1169149027710\">Solve: \\(\\sqrt{2r-3}+5=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148915761\"><p id=\"fs-id1169149289222\">\\(\\text{no solution}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149026518\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169146666654\"><div data-type=\"problem\" id=\"fs-id1169149361766\"><p id=\"fs-id1169144378163\">Solve: \\(\\sqrt{7s-3}+2=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144642391\"><p id=\"fs-id1169149367840\">\\(\\text{no solution}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169144555878\">If one side of an equation with a square root is a binomial, we use the Product of Binomial Squares Pattern when we square it.<\/p><div data-type=\"note\" id=\"fs-id1169149319586\"><div data-type=\"title\">Binomial Squares<\/div><div data-type=\"equation\" id=\"fs-id1169148998889\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{}\\\\ \\\\ {\\left(a+b\\right)}^{2}={a}^{2}+2ab+{b}^{2}\\\\ {\\left(a-b\\right)}^{2}={a}^{2}-2ab+{b}^{2}\\end{array}\\)<\/div><\/div><p id=\"fs-id1169140091544\">Don\u2019t forget the middle term!<\/p><div data-type=\"example\" id=\"fs-id1169144365771\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169144516404\"><div data-type=\"problem\"><p id=\"fs-id1169149121716\">Solve: \\(\\sqrt{p-1}+1=p.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144687863\"><table id=\"fs-id1169149215106\" class=\"unnumbered unstyled can-break\" summary=\"To isolate the radical, subtract 1 from both sides. The resulting equation is square root of the quantity p minus 1 in parentheses plus 1 minus 1 equals p minus 1. This simplifies to is square root of the quantity p minus 1 in parentheses equals p minus 1. Squaring both sides of the equation we get the square of the square root of the quantity p minus 1 in parentheses equals the square of the quantity p minus 1. Simplify using the binomial squares pattern on the right. The simplified equation is p minus 1 equals p squared minus 2 p plus 1. This is a quadratic equation, so get zero on one side. 0 equals p squared minus 3 p plus 2. Factor the right side. 0 equals the product of the quantity p minus 1 in parentheses with the quantity p minus 2 in parentheses. Use the zero procuct property. 0 equals p minus 1 and 0 equals p minus 2. Solving each equation we get p equals 1 and p equals 2. Checking the answer p equals 1. Does the square root of the quantity 1 minus 1 in parentheses equal 1? The square root of zero plus 1 equals 1 so p equals 1 is a solution. Checking the answer p equals 2. Does the square root of the quantity 2 minus 1 in parentheses plus 1 equal 2. Since the square root of 1 plus 1 equals 2, p equals 2 is also a solution. The solutions are p equals 1 and p equals 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-id1169149297082\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To isolate the radical, subtract 1 from both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146742316\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Square both sides of the equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146617711\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify, using the Product of Binomial Squares Pattern on the<div data-type=\"newline\"><br><\/div>right. Then solve the new equation.<\/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_06_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">It is a quadratic equation, so get zero on one side.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148956042\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the right side.<\/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_06_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the Zero Product Property.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149373628\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve each equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147107710\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check the answers.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149011207\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The solutions are \\(p=1,\\phantom{\\rule{0.5em}{0ex}}p=2.\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169144560422\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1169146667843\"><p id=\"fs-id1169144876584\">Solve: \\(\\sqrt{x-2}+2=x.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144892743\"><p id=\"fs-id1169148950842\">\\(x=2,x=3\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148966020\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148932918\"><div data-type=\"problem\" id=\"fs-id1169149040869\"><p id=\"fs-id1169146656984\">Solve: \\(\\sqrt{y-5}+5=y.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149211148\"><p id=\"fs-id1169149014631\">\\(y=5,y=6\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169146645827\">When the index of the radical is 3, we cube both sides to remove the radical.<\/p><div data-type=\"equation\" id=\"fs-id1168040445163\" class=\"unnumbered\" data-label=\"\">\\({\\left(\\sqrt[3]{a}\\right)}^{3}=a\\)<\/div><div data-type=\"example\" id=\"fs-id1169149212535\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169149308141\"><div data-type=\"problem\" id=\"fs-id1169146652198\"><p id=\"fs-id1169144379849\">Solve: \\(\\sqrt[3]{5x+1}+8=4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149095264\"><table id=\"fs-id1169139903023\" class=\"unnumbered unstyled can-break\" summary=\"To isolate the radical, subtract 8 from both sides. The resulting equation is cube root of the quantity 5 x plus 1 equals negative 4. Cubeing both sides of the equation we get the cube of the cube root of the quantity 5 x plus 1 equals the cube of negative 4. The simplified equation is 5 x plus 1 equals negative 64. This simplifies to 5 x equals negative 65. So x equals negative 13. Checking the answer x equals negative 13. Does the cube root of the quantity 5 times negative 13 plus 1 in parentheses plus 8 equal 4? The cube root of negative 64 plus 8 equals negative 4 plus 8 which equals 4 so the solution is x equals negative 13.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><\/td><td data-valign=\"top\" data-align=\"right\">\\(\\sqrt[3]{5x+1}+8=4\\phantom{\\rule{1.8em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To isolate the radical, subtract 8 from both sides.<\/td><td data-valign=\"top\" data-align=\"right\">\\(\\sqrt[3]{5x+1}=-4\\phantom{\\rule{1.2em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Cube both sides of the equation.<\/td><td data-valign=\"top\" data-align=\"right\">\\({\\left(\\sqrt[3]{5x+1}\\right)}^{3}={\\left(-4\\right)}^{3}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"right\">\\(5x+1=-64\\phantom{\\rule{0.6em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve the equation.<\/td><td data-valign=\"top\" data-align=\"right\">\\(5x=-65\\phantom{\\rule{0.6em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">\\(x=-13\\phantom{\\rule{0.6em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check the answer.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146655896\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">The solution is \\(x=-13.\\phantom{\\rule{0.35em}{0ex}}\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169144601269\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169149215173\"><div data-type=\"problem\" id=\"fs-id1169148956225\"><p id=\"fs-id1169147087056\">Solve: \\(\\sqrt[3]{4x-3}+8=5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148994363\"><p id=\"fs-id1169149154688\">\\(x=-6\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169144377434\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169149212463\"><div data-type=\"problem\" id=\"fs-id1169149343472\"><p id=\"fs-id1169144490955\">Solve: \\(\\sqrt[3]{6x-10}+1=-3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144559943\"><p id=\"fs-id1169149095754\">\\(x=-9\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169149107852\">Sometimes an equation will contain rational exponents instead of a radical. We use the same techniques to solve the equation as when we have a radical. We raise each side of the equation to the power of the denominator of the rational exponent. Since \\({\\left({a}^{m}\\right)}^{n}={a}^{m\u00b7n},\\) we have for example,<\/p><div data-type=\"equation\" id=\"fs-id1169144451236\" class=\"unnumbered\" data-label=\"\">\\({\\left({x}^{\\frac{1}{2}}\\right)}^{2}=x,\\phantom{\\rule{0.5em}{0ex}}{\\left({x}^{\\frac{1}{3}}\\right)}^{3}=x\\)<\/div><p id=\"fs-id1169147109917\">Remember, \\({x}^{\\frac{1}{2}}=\\sqrt{x}\\) and \\({x}^{\\frac{1}{3}}=\\sqrt[3]{x}.\\)<\/p><div data-type=\"example\" id=\"fs-id1169148952413\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169149297527\"><div data-type=\"problem\" id=\"fs-id1169146732624\"><p id=\"fs-id1169144604112\">Solve: \\({\\left(3x-2\\right)}^{\\frac{1}{4}}+3=5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144555211\"><table id=\"fs-id1169146594997\" class=\"unnumbered unstyled can-break\" summary=\"To isolate the term with the rational exponent, subtract 3 from both sides. The resulting equation is the quantity 3 x minus 2 raised to the power of one fourth equals 2. Raising each side to the fourth power we get the fourth power of the quantity 3 x minus 2 in parentheses raised to the power of one fourth in parentheses equals 2 to the fourth power. The simplified equation is 3 x minus 2 equals 16. This simplifies to 3 x equals 18. So x equals 6. Checking the answer x equals 6. Does the quantity 3 times 6 minus 2 in parentheses raised to the one-fourth power plus 3 equal 5? The quantity 18 minus 2 in parentheses raised to the one-fourth power plus 3 equals 16 to the one fourth power plus 3 which equals 2 plus 3 which does equal 5 so the solution is x equals 6.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">\\({\\left(3x-2\\right)}^{\\frac{1}{4}}+3=5\\phantom{\\rule{1.1em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To isolate the term with the rational exponent,<div data-type=\"newline\"><br><\/div>subtract 3 from both sides.<\/td><td data-valign=\"top\" data-align=\"right\">\\({\\left(3x-2\\right)}^{\\frac{1}{4}}=2\\phantom{\\rule{1.1em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Raise each side of the equation to the fourth power.<\/td><td data-valign=\"top\" data-align=\"right\">\\({\\left({\\left(3x-2\\right)}^{\\frac{1}{4}}\\right)}^{4}={\\left(2\\right)}^{4}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"right\">\\(3x-2=16\\phantom{\\rule{0.6em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve the equation.<\/td><td data-valign=\"top\" data-align=\"right\">\\(3x=18\\phantom{\\rule{0.6em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">\\(x=6\\phantom{\\rule{1.17em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check the answer.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147115372\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">The solution is \\(x=6.\\phantom{\\rule{0.95em}{0ex}}\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149042758\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169149030038\"><div data-type=\"problem\" id=\"fs-id1169149030828\"><p id=\"fs-id1169149374970\">Solve: \\({\\left(9x+9\\right)}^{\\frac{1}{4}}-2=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149006444\"><p id=\"fs-id1169149012975\">\\(x=8\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169138877440\"><div data-type=\"problem\" id=\"fs-id1169149088428\"><p id=\"fs-id1169149013626\">Solve: \\({\\left(4x-8\\right)}^{\\frac{1}{4}}+5=7.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146668137\"><p id=\"fs-id1169146816710\">\\(x=6\\)<\/p><\/div><\/div><\/div><p>Sometimes the solution of a radical equation results in two algebraic solutions, but one of them may be an <span data-type=\"term\" class=\"no-emphasis\">extraneous solution<\/span>!<\/p><div data-type=\"example\" id=\"fs-id1169148939142\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148850042\"><div data-type=\"problem\"><p id=\"fs-id1169149012026\">Solve: \\(\\sqrt{r+4}-r+2=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149012422\"><table id=\"fs-id1169148972821\" class=\"unnumbered unstyled can-break\" summary=\"First, isolate the radical. The resulting equation is square root of the quantity r plus 4 in parentheses equals r minus 2. Squaring both sides of the equation we get the square of the square root of the quantity r plus 4 in parentheses equals the square of the quantity r minus 2 in parentheses. Simplify using the binomial squares pattern on the right. The simplified equation is r plus 4 equals r squared minus 4 r plus 4. This is a quadratic equation, so get zero on one side. 0 equals r squared minus 5 r. Factor the right side. 0 equals r times the quantity r minus 5 in parentheses. Use the zero procuct property. 0 equals r and 0 equals r minus 5. Solving each equation we get r equals 0 and r equals 5. Checking the answer r equals 0. Does the square root of the quantity 0 plus 4 in parentheses minus 0 plus 2 equal 0? The square root of 4 plus 2 equals 0 so r equals 0 is a solution. Checking the answer r equals 5. Does the square root of the quantity 5 plus 4 in parentheses minus 5 plus 2 equal 0. Since the square root of 9 minus 3 equals 0, r equals 5 is also a solution. The solutions are r equals 0 and r equals 5.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">\\(\\sqrt{r+4}-r+2=0\\phantom{\\rule{4.8em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Isolate the radical.<\/td><td data-valign=\"top\" data-align=\"right\">\\(\\sqrt{r+4}=r-2\\phantom{\\rule{3.2em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Square both sides of the equation.<\/td><td data-valign=\"top\" data-align=\"right\">\\({\\left(\\sqrt{r+4}\\right)}^{2}={\\left(r-2\\right)}^{2}\\phantom{\\rule{2em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify and then solve the equation<\/td><td data-valign=\"top\" data-align=\"right\">\\(r+4={r}^{2}-4r+4\\phantom{\\rule{0.7em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">It is a quadratic equation, so get zero on<div data-type=\"newline\"><br><\/div>one side.<\/td><td data-valign=\"top\" data-align=\"right\">\\(0={r}^{2}-5r\\phantom{\\rule{2.4em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the right side.<\/td><td data-valign=\"top\" data-align=\"right\">\\(0=r\\left(r-5\\right)\\phantom{\\rule{2.1em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the Zero Product Property.<\/td><td data-valign=\"top\" data-align=\"right\">\\(0=r\\phantom{\\rule{1.5em}{0ex}}0=r-5\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve the equation.<\/td><td data-valign=\"top\" data-align=\"right\">\\(r=0\\phantom{\\rule{1em}{0ex}}r=5\\phantom{\\rule{1.5em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check your answer.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148859069\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"bottom\" data-align=\"left\">The solution is <em data-effect=\"italics\">r<\/em> = 5.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">\\(r=0\\) is an extraneous solution.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149224379\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148858909\"><div data-type=\"problem\" id=\"fs-id1169149028999\"><p id=\"fs-id1169146627237\">Solve: \\(\\sqrt{m+9}-m+3=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149122786\"><p id=\"fs-id1169147085867\">\\(m=7\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149144336\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148994616\"><div data-type=\"problem\"><p id=\"fs-id1169149140803\">Solve: \\(\\sqrt{n+1}-n+1=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149010540\"><p id=\"fs-id1169149346391\">\\(n=3\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169139960604\">When there is a coefficient in front of the radical, we must raise it to the power of the index, too.<\/p><div data-type=\"example\" id=\"fs-id1169149294243\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169144557460\"><div data-type=\"problem\" id=\"fs-id1169146643951\"><p id=\"fs-id1169149102502\">Solve: \\(\\text{3}\\phantom{\\rule{0.2em}{0ex}}\\sqrt{3x-5}-8=4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146738556\"><table id=\"fs-id1169149115633\" class=\"unnumbered unstyled can-break\" summary=\"First, isolate the radical term by adding 8 to both sides. 3 times square root of the quantity 3 x minus 5 in parentheses equals 12. Isolate the radical by dividing both sides by 3. Square root of the quantity 3 x minus 5 in parentheses equals 4. Square both sides of the equation. The square of the square root of the quantity 3 x minus 5 in parentheses equals 4 squared. Simplify, then solve the new equation. 3 x minus 5 equals 16. 3 x equals 21. x equals 7. Check the answer x equals 7. Does 3 times the square root of the quantity 3 times 7 minus 5 in parentheses minus 8 equal 4? Simplifying the left side we get 3 times the square root of the quantity 21 minus 5 in parentheses minus 8 which equals 3 times the square root of 16 minus 8 which equals 3 times 4 minus 8 which equals 4. The solution is x equals 7.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">\\(\\text{3}\\phantom{\\rule{0.2em}{0ex}}\\sqrt{3x-5}-8=4\\phantom{\\rule{1.05em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Isolate the radical term.<\/td><td data-valign=\"top\" data-align=\"right\">\\(3\\sqrt{3x-5}=12\\phantom{\\rule{0.6em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Isolate the radical by dividing both sides by 3.<\/td><td data-valign=\"top\" data-align=\"right\">\\(\\sqrt{3x-5}=4\\phantom{\\rule{1.05em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Square both sides of the equation.<\/td><td data-valign=\"top\" data-align=\"right\">\\({\\left(\\sqrt{3x-5}\\right)}^{2}={\\left(4\\right)}^{2}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify, then solve the new equation.<\/td><td data-valign=\"top\" data-align=\"right\">\\(3x-5=16\\phantom{\\rule{0.6em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">\\(3x=21\\phantom{\\rule{0.6em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Solve the equation.<\/td><td data-valign=\"top\" data-align=\"right\">\\(x=7\\phantom{\\rule{1.05em}{0ex}}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check the answer.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144684619\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"right\">The solution is \\(x=7.\\phantom{\\rule{1.05em}{0ex}}\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148844737\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169144615527\"><div data-type=\"problem\" id=\"fs-id1169149308487\"><p id=\"fs-id1169140055625\">Solve: \\(2\\sqrt{4a+4}-16=16.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1169149370615\">\\(a=63\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149329127\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169149117576\"><div data-type=\"problem\" id=\"fs-id1169148859787\"><p id=\"fs-id1169149096759\">Solve: \\(3\\sqrt{2b+3}-25=50.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144377869\"><p id=\"fs-id1169144432792\">\\(b=311\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149230183\"><h3 data-type=\"title\">Solve Radical Equations with Two Radicals<\/h3><p id=\"fs-id1169149144685\">If the radical equation has two radicals, we start out by isolating one of them. It often works out easiest to isolate the more complicated radical first.<\/p><p id=\"fs-id1169148968522\">In the next example, when one radical is isolated, the second radical is also isolated.<\/p><div data-type=\"example\" id=\"fs-id1169148915268\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148865471\"><div data-type=\"problem\" id=\"fs-id1169149285487\"><p id=\"fs-id1169149311031\">Solve: \\(\\sqrt[3]{4x-3}=\\sqrt[3]{3x+2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148926116\"><p id=\"fs-id1169149351070\">\\(\\begin{array}{cccccc}\\text{The radical terms are isolated.}\\hfill &amp; &amp; &amp; \\hfill \\sqrt[3]{4x-3}&amp; =\\hfill &amp; \\sqrt[3]{3x+2}\\hfill \\\\ \\begin{array}{c}\\text{Since the index is 3, cube both sides of the}\\hfill \\\\ \\text{equation.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill {\\left(\\sqrt[3]{4x-3}\\right)}^{3}&amp; =\\hfill &amp; {\\left(\\sqrt[3]{3x+2}\\right)}^{3}\\hfill \\\\ \\text{Simplify, then solve the new equation.}\\hfill &amp; &amp; &amp; \\hfill 4x-3&amp; =\\hfill &amp; 3x+2\\hfill \\\\ &amp; &amp; &amp; \\hfill x-3&amp; =\\hfill &amp; 2\\hfill \\\\ &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; 5\\hfill \\\\ &amp; &amp; &amp; \\hfill \\text{The solution is}\\phantom{\\rule{0.2em}{0ex}}x&amp; =\\hfill &amp; 5.\\hfill \\\\ \\text{Check the answer.}\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\text{We leave it to you to show that 5 checks!}\\hfill &amp; &amp; &amp; &amp; &amp; \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169146612764\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169149220632\"><div data-type=\"problem\" id=\"fs-id1169146594078\"><p id=\"fs-id1169147088008\">Solve: \\(\\sqrt[3]{5x-4}=\\sqrt[3]{2x+5}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146745629\"><p id=\"fs-id1169146817003\">\\(x=3\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169146665288\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148974254\"><div data-type=\"problem\" id=\"fs-id1169146652924\"><p id=\"fs-id1169146608926\">Solve: \\(\\sqrt[3]{7x+1}=\\sqrt[3]{2x-5}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149350577\"><p id=\"fs-id1169149222427\">\\(x=-\\frac{6}{5}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169149285371\">Sometimes after raising both sides of an equation to a power, we still have a variable inside a radical. When that happens, we repeat Step 1 and Step 2 of our procedure. We isolate the radical and raise both sides of the equation to the power of the index again.<\/p><div data-type=\"example\" id=\"fs-id1169149376671\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a Radical Equation<\/div><div data-type=\"exercise\" id=\"fs-id1169149010550\"><div data-type=\"problem\" id=\"fs-id1169146668708\"><p>Solve: \\(\\sqrt{m}+1=\\sqrt{m+9}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149343510\"><span data-type=\"media\" id=\"fs-id1169149169645\" data-alt=\"Step 1 is to isolate one of the radical terms on one side of the equation. The radical on the right is isolated.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_008a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to isolate one of the radical terms on one side of the equation. The radical on the right is isolated.\"><\/span><span data-type=\"media\" id=\"fs-id1169144788286\" data-alt=\"Step 2 is to raise both sides of the equation to the power of the index. We square both sides. The equation that results is the square of the quantity square root of m plus 1 in parentheses equals the square of the square root of the quantity m plus 9 in parentheses. Simplify \u2013 be very careful as you multiply! This simplifies to m plus 2 times square root m plus 1 equals m plus 9.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_008b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to raise both sides of the equation to the power of the index. We square both sides. The equation that results is the square of the quantity square root of m plus 1 in parentheses equals the square of the square root of the quantity m plus 9 in parentheses. Simplify \u2013 be very careful as you multiply! This simplifies to m plus 2 times square root m plus 1 equals m plus 9.\"><\/span><span data-type=\"media\" id=\"fs-id1169146731390\" data-alt=\"Step 3 is to repeat steps 1 and 2 again if there are any more radicals. There is still a radical in the equation. So we must repeat the previous steps. Isolate the radical term. 2 times square root m equals 8. Here, we can easily isolate the radical by dividing both sides by 2. We get square root m equals 4. Squaring both sides we get the square of the square root of m equals 4 squared. m equals 16.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_008c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to repeat steps 1 and 2 again if there are any more radicals. There is still a radical in the equation. So we must repeat the previous steps. Isolate the radical term. 2 times square root m equals 8. Here, we can easily isolate the radical by dividing both sides by 2. We get square root m equals 4. Squaring both sides we get the square of the square root of m equals 4 squared. m equals 16.\"><\/span><span data-type=\"media\" id=\"fs-id1169146738581\" data-alt=\"Step 4 is to check the answer in the original equation. Does the square root of 16 plus 1 equal the square root of the quantity 16 plus 9? Simplifying both sides we get 4 plus 1 equals 5. This verifies that the solution is m equals 16.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_008d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to check the answer in the original equation. Does the square root of 16 plus 1 equal the square root of the quantity 16 plus 9? Simplifying both sides we get 4 plus 1 equals 5. This verifies that the solution is m equals 16.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149011708\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169144885903\"><div data-type=\"problem\" id=\"fs-id1169146660138\"><p id=\"fs-id1169144373778\">Solve: \\(3-\\sqrt{x}=\\sqrt{x-3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148875002\"><p id=\"fs-id1169148820998\">\\(x=4\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148875485\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169147027836\"><div data-type=\"problem\" id=\"fs-id1169149003300\"><p id=\"fs-id1169148826041\">Solve: \\(\\sqrt{x}+2=\\sqrt{x+16}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149295810\"><p id=\"fs-id1169148894141\">\\(x=9\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1169149329150\">We summarize the steps here. We have adjusted our previous steps to include more than one radical in the equation This procedure will now work for any radical equations.<\/p><div data-type=\"note\" id=\"fs-id1169148926139\" class=\"howto\"><div data-type=\"title\">Solve a radical equation.<\/div><ol id=\"fs-id1169148866012\" type=\"1\" class=\"stepwise\"><li>Isolate one of the radical terms on one side of the equation.<\/li><li>Raise both sides of the equation to the power of the index.<\/li><li>Are there any more radicals?<div data-type=\"newline\"><br><\/div> If yes, repeat Step 1 and Step 2 again.<div data-type=\"newline\"><br><\/div> If no, solve the new equation.<\/li><li>Check the answer in the original equation.<\/li><\/ol><\/div><p id=\"fs-id1169146647278\">Be careful as you square binomials in the next example. Remember the pattern is \\({\\left(a+b\\right)}^{2}={a}^{2}+2ab+{b}^{2}\\) or \\({\\left(a-b\\right)}^{2}={a}^{2}-2ab+{b}^{2}.\\)<\/p><div data-type=\"example\" id=\"fs-id1169146664073\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148995792\"><div data-type=\"problem\" id=\"fs-id1169149284527\"><p id=\"fs-id1169148835927\">Solve: \\(\\sqrt{q-2}+3=\\sqrt{4q+1}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144744387\"><table id=\"fs-id1169144715813\" class=\"unnumbered unstyled can-break\" summary=\"The radical on the right is isolated. Square both sides. The equation that results is the square of the sum of the square root of the quantity q minus 2 in parentheses and 3 in parentheses equals the square of the square root of the quantity 4 q plus 1 in parentheses. This simplifies to q minus 2 plus 6 times square root of the quantity q minus 2 in parentheses plus 9 equals 4 q plus 1. There is still a radical in the equation. So we must repeat the previous steps. Isolate the radical term. 6 times square root of the quantity q minus 2 in parentheses equals 3 q minus 6. It would not help to divide both sides by 6. Squaring both sides we get the square of the product of 6 and the square root of the quantity q minus 2 in parentheses the square of the quantity 3 q minus 6 in parentheses. Remember to square both the 6 and the square root of the quantity q minus 2. When squaring the right side use the formula the quantity a minus b in parentheses squared equals a squared minus 2 a b plus b squared. The resulting equation is 6 squared times the square of the square root of the quantity q minus 2 in parentheses equals the quantity 3 q in parentheses squared minus 2 times 3 q times 6 plus 6 squared. Simplifying we get 36 times the quantity q minus 2 in parentheses equals 9 q squared minus 36 q plus 36. Distributing we get 36 q minus 72 equals 9 q squared minus 36 q plus 36. It is a quadratic equation, so get zero on one side. 0 equals 9 q squared minus 72 q plus 108. Factor the right side to get 0 equals 9 times the quantity q minus 6 in parentheses times the quantity q minus 2 in parentheses. Use the zero product property to get the equations q minus 6 equals 0 and q minus 2 equals 0. Solving eah equation we get q equals 6 and q equals 2. The checks are left to you. The solutions are q equals 6 and q equals 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-id1169146613138\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The radical on the right is isolated. Square<div data-type=\"newline\"><br><\/div>both sides.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148869414\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009b_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-id1169144551290\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">There is still a radical in the equation so<div data-type=\"newline\"><br><\/div>we must repeat the previous steps. Isolate<div data-type=\"newline\"><br><\/div>the radical.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144796710\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Square both sides. It would not help to<div data-type=\"newline\"><br><\/div>divide both sides by 6. Remember to<div data-type=\"newline\"><br><\/div>square both the 6 and the \\(\\sqrt{q-2}.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146661975\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify, then solve the new equation.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148911599\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148924892\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">It is a quadratic equation, so get zero on<div data-type=\"newline\"><br><\/div>one side.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148967666\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the right side.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146637367\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the Zero Product Property.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146627489\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The checks are left to you.<\/td><td data-valign=\"top\" data-align=\"center\">The solutions are \\(q=6\\) and \\(q=2.\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149121469\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1169144568120\"><p id=\"fs-id1169146740691\">Solve: \\(\\sqrt{x-1}+2=\\sqrt{2x+6}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146740572\"><p id=\"fs-id1169149172270\">\\(x=5\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148957835\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148929863\"><div data-type=\"problem\" id=\"fs-id1169144377478\"><p id=\"fs-id1169149118407\">Solve: \\(\\sqrt{x}+2=\\sqrt{3x+4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146609045\"><p id=\"fs-id1169148984154\">\\(x=0\\phantom{\\rule{0.2em}{0ex}}x=4\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149025241\"><h3 data-type=\"title\">Use Radicals in Applications<\/h3><p id=\"fs-id1169148985032\">As you progress through your college courses, you\u2019ll encounter formulas that include radicals in many disciplines. We will modify our Problem Solving Strategy for Geometry Applications slightly to give us a plan for solving applications with formulas from any discipline.<\/p><div data-type=\"note\" id=\"fs-id1169149040822\" class=\"howto\"><div data-type=\"title\">Use a problem solving strategy for applications with formulas.<\/div><ol id=\"fs-id1169144543890\" type=\"1\" class=\"stepwise\"><li><strong data-effect=\"bold\">Read<\/strong> the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.<\/li><li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li><li><strong data-effect=\"bold\">Name<\/strong> what we are looking for by choosing a variable to represent it.<\/li><li><strong data-effect=\"bold\">Translate<\/strong> into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li><li><strong data-effect=\"bold\">Solve the equation<\/strong> using good algebra techniques.<\/li><li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li><li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li><\/ol><\/div><p id=\"fs-id1169149118428\">One application of radicals has to do with the effect of <span data-type=\"term\" class=\"no-emphasis\">gravity<\/span> on falling objects. The formula allows us to determine how long it will take a fallen object to hit the gound.<\/p><div data-type=\"note\" id=\"fs-id1169148957158\"><div data-type=\"title\">Falling Objects<\/div><p id=\"fs-id1169144729653\">On Earth, if an object is dropped from a height of <em data-effect=\"italics\">h<\/em> feet, the time in seconds it will take to reach the ground is found by using the formula<\/p><div data-type=\"equation\" id=\"fs-id1169149103897\" class=\"unnumbered\" data-label=\"\">\\(t=\\frac{\\sqrt{h}}{4}.\\)<\/div><\/div><p id=\"fs-id1169146628740\">For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by substituting \\(h=64\\) into the formula.<\/p><table class=\"unnumbered unstyled\" summary=\"Since h equals 64 we rewrite the formula, replacing h with the number 64. The formula then becomes t equals square root of 64 divided by 4. Taking the square root of 64 we get t equals 8 divided by 4. Simplifying the fraction we get t equals 2. It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.\" 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-id1169149065270\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149032044\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Take the square root of 64.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149342082\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_010c_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\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1169149190917\">It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.<\/p><div data-type=\"example\" id=\"fs-id1169149292896\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1169148889574\"><div data-type=\"problem\" id=\"fs-id1169149120935\"><p id=\"fs-id1169148973896\">Marissa dropped her sunglasses from a bridge 400 feet above a river. Use the formula \\(t=\\frac{\\sqrt{h}}{4}\\) to find how many seconds it took for the sunglasses to reach the river.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148890736\"><table id=\"fs-id1169149376727\" class=\"unnumbered unstyled can-break\" summary=\"The first step in the process is to read the problem. Step 2 is to identify what we are looking for. We are looking for the time it takes the sunglasses to reach the river. Step 3 is to name what we are looking for. Let t equal the time. Step 4 is to translate into an equation by writing the appropriate formula and substitute in the given information. t equals the square root of h divided by 4 and h equals 400. So t equals the square root of 400 divided by 4. Step 5 is to solve the equation. So t equals 20 divided by 4. So t equals 5. Step 6 is to check the answer in the problem and make sure it makes sense. Does 5 equal the square root of 400 divided 4. Since 5 equals 20 divided by 4, the answer is a solution to the equation. Does 5 seconds seem like a reasonable length of time? Yes. Step 7 is to answer the question. It will take 5 seconds for the sunglasses to reach the river.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td><td data-valign=\"top\" data-align=\"left\">the time it takes for the<div data-type=\"newline\"><br><\/div>sunglasses to reach the river<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking.<\/td><td data-valign=\"top\" data-align=\"left\">Let \\(t=\\) time.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into an equation by writing the<div data-type=\"newline\"><br><\/div>appropriate formula. Substitute in the given<div data-type=\"newline\"><br><\/div>information.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148938836\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve the equation.<\/strong><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146744203\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_011c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144561652\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_011d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem and make<div data-type=\"newline\"><br><\/div>sure it makes sense.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149157046\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Does 5 seconds seem like a reasonable length of<div data-type=\"newline\"><br><\/div>time?<\/td><td data-valign=\"top\" data-align=\"left\">Yes.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td><td data-valign=\"top\" data-align=\"left\">It will take 5 seconds for the<div data-type=\"newline\"><br><\/div>sunglasses to reach the river.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148879930\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169146645348\"><div data-type=\"problem\" id=\"fs-id1169149115845\"><p id=\"fs-id1169149089351\">A helicopter dropped a rescue package from a height of 1,296 feet. Use the formula \\(t=\\frac{\\sqrt{h}}{4}\\) to find how many seconds it took for the package to reach the ground.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144450967\"><p id=\"fs-id1169138945430\">9 seconds<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169149028779\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169148933988\"><div data-type=\"problem\" id=\"fs-id1169149004636\"><p id=\"fs-id1169149293850\">A window washer dropped a squeegee from a platform 196 feet above the sidewalk Use the formula \\(t=\\frac{\\sqrt{h}}{4}\\) to find how many seconds it took for the squeegee to reach the sidewalk.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144555964\"><p id=\"fs-id1169144614940\">\\(3.5\\) seconds<\/p><\/div><\/div><\/div><p id=\"fs-id1169148957152\">Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the <span data-type=\"term\" class=\"no-emphasis\">speed<\/span>, in miles per hour, a car was going before applying the brakes.<\/p><div data-type=\"note\" id=\"fs-id1169149012194\"><div data-type=\"title\">Skid Marks and Speed of a Car<\/div><p id=\"fs-id1169144555978\">If the length of the skid marks is <em data-effect=\"italics\">d<\/em> feet, then the speed, <em data-effect=\"italics\">s<\/em>, of the car before the brakes were applied can be found by using the formula<\/p><div data-type=\"equation\" id=\"fs-id1169149092734\" class=\"unnumbered\" data-label=\"\">\\(s=\\sqrt{24d}\\)<\/div><\/div><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1169149311099\"><p id=\"fs-id1169147133499\">After a car accident, the skid marks for one car measured 190 feet. Use the formula \\(s=\\sqrt{24d}\\) to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148993220\"><table id=\"fs-id1169144381624\" class=\"unnumbered unstyled can-break\" summary=\"The first step in the process is to read the problem. Step 2 is to identify what we are looking for. We are looking for the speed of the car. Step 3 is to name what we are looking for. Let s equal the speed. Step 4 is to translate into an equation by writing the appropriate formula and substitute in the given information. s equals the square root of the quantity 24 d in parentheses, and d equals 190. So s equals the square root of the quantity 24 times 190 in parentheses. Step 5 is to solve the equation. So s equals the square root of 4560. So s is approximately equal to 67.52777. Rounding to 1 decimal place we et s equal to 67.5. Step 6 is to check the answer in the problem and make sure it makes sense. Does the square root of 4560 equal the square root of the quantity 24 times 190 in parentheses? It does. Does 67.5 mph seem like a reasonable speed? Yes. Step 7 is to answer the question. The car was traveling approximately 67.5 mph before the brakes were applied.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td><td data-valign=\"top\" data-align=\"left\">the speed of a car<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what weare looking for,<\/td><td data-valign=\"top\" data-align=\"left\">Let \\(s=\\) the speed.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into an equation by writing<div data-type=\"newline\"><br><\/div>the appropriate formula. Substitute in the<div data-type=\"newline\"><br><\/div>given information.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147109998\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_012a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 5. Solve the equation.<\/strong><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148971507\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149219635\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Round to 1 decimal place.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144558132\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_012d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146594591\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_012e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The speed of the car before the brakes were applied<div data-type=\"newline\"><br><\/div>was 67.5 miles per hour.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169144382040\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169144382958\"><div data-type=\"problem\" id=\"fs-id1169149330072\"><p id=\"fs-id1169149354829\">An accident investigator measured the skid marks of the car. The length of the skid marks was 76 feet. Use the formula \\(s=\\sqrt{24d}\\) to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1169146662927\">\\(42.7\\) feet<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169148984286\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1169149373592\"><div data-type=\"problem\" id=\"fs-id1169149065549\"><p id=\"fs-id1169149339440\">The skid marks of a vehicle involved in an accident were 122 feet long. Use the formula \\(s=\\sqrt{24d}\\) to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144615503\"><p id=\"fs-id1169148872024\">\\(54.1\\) feet<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1169144566399\" class=\"media-2\"><p id=\"fs-id1169149329664\">Access these online resources for additional instruction and practice with solving radical equations.<\/p><ul id=\"fs-id1169147089449\" data-bullet-style=\"bullet\"><li><a href=\"https:\/\/openstax.org\/l\/37RadEquat1\">Solving an Equation Involving a Single Radical<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37RadEquat2\">Solving Equations with Radicals and Rational Exponents<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37RadEquat3\">Solving Radical Equations<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37RadEquat4\">Solve Radical Equations<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37RadEquat5\">Radical Equation Application<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169149092589\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1169149351406\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Binomial Squares<\/strong><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{c}{\\left(a+b\\right)}^{2}={a}^{2}+2ab+{b}^{2}\\\\ {\\left(a-b\\right)}^{2}={a}^{2}-2ab+{b}^{2}\\end{array}\\)<\/li><li><strong data-effect=\"bold\">Solve a Radical Equation<\/strong><ol id=\"fs-id1169149356299\" type=\"1\" class=\"stepwise\"><li>Isolate one of the radical terms on one side of the equation.<\/li><li>Raise both sides of the equation to the power of the index.<\/li><li>Are there any more radicals?<div data-type=\"newline\"><br><\/div> If yes, repeat Step 1 and Step 2 again.<div data-type=\"newline\"><br><\/div> If no, solve the new equation.<\/li><li>Check the answer in the original equation.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">Problem Solving Strategy for Applications with Formulas<\/strong><ol id=\"fs-id1169144768017\" type=\"1\" class=\"stepwise\"><li>Read the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.<\/li><li>Identify what we are looking for.<\/li><li>Name what we are looking for by choosing a variable to represent it.<\/li><li>Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li><li>Solve the equation using good algebra techniques.<\/li><li>Check the answer in the problem and make sure it makes sense.<\/li><li>Answer the question with a complete sentence.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">Falling Objects<\/strong><ul id=\"fs-id1169149361874\" data-bullet-style=\"bullet\"><li>On Earth, if an object is dropped from a height of <em data-effect=\"italics\">h<\/em> feet, the time in seconds it will take to reach the ground is found by using the formula \\(t=\\frac{\\sqrt{h}}{4}.\\)<\/li><\/ul><\/li><li><strong data-effect=\"bold\">Skid Marks and Speed of a Car<\/strong><ul id=\"fs-id1169144615595\" data-bullet-style=\"bullet\"><li>If the length of the skid marks is <em data-effect=\"italics\">d<\/em> feet, then the speed, <em data-effect=\"italics\">s<\/em>, of the car before the brakes were applied can be found by using the formula \\(s=\\sqrt{24d}.\\)<\/li><\/ul><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169148889368\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169148955469\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1169148967553\"><strong data-effect=\"bold\">Solve Radical Equations<\/strong><\/p><p id=\"fs-id1169146645835\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1169144767986\"><div data-type=\"problem\" id=\"fs-id1169146645006\"><p id=\"fs-id1169144892849\">\\(\\sqrt{5x-6}=8\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146611138\"><p id=\"fs-id1169149008350\">\\(m=14\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149030412\"><div data-type=\"problem\" id=\"fs-id1169149375804\"><p id=\"fs-id1169148997760\">\\(\\sqrt{4x-3}=7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144544993\"><div data-type=\"problem\" id=\"fs-id1169146669980\"><p id=\"fs-id1169146660926\">\\(\\sqrt{5x+1}=-3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148868701\"><p id=\"fs-id1169144565248\">no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144421298\"><div data-type=\"problem\" id=\"fs-id1169146609692\"><p id=\"fs-id1169148924664\">\\(\\sqrt{3y-4}=-2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148884439\"><div data-type=\"problem\" id=\"fs-id1169146745549\"><p id=\"fs-id1169148917729\">\\(\\sqrt[3]{2x}=-2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146669918\"><p id=\"fs-id1169149113769\">\\(x=-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149086617\"><div data-type=\"problem\" id=\"fs-id1169147027817\"><p id=\"fs-id1169148870239\">\\(\\sqrt[3]{4x-1}=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149220779\"><div data-type=\"problem\" id=\"fs-id1169146630929\"><p id=\"fs-id1169144381891\">\\(\\sqrt{2m-3}-5=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149006802\"><p id=\"fs-id1169144545042\">\\(m=14\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144516801\"><div data-type=\"problem\" id=\"fs-id1169146741588\"><p id=\"fs-id1169149140544\">\\(\\sqrt{2n-1}-3=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149039277\"><div data-type=\"problem\" id=\"fs-id1169146665818\"><p id=\"fs-id1169146660008\">\\(\\sqrt{6v-2}-10=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146629385\"><p id=\"fs-id1169149115657\">\\(v=17\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149102979\"><div data-type=\"problem\" id=\"fs-id1169149285826\"><p id=\"fs-id1169146642595\">\\(\\sqrt{12u+1}-11=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148995897\"><div data-type=\"problem\" id=\"fs-id1169149140501\"><p id=\"fs-id1169144560794\">\\(\\sqrt{4m+2}+2=6\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1169146611236\">\\(m=\\frac{7}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149108832\"><div data-type=\"problem\" id=\"fs-id1169149040777\"><p id=\"fs-id1169148992914\">\\(\\sqrt{6n+1}+4=8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149037013\"><div data-type=\"problem\" id=\"fs-id1169146610982\"><p id=\"fs-id1169149094596\">\\(\\sqrt{2u-3}+2=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149189792\"><p id=\"fs-id1169148958541\">no solution<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149319612\"><div data-type=\"problem\" id=\"fs-id1169149172684\"><p id=\"fs-id1169149336271\">\\(\\sqrt{5v-2}+5=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169140166746\"><div data-type=\"problem\" id=\"fs-id1169149210689\"><p id=\"fs-id1169144875868\">\\(\\sqrt{u-3}-3=u\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169148895758\"><p id=\"fs-id1169149230001\">\\(u=3,u=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148951939\"><div data-type=\"problem\" id=\"fs-id1169144746311\"><p id=\"fs-id1169148994556\">\\(\\sqrt{v-10}+10=v\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169146609259\"><div data-type=\"problem\" id=\"fs-id1169147028026\"><p id=\"fs-id1169146595086\">\\(\\sqrt{r-1}=r-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149328130\"><p id=\"fs-id1169144381096\">\\(r=1,r=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169140091631\"><div data-type=\"problem\" id=\"fs-id1169144604107\"><p id=\"fs-id1169144604073\">\\(\\sqrt{s-8}=s-8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144744651\"><div data-type=\"problem\" id=\"fs-id1169144744653\"><p id=\"fs-id1169144416352\">\\(\\sqrt[3]{6x+4}=4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144379926\"><p id=\"fs-id1169140166884\">\\(x=10\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149351498\"><div data-type=\"problem\" id=\"fs-id1169146632531\"><p id=\"fs-id1169146632533\">\\(\\sqrt[3]{11x+4}=5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169148871204\"><div data-type=\"problem\" id=\"fs-id1169144417097\"><p id=\"fs-id1169144604691\">\\(\\sqrt[3]{4x+5}-2=-5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144744577\"><p id=\"fs-id1169149326524\">\\(x=-8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144375789\"><div data-type=\"problem\" id=\"fs-id1169144744881\"><p id=\"fs-id1169144744883\">\\(\\sqrt[3]{9x-1}-1=-5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144453078\"><div data-type=\"problem\" id=\"fs-id1169146833202\"><p id=\"fs-id1169146833204\">\\({\\left(6x+1\\right)}^{\\frac{1}{2}}-3=4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149357243\"><p id=\"fs-id1169149357245\">\\(x=8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144744499\"><div data-type=\"problem\" id=\"fs-id1169149102314\"><p id=\"fs-id1169149102316\">\\({\\left(3x-2\\right)}^{\\frac{1}{2}}+1=6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144452942\"><div data-type=\"problem\" id=\"fs-id1169144416963\"><p id=\"fs-id1169144416966\">\\({\\left(8x+5\\right)}^{\\frac{1}{3}}+2=-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144417003\"><p id=\"fs-id1169144417005\">\\(x=-4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169146732087\"><div data-type=\"problem\" id=\"fs-id1169146732089\"><p id=\"fs-id1169146732091\">\\({\\left(12x-5\\right)}^{\\frac{1}{3}}+8=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149102331\"><div data-type=\"problem\" id=\"fs-id1169149102333\"><p id=\"fs-id1169149102335\">\\({\\left(12x-3\\right)}^{\\frac{1}{4}}-5=-2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149102373\"><p id=\"fs-id1169149102375\">\\(x=7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147082537\"><div data-type=\"problem\" id=\"fs-id1169147082540\"><p id=\"fs-id1169147082542\">\\({\\left(5x-4\\right)}^{\\frac{1}{4}}+7=9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169147082594\"><div data-type=\"problem\" id=\"fs-id1169147082596\"><p id=\"fs-id1169147082598\">\\(\\sqrt{x+1}-x+1=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144604719\"><p id=\"fs-id1169144604721\">\\(x=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144604734\"><div data-type=\"problem\" id=\"fs-id1169144604736\"><p id=\"fs-id1169144604738\">\\(\\sqrt{y+4}-y+2=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144376883\"><div data-type=\"problem\" id=\"fs-id1169144376885\"><p id=\"fs-id1169144376887\">\\(\\sqrt{z+100}-z=-10\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144376910\"><p id=\"fs-id1169144376912\">\\(z=21\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144376925\"><div data-type=\"problem\" id=\"fs-id1169144376927\"><p id=\"fs-id1169144376929\">\\(\\sqrt{w+25}-w=-5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144376966\"><div data-type=\"problem\" id=\"fs-id1169144376968\"><p id=\"fs-id1169144376970\">\\(3\\sqrt{2x-3}-20=7\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149355571\"><p id=\"fs-id1169149355573\">\\(x=42\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149355586\"><div data-type=\"problem\" id=\"fs-id1169149355588\"><p id=\"fs-id1169149355590\">\\(2\\sqrt{5x+1}-8=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149355632\"><div data-type=\"problem\" id=\"fs-id1169149355634\"><p id=\"fs-id1169149355636\">\\(2\\sqrt{8r+1}-8=2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149221178\"><p id=\"fs-id1169149221180\">\\(r=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149221193\"><div data-type=\"problem\" id=\"fs-id1169149221195\"><p id=\"fs-id1169149221197\">\\(3\\sqrt{7y+1}-10=8\\)<\/p><\/div><\/div><p id=\"fs-id1169149221240\"><strong data-effect=\"bold\">Solve Radical Equations with Two Radicals<\/strong><\/p><p id=\"fs-id1169149221245\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1169149221249\"><div data-type=\"problem\" id=\"fs-id1169149221251\"><p id=\"fs-id1169149221253\">\\(\\sqrt{3u+7}=\\sqrt{5u+1}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169139890415\"><p id=\"fs-id1169139890417\">\\(u=3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169139890430\"><div data-type=\"problem\" id=\"fs-id1169139890432\"><p id=\"fs-id1169139890434\">\\(\\sqrt{4v+1}=\\sqrt{3v+3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169139890478\"><div data-type=\"problem\" id=\"fs-id1169139890480\"><p id=\"fs-id1169139890482\">\\(\\sqrt{8+2r}=\\sqrt{3r+10}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169139890511\"><p id=\"fs-id1169139890513\">\\(r=-2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169146651576\"><div data-type=\"problem\" id=\"fs-id1169146651578\"><p id=\"fs-id1169146651580\">\\(\\sqrt{10+2c}=\\sqrt{4c+16}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169146651624\"><div data-type=\"problem\" id=\"fs-id1169146651626\"><p id=\"fs-id1169146651628\">\\(\\sqrt[3]{5x-1}=\\sqrt[3]{x+3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169146651659\"><p id=\"fs-id1169146651662\">\\(x=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144684269\"><div data-type=\"problem\" id=\"fs-id1169144684271\"><p id=\"fs-id1169144684273\">\\(\\sqrt[3]{8x-5}=\\sqrt[3]{3x+5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144684321\"><div data-type=\"problem\" id=\"fs-id1169144684323\"><p id=\"fs-id1169144684325\">\\(\\sqrt[3]{2{x}^{2}+9x-18}=\\sqrt[3]{{x}^{2}+3x-2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144684375\"><p id=\"fs-id1169149172962\">\\(x=-8,x=2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149172984\"><div data-type=\"problem\" id=\"fs-id1169149172986\"><p id=\"fs-id1169149172988\">\\(\\sqrt[3]{{x}^{2}-x+18}=\\sqrt[3]{2{x}^{2}-3x-6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149173059\"><div data-type=\"problem\" id=\"fs-id1169149173062\"><p id=\"fs-id1169149173064\">\\(\\sqrt{a}+2=\\sqrt{a+4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149359734\"><p id=\"fs-id1169149359736\">\\(a=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149359749\"><div data-type=\"problem\" id=\"fs-id1169149359751\"><p id=\"fs-id1169149359753\">\\(\\sqrt{r}+6=\\sqrt{r+8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149359784\"><div data-type=\"problem\" id=\"fs-id1169149359786\"><p id=\"fs-id1169149359788\">\\(\\sqrt{u}+1=\\sqrt{u+4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149359812\"><p id=\"fs-id1169149359814\">\\(u=\\frac{9}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144557874\"><div data-type=\"problem\" id=\"fs-id1169144557876\"><p id=\"fs-id1169144557878\">\\(\\sqrt{x}+1=\\sqrt{x+2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144557920\"><div data-type=\"problem\" id=\"fs-id1169144557922\"><p id=\"fs-id1169144557924\">\\(\\sqrt{a+5}-\\sqrt{a}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144557948\"><p id=\"fs-id1169144557950\">\\(a=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144557963\"><div data-type=\"problem\" id=\"fs-id1169144557965\"><p id=\"fs-id1169144557967\">\\(-2=\\sqrt{d-20}-\\sqrt{d}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144744956\"><div data-type=\"problem\" id=\"fs-id1169144744958\"><p id=\"fs-id1169144744960\">\\(\\sqrt{2x+1}=1+\\sqrt{x}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169144744986\"><p id=\"fs-id1169144744988\">\\(x=0\\phantom{\\rule{0.2em}{0ex}}x=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169144745009\"><div data-type=\"problem\" id=\"fs-id1169144745011\"><p id=\"fs-id1169144745013\">\\(\\sqrt{3x+1}=1+\\sqrt{2x-1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149327613\"><div data-type=\"problem\" id=\"fs-id1169149327615\"><p id=\"fs-id1169149327617\">\\(\\sqrt{2x-1}-\\sqrt{x-1}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149327648\"><p id=\"fs-id1169149327650\">\\(x=1\\phantom{\\rule{0.2em}{0ex}}x=5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149327671\"><div data-type=\"problem\" id=\"fs-id1169149327674\"><p id=\"fs-id1169149327676\">\\(\\sqrt{x+1}-\\sqrt{x-2}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149335518\"><div data-type=\"problem\" id=\"fs-id1169149335520\"><p id=\"fs-id1169149335522\">\\(\\sqrt{x+7}-\\sqrt{x-5}=2\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149335551\"><p id=\"fs-id1169149335553\">\\(x=9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149335566\"><div data-type=\"problem\" id=\"fs-id1169149335568\"><p id=\"fs-id1169149335570\">\\(\\sqrt{x+5}-\\sqrt{x-3}=2\\)<\/p><\/div><\/div><p id=\"fs-id1169140418470\"><strong data-effect=\"bold\">Use Radicals in Applications<\/strong><\/p><p id=\"fs-id1169140418475\">In the following exercises, solve. Round approximations to one decimal place.<\/p><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1169140418482\"><strong data-effect=\"bold\">Landscaping<\/strong> Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula \\(s=\\sqrt{A}\\) to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169140418502\"><p id=\"fs-id1169140418504\">\\(8.7\\) feet<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169140418514\"><div data-type=\"problem\" id=\"fs-id1169140418516\"><p id=\"fs-id1169140418518\"><strong data-effect=\"bold\">Landscaping<\/strong> Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130 square feet. Use the formula \\(s=\\sqrt{A}\\) to find the length of each side of his patio. Round your answer to the nearest tenth of a foot.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169140418549\"><div data-type=\"problem\" id=\"fs-id1169140418551\"><p id=\"fs-id1169140418553\"><strong data-effect=\"bold\">Gravity<\/strong> A hang glider dropped his cell phone from a height of 350 feet. Use the formula \\(t=\\frac{\\sqrt{h}}{4}\\) to find how many seconds it took for the cell phone to reach the ground.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149235355\"><p id=\"fs-id1169149235357\">\\(4.7\\) seconds<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149235366\"><div data-type=\"problem\" id=\"fs-id1169149235368\"><p id=\"fs-id1169149235371\"><strong data-effect=\"bold\">Gravity<\/strong> A construction worker dropped a hammer while building the Grand Canyon skywalk, 4000 feet above the Colorado River. Use the formula \\(t=\\frac{\\sqrt{h}}{4}\\) to find how many seconds it took for the hammer to reach the river.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149235406\"><div data-type=\"problem\" id=\"fs-id1169149235408\"><p id=\"fs-id1169149235410\"><strong data-effect=\"bold\">Accident investigation<\/strong> The skid marks for a car involved in an accident measured 216 feet. Use the formula \\(s=\\sqrt{24d}\\) to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149235432\"><p id=\"fs-id1169149235434\">72 feet<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149235439\"><div data-type=\"problem\" id=\"fs-id1169149235441\"><p id=\"fs-id1169149235443\"><strong data-effect=\"bold\">Accident investigation<\/strong> An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. Use the formula \\(s=\\sqrt{24d}\\) to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1169149335189\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1169149335196\"><div data-type=\"problem\" id=\"fs-id1169149335198\"><p id=\"fs-id1169149335200\">Explain why an equation of the form \\(\\sqrt{x}+1=0\\) has no solution.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1169149335219\"><p id=\"fs-id1169149335221\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1169149335226\"><div data-type=\"problem\" id=\"fs-id1169149335229\"><p id=\"fs-id1169149335231\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> Solve the equation \\(\\sqrt{r+4}-r+2=0.\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Explain why one of the \u201csolutions\u201d that was found was not actually a solution to the equation.<\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169149335281\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1169144746564\"><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-id1169144746579\" data-alt=\"The table has 4 columns and 4 rows. The first row is a header row with the headers \u201cI can\u2026\u201d, \u201cConfidently\u201d, \u201cWith some help.\u201d, and \u201cNo \u2013 I don\u2019t get it!\u201d. The first column contains the phrases \u201cSolve radical equations\u201d, \u201csolve radical equations with two radicals\u201d, and \u201cuse radicals in applications\u201d. The other columns are left blank so the learner can indicate their level of understanding.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The table has 4 columns and 4 rows. The first row is a header row with the headers \u201cI can\u2026\u201d, \u201cConfidently\u201d, \u201cWith some help.\u201d, and \u201cNo \u2013 I don\u2019t get it!\u201d. The first column contains the phrases \u201cSolve radical equations\u201d, \u201csolve radical equations with two radicals\u201d, and \u201cuse radicals in applications\u201d. The other columns are left blank so the learner can indicate their level of understanding.\"><\/span><p id=\"fs-id1169144746585\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1169144746596\"><dt>radical equation<\/dt><dd id=\"fs-id1169144746602\">An equation in which a variable is in the radicand of a radical expression is called a radical equation.<\/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>Solve radical equations<\/li>\n<li>Solve radical equations with two radicals<\/li>\n<li>Use radicals in applications<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149032213\" class=\"be-prepared\">\n<p id=\"fs-id1169144829761\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1169146738034\" type=\"1\">\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6981d38b1346148fbe7ba0244d3b269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"64\" 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-id1167836660220\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62debeb15cb01fa04cb6836beff0390b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;&#45;&#53;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"86\" style=\"vertical-align: 0px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836432956\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc86700d5437fb9a6a0a0070a9679b6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#110;&#43;&#56;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"127\" style=\"vertical-align: -2px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/da8478b4-93bc-4919-81a1-5e3267050e7e#fs-id1167836625705\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169146610807\">\n<h3 data-type=\"title\">Solve Radical Equations<\/h3>\n<p id=\"fs-id1169148909008\">In this section we will solve equations that have a variable in the radicand of a radical expression. An equation of this type is called a <span data-type=\"term\">radical equation<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1169149330689\">\n<div data-type=\"title\">Radical Equation<\/div>\n<p id=\"fs-id1169148989002\">An equation in which a variable is in the radicand of a radical expression is called a <strong data-effect=\"bold\">radical equation<\/strong>.<\/p>\n<\/div>\n<p id=\"fs-id1169146646052\">As usual, when solving these equations, what we do to one side of an equation we must do to the other side as well. Once we isolate the radical, our strategy will be to raise both sides of the equation to the power of the index. This will eliminate the radical.<\/p>\n<p id=\"fs-id1169149319453\">Solving radical equations containing an even index by raising both sides to the power of the index may introduce an algebraic solution that would not be a solution to the original radical equation. Again, we call this an extraneous solution as we did when we solved rational equations.<\/p>\n<p id=\"fs-id1169149011290\">In the next example, we will see how to solve a radical equation. Our strategy is based on raising a radical with index <em data-effect=\"italics\">n<\/em> to the <em data-effect=\"italics\">n<\/em><sup>th<\/sup> power. This will eliminate the radical.<\/p>\n<div data-type=\"equation\" id=\"fs-id1169146621794\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8223c6470025e204e704cbd1a1c38042_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#92;&#103;&#101;&#32;&#48;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#110;&#93;&#123;&#97;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#61;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"169\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"example\" id=\"fs-id1169149296250\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a Radical Equation<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148959770\">\n<div data-type=\"problem\" id=\"fs-id1169149340153\">\n<p id=\"fs-id1169147107160\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c615b159b9c71c15a882e59ec64d72e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#110;&#45;&#52;&#125;&#45;&#57;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144382249\"><span data-type=\"media\" id=\"fs-id1169149094355\" data-alt=\"Step 1 is to isolate the radical on one side of the equation. To isolate the radical add 9 to both sides. The resulting equation is square root of the quantity 5 n minus 4 in parentheses minus 9 plus 9 equals 0 plus 9. This simplifies to square root of the quantity 5 n minus 4 in parentheses equals 9.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_001a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to isolate the radical on one side of the equation. To isolate the radical add 9 to both sides. The resulting equation is square root of the quantity 5 n minus 4 in parentheses minus 9 plus 9 equals 0 plus 9. This simplifies to square root of the quantity 5 n minus 4 in parentheses equals 9.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169146741252\" data-alt=\"Step 2 is to raise both sides of the equation to the power of the index. Since the index of a square root is 2, we square both sides. Remember that the square of the square root of \u201ca\u201d is equal to \u201ca\u201d. The equation that results is the square of the square root of the quantity 5 n minus 4 in parentheses equals 9 squared. This simplifies to 5 n minus 4 equals 81.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_001b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to raise both sides of the equation to the power of the index. Since the index of a square root is 2, we square both sides. Remember that the square of the square root of \u201ca\u201d is equal to \u201ca\u201d. The equation that results is the square of the square root of the quantity 5 n minus 4 in parentheses equals 9 squared. This simplifies to 5 n minus 4 equals 81.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169149169210\" data-alt=\"Step 3 is to solve the new equation. We get 5 n equals 85 and then n equals 17.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_001c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to solve the new equation. We get 5 n equals 85 and then n equals 17.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169148971297\" data-alt=\"Step 4 is to check the answer in the original equation. Does the square root of the quantity 5 times 17 minus 4 in parentheses minus 9 equal zero? Simplifying the left side we get square root of the quantity 85 minus 4 in parentheses minus 9 and then square root of 81 minus 9 and then 9 minus 9 which does equal 0. This verifies that the solution is n equals 17.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_001d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to check the answer in the original equation. Does the square root of the quantity 5 times 17 minus 4 in parentheses minus 9 equal zero? Simplifying the left side we get square root of the quantity 85 minus 4 in parentheses minus 9 and then square root of 81 minus 9 and then 9 minus 9 which does equal 0. This verifies that the solution is n equals 17.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144560187\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147133968\">\n<div data-type=\"problem\" id=\"fs-id1169146632210\">\n<p id=\"fs-id1169148858103\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79302f135cc8ed9b1534ea2e9d38e311_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#109;&#43;&#50;&#125;&#45;&#53;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"136\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148965261\">\n<p id=\"fs-id1169149223370\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb3f6e57e9f4e7b8131b63d632802c6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#51;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148939407\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149144348\">\n<div data-type=\"problem\" id=\"fs-id1169146648761\">\n<p id=\"fs-id1169146645178\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-403b5d81caa1c6160522d2bf75d26b37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#122;&#43;&#49;&#125;&#45;&#50;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"138\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149196136\">\n<p id=\"fs-id1169144383780\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-944393b942245d7e36307ef1f7b1e012_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"49\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149319438\" class=\"howto\">\n<div data-type=\"title\">Solve a radical equation with one radical.<\/div>\n<ol id=\"fs-id1169149302699\" type=\"1\" class=\"stepwise\">\n<li>Isolate the radical on one side of the equation.<\/li>\n<li>Raise both sides of the equation to the power of the index.<\/li>\n<li>Solve the new equation.<\/li>\n<li>Check the answer in the original equation.<\/li>\n<\/ol>\n<\/div>\n<p>When we use a radical sign, it indicates the principal or positive root. If an equation has a radical with an even index equal to a negative number, that equation will have no solution.<\/p>\n<div data-type=\"example\" id=\"fs-id1169144604426\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169146719065\">\n<div data-type=\"problem\" id=\"fs-id1169149004291\">\n<p id=\"fs-id1169149103407\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7c741043db744143df7bca66727e4285_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#57;&#107;&#45;&#50;&#125;&#43;&#49;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"130\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149196386\">\n<table id=\"fs-id1169148968245\" class=\"unnumbered unstyled\" summary=\"To isolate the radical, subtract 1 from both sides. The resulting equation is square root of the quantity 9 k minus 2 in parentheses plus 1 minus 1 equals 0 minus 1. Simplifying this we get square root of the quantity 9 k minus 2 in parentheses equals negative 1. Since the square root of a real number is always positive there is no solution to the equation.\" 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-id1169149194537\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\">To isolate the radical, subtract 1 to both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149358284\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144383049\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169149094818\">Because the square root is equal to a negative number, the equation has no solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144523599\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169146799266\">\n<div data-type=\"problem\" id=\"fs-id1169144359277\">\n<p id=\"fs-id1169149027710\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29ac69673208feeffee5a224fe44443a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#114;&#45;&#51;&#125;&#43;&#53;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148915761\">\n<p id=\"fs-id1169149289222\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43906dc665a610285c8bc07c365825b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"87\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149026518\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169146666654\">\n<div data-type=\"problem\" id=\"fs-id1169149361766\">\n<p id=\"fs-id1169144378163\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-82db2b3e07c627245723091d7e692731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#115;&#45;&#51;&#125;&#43;&#50;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144642391\">\n<p id=\"fs-id1169149367840\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43906dc665a610285c8bc07c365825b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"87\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169144555878\">If one side of an equation with a square root is a binomial, we use the Product of Binomial Squares Pattern when we square it.<\/p>\n<div data-type=\"note\" id=\"fs-id1169149319586\">\n<div data-type=\"title\">Binomial Squares<\/div>\n<div data-type=\"equation\" id=\"fs-id1169148998889\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7374aef25bc7418091e811b63f5782ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#92;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"103\" style=\"vertical-align: -146px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169140091544\">Don\u2019t forget the middle term!<\/p>\n<div data-type=\"example\" id=\"fs-id1169144365771\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169144516404\">\n<div data-type=\"problem\">\n<p id=\"fs-id1169149121716\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a5850421605f479813dbf2d768b691b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#112;&#45;&#49;&#125;&#43;&#49;&#61;&#112;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144687863\">\n<table id=\"fs-id1169149215106\" class=\"unnumbered unstyled can-break\" summary=\"To isolate the radical, subtract 1 from both sides. The resulting equation is square root of the quantity p minus 1 in parentheses plus 1 minus 1 equals p minus 1. This simplifies to is square root of the quantity p minus 1 in parentheses equals p minus 1. Squaring both sides of the equation we get the square of the square root of the quantity p minus 1 in parentheses equals the square of the quantity p minus 1. Simplify using the binomial squares pattern on the right. The simplified equation is p minus 1 equals p squared minus 2 p plus 1. This is a quadratic equation, so get zero on one side. 0 equals p squared minus 3 p plus 2. Factor the right side. 0 equals the product of the quantity p minus 1 in parentheses with the quantity p minus 2 in parentheses. Use the zero procuct property. 0 equals p minus 1 and 0 equals p minus 2. Solving each equation we get p equals 1 and p equals 2. Checking the answer p equals 1. Does the square root of the quantity 1 minus 1 in parentheses equal 1? The square root of zero plus 1 equals 1 so p equals 1 is a solution. Checking the answer p equals 2. Does the square root of the quantity 2 minus 1 in parentheses plus 1 equal 2. Since the square root of 1 plus 1 equals 2, p equals 2 is also a solution. The solutions are p equals 1 and p equals 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-id1169149297082\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\">To isolate the radical, subtract 1 from both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146742316\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Square both sides of the equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146617711\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003e_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, using the Product of Binomial Squares Pattern on the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>right. Then solve the new equation.<\/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_06_003f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">It is a quadratic equation, so get zero on one side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148956042\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the right side.<\/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_06_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the Zero Product Property.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149373628\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve each equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147107710\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check the answers.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149011207\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The solutions are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a89e9d48a22a11e323d1828285629bbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#49;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#112;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144560422\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1169146667843\">\n<p id=\"fs-id1169144876584\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ff66ff4534ae741d49cc1db97cc2ffd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#45;&#50;&#125;&#43;&#50;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144892743\">\n<p id=\"fs-id1169148950842\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-224694a083806d414fab5eec8a159cce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#44;&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"93\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148966020\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148932918\">\n<div data-type=\"problem\" id=\"fs-id1169149040869\">\n<p id=\"fs-id1169146656984\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd0737a3899f98a030dd4b110140dd15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#121;&#45;&#53;&#125;&#43;&#53;&#61;&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149211148\">\n<p id=\"fs-id1169149014631\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94178b522af9e947b1e8aebd9adfa6fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;&#44;&#121;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169146645827\">When the index of the radical is 3, we cube both sides to remove the radical.<\/p>\n<div data-type=\"equation\" id=\"fs-id1168040445163\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1183d5a8f17ee1e5d3e8d13b722ab6d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#97;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#61;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"example\" id=\"fs-id1169149212535\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149308141\">\n<div data-type=\"problem\" id=\"fs-id1169146652198\">\n<p id=\"fs-id1169144379849\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4316170d8561be6d3f848b7a27b8b625_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#120;&#43;&#49;&#125;&#43;&#56;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"130\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149095264\">\n<table id=\"fs-id1169139903023\" class=\"unnumbered unstyled can-break\" summary=\"To isolate the radical, subtract 8 from both sides. The resulting equation is cube root of the quantity 5 x plus 1 equals negative 4. Cubeing both sides of the equation we get the cube of the cube root of the quantity 5 x plus 1 equals the cube of negative 4. The simplified equation is 5 x plus 1 equals negative 64. This simplifies to 5 x equals negative 65. So x equals negative 13. Checking the answer x equals negative 13. Does the cube root of the quantity 5 times negative 13 plus 1 in parentheses plus 8 equal 4? The cube root of negative 64 plus 8 equals negative 4 plus 8 which equals 4 so the solution is x equals negative 13.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14760efb5fbfcdee92935524e8a3f25c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#120;&#43;&#49;&#125;&#43;&#56;&#61;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To isolate the radical, subtract 8 from both sides.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fa1c5f0db34f4f49ad5173de4c07287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#120;&#43;&#49;&#125;&#61;&#45;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Cube both sides of the equation.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af87100d850a737a1a41b556d1c8c0ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#120;&#43;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"154\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-658aecedb974b1b8454d25c8fe29beae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#43;&#49;&#61;&#45;&#54;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"105\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve the equation.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68dbc0b9c07c7e85c260408f4ba45c48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#120;&#61;&#45;&#54;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"74\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67970bc08b9ff1c694b081a220f1b807_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"66\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check the answer.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146655896\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\">The solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22b03d34ee0d0f721fe176d5a1c7a635_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#49;&#51;&#46;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"70\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144601269\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149215173\">\n<div data-type=\"problem\" id=\"fs-id1169148956225\">\n<p id=\"fs-id1169147087056\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1bff8900d187e4f3caa213dc1626db13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#120;&#45;&#51;&#125;&#43;&#56;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148994363\">\n<p id=\"fs-id1169149154688\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77758982e3a224cd6ee6269fafa47055_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144377434\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149212463\">\n<div data-type=\"problem\" id=\"fs-id1169149343472\">\n<p id=\"fs-id1169144490955\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01947044e949aedfec637115d6d47fe7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#54;&#120;&#45;&#49;&#48;&#125;&#43;&#49;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144559943\">\n<p id=\"fs-id1169149095754\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7bbade6e6bfe4cc0591578b92c63062b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149107852\">Sometimes an equation will contain rational exponents instead of a radical. We use the same techniques to solve the equation as when we have a radical. We raise each side of the equation to the power of the denominator of the rational exponent. Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ffe54a0a065c4f2fd6ee0a1d7df1a5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#109;&middot;&#110;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"103\" style=\"vertical-align: -4px;\" \/> we have for example,<\/p>\n<div data-type=\"equation\" id=\"fs-id1169144451236\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68191ca79516f4fc77a25ced0c6791cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#120;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"180\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"fs-id1169147109917\">Remember, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-958567417144abef7df581b7b13f22e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8896653fbf373dcb86ff12db306a5569_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"example\" id=\"fs-id1169148952413\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169149297527\">\n<div data-type=\"problem\" id=\"fs-id1169146732624\">\n<p id=\"fs-id1169144604112\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9daa6976c3a4aee84343315bccf0943_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#43;&#51;&#61;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144555211\">\n<table id=\"fs-id1169146594997\" class=\"unnumbered unstyled can-break\" summary=\"To isolate the term with the rational exponent, subtract 3 from both sides. The resulting equation is the quantity 3 x minus 2 raised to the power of one fourth equals 2. Raising each side to the fourth power we get the fourth power of the quantity 3 x minus 2 in parentheses raised to the power of one fourth in parentheses equals 2 to the fourth power. The simplified equation is 3 x minus 2 equals 16. This simplifies to 3 x equals 18. So x equals 6. Checking the answer x equals 6. Does the quantity 3 times 6 minus 2 in parentheses raised to the one-fourth power plus 3 equal 5? The quantity 18 minus 2 in parentheses raised to the one-fourth power plus 3 equals 16 to the one fourth power plus 3 which equals 2 plus 3 which does equal 5 so the solution is x equals 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=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-184c1dcbd0f3a082740b2cb0c07d3192_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#43;&#51;&#61;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"135\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To isolate the term with the rational exponent,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>subtract 3 from both sides.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3c92fad37116cec60191b9318ec1b9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#61;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Raise each side of the equation to the fourth power.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-63609a1af9f5cc66ac902020a05458b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"153\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e2e558aaf2b92a16a3e1b9fa3c091f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#50;&#61;&#49;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"91\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve the equation.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fc853e3249252eacf8d155885386c48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#61;&#49;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-319e31e8e53ee9f49d9cd16c28df6b30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#49;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check the answer.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147115372\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\">The solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1a0c18aca243df63c2c8c1acf2973c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#54;&#46;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#57;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149042758\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149030038\">\n<div data-type=\"problem\" id=\"fs-id1169149030828\">\n<p id=\"fs-id1169149374970\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc9d545f2b9e504f9113ddf68d954b23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#45;&#50;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149006444\">\n<p id=\"fs-id1169149012975\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1dc438da70f358c2eb1bf64a8b7ea4b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169138877440\">\n<div data-type=\"problem\" id=\"fs-id1169149088428\">\n<p id=\"fs-id1169149013626\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-327b2f332c75739e730017609afc8693_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#43;&#53;&#61;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146668137\">\n<p id=\"fs-id1169146816710\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-116dfacb4c2a32bdfb5ab9e852f29986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Sometimes the solution of a radical equation results in two algebraic solutions, but one of them may be an <span data-type=\"term\" class=\"no-emphasis\">extraneous solution<\/span>!<\/p>\n<div data-type=\"example\" id=\"fs-id1169148939142\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148850042\">\n<div data-type=\"problem\">\n<p id=\"fs-id1169149012026\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de5762b213341fbe4e1d316c98e5010e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#43;&#52;&#125;&#45;&#114;&#43;&#50;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149012422\">\n<table id=\"fs-id1169148972821\" class=\"unnumbered unstyled can-break\" summary=\"First, isolate the radical. The resulting equation is square root of the quantity r plus 4 in parentheses equals r minus 2. Squaring both sides of the equation we get the square of the square root of the quantity r plus 4 in parentheses equals the square of the quantity r minus 2 in parentheses. Simplify using the binomial squares pattern on the right. The simplified equation is r plus 4 equals r squared minus 4 r plus 4. This is a quadratic equation, so get zero on one side. 0 equals r squared minus 5 r. Factor the right side. 0 equals r times the quantity r minus 5 in parentheses. Use the zero procuct property. 0 equals r and 0 equals r minus 5. Solving each equation we get r equals 0 and r equals 5. Checking the answer r equals 0. Does the square root of the quantity 0 plus 4 in parentheses minus 0 plus 2 equal 0? The square root of 4 plus 2 equals 0 so r equals 0 is a solution. Checking the answer r equals 5. Does the square root of the quantity 5 plus 4 in parentheses minus 5 plus 2 equal 0. Since the square root of 9 minus 3 equals 0, r equals 5 is also a solution. The solutions are r equals 0 and r equals 5.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d34fbac258b4f162146bf856934ee747_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#43;&#52;&#125;&#45;&#114;&#43;&#50;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"146\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Isolate the radical.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a4c1f087cfd64f1a5b8cc9b965fb70f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#43;&#52;&#125;&#61;&#114;&#45;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Square both sides of the equation.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4246a5f752bc0e7d5e30ee732e1c728d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#43;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"159\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify and then solve the equation<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16e5cb9a4b810f9266d321bd3a213894_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#43;&#52;&#61;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#114;&#43;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"149\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">It is a quadratic equation, so get zero on<\/p>\n<div data-type=\"newline\"><\/div>\n<p>one side.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3633280155320bcabdbd13c3fd5d866_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#61;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#114;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"87\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the right side.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7fb513a39f29d8df1887f16edb63d68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#61;&#114;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the Zero Product Property.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f86b6047e4e22fe3d4d803a90ff532f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#61;&#114;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#61;&#114;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"138\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve the equation.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f697ba54eae71f753c0693b4b2ef31b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#48;&#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;&#114;&#61;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"99\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check your answer.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148859069\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"bottom\" data-align=\"left\">The solution is <em data-effect=\"italics\">r<\/em> = 5.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0844d5b60655fd4a6e477dfcceba4341_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/> is an extraneous solution.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149224379\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148858909\">\n<div data-type=\"problem\" id=\"fs-id1169149028999\">\n<p id=\"fs-id1169146627237\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e03d6196bc3b2d5260f0eaf53df59e37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#43;&#57;&#125;&#45;&#109;&#43;&#51;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"164\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149122786\">\n<p id=\"fs-id1169147085867\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fae83aecfe55db8b07183bb7fb07150c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149144336\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148994616\">\n<div data-type=\"problem\">\n<p id=\"fs-id1169149140803\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa94c45c4cf504e173d141d21836cfee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#43;&#49;&#125;&#45;&#110;&#43;&#49;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"154\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149010540\">\n<p id=\"fs-id1169149346391\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-85761a607075b960ff00638d721cfe9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169139960604\">When there is a coefficient in front of the radical, we must raise it to the power of the index, too.<\/p>\n<div data-type=\"example\" id=\"fs-id1169149294243\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169144557460\">\n<div data-type=\"problem\" id=\"fs-id1169146643951\">\n<p id=\"fs-id1169149102502\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e38b1c641bf7b59ce611e8535b12984_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#120;&#45;&#53;&#125;&#45;&#56;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146738556\">\n<table id=\"fs-id1169149115633\" class=\"unnumbered unstyled can-break\" summary=\"First, isolate the radical term by adding 8 to both sides. 3 times square root of the quantity 3 x minus 5 in parentheses equals 12. Isolate the radical by dividing both sides by 3. Square root of the quantity 3 x minus 5 in parentheses equals 4. Square both sides of the equation. The square of the square root of the quantity 3 x minus 5 in parentheses equals 4 squared. Simplify, then solve the new equation. 3 x minus 5 equals 16. 3 x equals 21. x equals 7. Check the answer x equals 7. Does 3 times the square root of the quantity 3 times 7 minus 5 in parentheses minus 8 equal 4? Simplifying the left side we get 3 times the square root of the quantity 21 minus 5 in parentheses minus 8 which equals 3 times the square root of 16 minus 8 which equals 3 times 4 minus 8 which equals 4. The solution is x equals 7.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3312a4cd74f33d07ebead3ba2aa11c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#120;&#45;&#53;&#125;&#45;&#56;&#61;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"140\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Isolate the radical term.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3fba70dc3fb0cafcd09804d01cc42599_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#120;&#45;&#53;&#125;&#61;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Isolate the radical by dividing both sides by 3.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d727e1656fd8d50cb8fc797abf7372ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#120;&#45;&#53;&#125;&#61;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Square both sides of the equation.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ba1af4b8c372cb7a4efa72b9ced0b6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#120;&#45;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"140\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify, then solve the new equation.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d45cecfb9fc47f1c97dc5f828eeed4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#45;&#53;&#61;&#49;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"91\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd59fe6ab8a8043ada51e15b34a20476_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#61;&#50;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Solve the equation.<\/td>\n<td data-valign=\"top\" data-align=\"right\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1ee6f42b11ae5593c449d12a1c4f4c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#55;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check the answer.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144684619\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"right\">The solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b0624b180456707c99bcbbc576710d93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#55;&#46;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#48;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148844737\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169144615527\">\n<div data-type=\"problem\" id=\"fs-id1169149308487\">\n<p id=\"fs-id1169140055625\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d48e56326700c668e70cf1070a654160_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#97;&#43;&#52;&#125;&#45;&#49;&#54;&#61;&#49;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"157\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1169149370615\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97b59af4b38dfd7b51199ac6c2744c99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#54;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149329127\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149117576\">\n<div data-type=\"problem\" id=\"fs-id1169148859787\">\n<p id=\"fs-id1169149096759\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa52ff0f20a956f42923a08d45d56964_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#98;&#43;&#51;&#125;&#45;&#50;&#53;&#61;&#53;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"156\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144377869\">\n<p id=\"fs-id1169144432792\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c9332297006171ed454dcbcd8ac7977_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#51;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149230183\">\n<h3 data-type=\"title\">Solve Radical Equations with Two Radicals<\/h3>\n<p id=\"fs-id1169149144685\">If the radical equation has two radicals, we start out by isolating one of them. It often works out easiest to isolate the more complicated radical first.<\/p>\n<p id=\"fs-id1169148968522\">In the next example, when one radical is isolated, the second radical is also isolated.<\/p>\n<div data-type=\"example\" id=\"fs-id1169148915268\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148865471\">\n<div data-type=\"problem\" id=\"fs-id1169149285487\">\n<p id=\"fs-id1169149311031\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d083acfd2d66c4fa081f81b6af2982c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#120;&#45;&#51;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#120;&#43;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"156\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148926116\">\n<p id=\"fs-id1169149351070\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29f077d07a0f64f2218c72401fadee54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#97;&#114;&#101;&#32;&#105;&#115;&#111;&#108;&#97;&#116;&#101;&#100;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#120;&#45;&#51;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#120;&#43;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#101;&#32;&#116;&#104;&#101;&#32;&#105;&#110;&#100;&#101;&#120;&#32;&#105;&#115;&#32;&#51;&#44;&#32;&#99;&#117;&#98;&#101;&#32;&#98;&#111;&#116;&#104;&#32;&#115;&#105;&#100;&#101;&#115;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#120;&#45;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#120;&#43;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#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;&#44;&#32;&#116;&#104;&#101;&#110;&#32;&#115;&#111;&#108;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#110;&#101;&#119;&#32;&#101;&#113;&#117;&#97;&#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;&#52;&#120;&#45;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#120;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#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;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#116;&#104;&#101;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#108;&#101;&#97;&#118;&#101;&#32;&#105;&#116;&#32;&#116;&#111;&#32;&#121;&#111;&#117;&#32;&#116;&#111;&#32;&#115;&#104;&#111;&#119;&#32;&#116;&#104;&#97;&#116;&#32;&#53;&#32;&#99;&#104;&#101;&#99;&#107;&#115;&#33;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"194\" width=\"662\" style=\"vertical-align: -91px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169146612764\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149220632\">\n<div data-type=\"problem\" id=\"fs-id1169146594078\">\n<p id=\"fs-id1169147088008\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d64d88e326b269911eb86448ef9df07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#120;&#45;&#52;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#120;&#43;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"156\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146745629\">\n<p id=\"fs-id1169146817003\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169146665288\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148974254\">\n<div data-type=\"problem\" id=\"fs-id1169146652924\">\n<p id=\"fs-id1169146608926\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b0b6344df716fadcf7437367919e134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#55;&#120;&#43;&#49;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#120;&#45;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"156\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149350577\">\n<p id=\"fs-id1169149222427\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acd63cd7241d63fe0b61797be711ee91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149285371\">Sometimes after raising both sides of an equation to a power, we still have a variable inside a radical. When that happens, we repeat Step 1 and Step 2 of our procedure. We isolate the radical and raise both sides of the equation to the power of the index again.<\/p>\n<div data-type=\"example\" id=\"fs-id1169149376671\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a Radical Equation<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149010550\">\n<div data-type=\"problem\" id=\"fs-id1169146668708\">\n<p>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc681b7aa79866393c9c12bb42ab43ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#125;&#43;&#49;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#109;&#43;&#57;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149343510\"><span data-type=\"media\" id=\"fs-id1169149169645\" data-alt=\"Step 1 is to isolate one of the radical terms on one side of the equation. The radical on the right is isolated.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_008a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to isolate one of the radical terms on one side of the equation. The radical on the right is isolated.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169144788286\" data-alt=\"Step 2 is to raise both sides of the equation to the power of the index. We square both sides. The equation that results is the square of the quantity square root of m plus 1 in parentheses equals the square of the square root of the quantity m plus 9 in parentheses. Simplify \u2013 be very careful as you multiply! This simplifies to m plus 2 times square root m plus 1 equals m plus 9.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_008b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to raise both sides of the equation to the power of the index. We square both sides. The equation that results is the square of the quantity square root of m plus 1 in parentheses equals the square of the square root of the quantity m plus 9 in parentheses. Simplify \u2013 be very careful as you multiply! This simplifies to m plus 2 times square root m plus 1 equals m plus 9.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169146731390\" data-alt=\"Step 3 is to repeat steps 1 and 2 again if there are any more radicals. There is still a radical in the equation. So we must repeat the previous steps. Isolate the radical term. 2 times square root m equals 8. Here, we can easily isolate the radical by dividing both sides by 2. We get square root m equals 4. Squaring both sides we get the square of the square root of m equals 4 squared. m equals 16.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_008c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to repeat steps 1 and 2 again if there are any more radicals. There is still a radical in the equation. So we must repeat the previous steps. Isolate the radical term. 2 times square root m equals 8. Here, we can easily isolate the radical by dividing both sides by 2. We get square root m equals 4. Squaring both sides we get the square of the square root of m equals 4 squared. m equals 16.\" \/><\/span><span data-type=\"media\" id=\"fs-id1169146738581\" data-alt=\"Step 4 is to check the answer in the original equation. Does the square root of 16 plus 1 equal the square root of the quantity 16 plus 9? Simplifying both sides we get 4 plus 1 equals 5. This verifies that the solution is m equals 16.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_008d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to check the answer in the original equation. Does the square root of 16 plus 1 equal the square root of the quantity 16 plus 9? Simplifying both sides we get 4 plus 1 equals 5. This verifies that the solution is m equals 16.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149011708\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169144885903\">\n<div data-type=\"problem\" id=\"fs-id1169146660138\">\n<p id=\"fs-id1169144373778\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2d117a84b5fdbd52e793563f34c1959_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#45;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148875002\">\n<p id=\"fs-id1169148820998\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2145acc2878ed61214887e120f2485b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148875485\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169147027836\">\n<div data-type=\"problem\" id=\"fs-id1169149003300\">\n<p id=\"fs-id1169148826041\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-313f4a4db7018159d1ef60ce98f7c2b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#43;&#50;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#43;&#49;&#54;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"146\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149295810\">\n<p id=\"fs-id1169148894141\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-300a345ef7b973d34879ac8e90555390_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149329150\">We summarize the steps here. We have adjusted our previous steps to include more than one radical in the equation This procedure will now work for any radical equations.<\/p>\n<div data-type=\"note\" id=\"fs-id1169148926139\" class=\"howto\">\n<div data-type=\"title\">Solve a radical equation.<\/div>\n<ol id=\"fs-id1169148866012\" type=\"1\" class=\"stepwise\">\n<li>Isolate one of the radical terms on one side of the equation.<\/li>\n<li>Raise both sides of the equation to the power of the index.<\/li>\n<li>Are there any more radicals?\n<div data-type=\"newline\"><\/div>\n<p> If yes, repeat Step 1 and Step 2 again.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> If no, solve the new equation.<\/li>\n<li>Check the answer in the original equation.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1169146647278\">Be careful as you square binomials in the next example. Remember the pattern is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1cfbdf72854a6dc63ba0f2ae17f46708_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3776a27018a543463fbf0547add5076c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"188\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"example\" id=\"fs-id1169146664073\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148995792\">\n<div data-type=\"problem\" id=\"fs-id1169149284527\">\n<p id=\"fs-id1169148835927\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e283fc4fd893e1ee0b84104874f5258_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#113;&#45;&#50;&#125;&#43;&#51;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#113;&#43;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"174\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144744387\">\n<table id=\"fs-id1169144715813\" class=\"unnumbered unstyled can-break\" summary=\"The radical on the right is isolated. Square both sides. The equation that results is the square of the sum of the square root of the quantity q minus 2 in parentheses and 3 in parentheses equals the square of the square root of the quantity 4 q plus 1 in parentheses. This simplifies to q minus 2 plus 6 times square root of the quantity q minus 2 in parentheses plus 9 equals 4 q plus 1. There is still a radical in the equation. So we must repeat the previous steps. Isolate the radical term. 6 times square root of the quantity q minus 2 in parentheses equals 3 q minus 6. It would not help to divide both sides by 6. Squaring both sides we get the square of the product of 6 and the square root of the quantity q minus 2 in parentheses the square of the quantity 3 q minus 6 in parentheses. Remember to square both the 6 and the square root of the quantity q minus 2. When squaring the right side use the formula the quantity a minus b in parentheses squared equals a squared minus 2 a b plus b squared. The resulting equation is 6 squared times the square of the square root of the quantity q minus 2 in parentheses equals the quantity 3 q in parentheses squared minus 2 times 3 q times 6 plus 6 squared. Simplifying we get 36 times the quantity q minus 2 in parentheses equals 9 q squared minus 36 q plus 36. Distributing we get 36 q minus 72 equals 9 q squared minus 36 q plus 36. It is a quadratic equation, so get zero on one side. 0 equals 9 q squared minus 72 q plus 108. Factor the right side to get 0 equals 9 times the quantity q minus 6 in parentheses times the quantity q minus 2 in parentheses. Use the zero product property to get the equations q minus 6 equals 0 and q minus 2 equals 0. Solving eah equation we get q equals 6 and q equals 2. The checks are left to you. The solutions are q equals 6 and q equals 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-id1169146613138\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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 radical on the right is isolated. Square<\/p>\n<div data-type=\"newline\"><\/div>\n<p>both sides.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148869414\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144551290\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\">There is still a radical in the equation so<\/p>\n<div data-type=\"newline\"><\/div>\n<p>we must repeat the previous steps. Isolate<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the radical.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144796710\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\">Square both sides. It would not help to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>divide both sides by 6. Remember to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>square both the 6 and the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-630737492b19ab314c050e808471597f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#113;&#45;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146661975\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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, then solve the new equation.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148911599\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148924892\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\">It is a quadratic equation, so get zero on<\/p>\n<div data-type=\"newline\"><\/div>\n<p>one side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148967666\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the right side.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146637367\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the Zero Product Property.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146627489\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_009j_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 checks are left to you.<\/td>\n<td data-valign=\"top\" data-align=\"center\">The solutions are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33a082aec42fb4613663bc3c562ba227_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b2497a35ac33deaad14714cb7a68bba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149121469\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1169144568120\">\n<p id=\"fs-id1169146740691\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f78ef6d4b7e99987e31eb3e3ed7296d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#45;&#49;&#125;&#43;&#50;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#43;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"173\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146740572\">\n<p id=\"fs-id1169149172270\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ddab230605c435eb8b7408a736d3e77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148957835\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148929863\">\n<div data-type=\"problem\" id=\"fs-id1169144377478\">\n<p id=\"fs-id1169149118407\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-debde5330aed4b0f69b0c36651247b72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#43;&#50;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#120;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146609045\">\n<p id=\"fs-id1169148984154\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88b35914324b9351eb5c681558266251_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"89\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1169149025241\">\n<h3 data-type=\"title\">Use Radicals in Applications<\/h3>\n<p id=\"fs-id1169148985032\">As you progress through your college courses, you\u2019ll encounter formulas that include radicals in many disciplines. We will modify our Problem Solving Strategy for Geometry Applications slightly to give us a plan for solving applications with formulas from any discipline.<\/p>\n<div data-type=\"note\" id=\"fs-id1169149040822\" class=\"howto\">\n<div data-type=\"title\">Use a problem solving strategy for applications with formulas.<\/div>\n<ol id=\"fs-id1169144543890\" type=\"1\" class=\"stepwise\">\n<li><strong data-effect=\"bold\">Read<\/strong> the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.<\/li>\n<li><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/li>\n<li><strong data-effect=\"bold\">Name<\/strong> what we are looking for by choosing a variable to represent it.<\/li>\n<li><strong data-effect=\"bold\">Translate<\/strong> into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\n<li><strong data-effect=\"bold\">Solve the equation<\/strong> using good algebra techniques.<\/li>\n<li><strong data-effect=\"bold\">Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n<li><strong data-effect=\"bold\">Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1169149118428\">One application of radicals has to do with the effect of <span data-type=\"term\" class=\"no-emphasis\">gravity<\/span> on falling objects. The formula allows us to determine how long it will take a fallen object to hit the gound.<\/p>\n<div data-type=\"note\" id=\"fs-id1169148957158\">\n<div data-type=\"title\">Falling Objects<\/div>\n<p id=\"fs-id1169144729653\">On Earth, if an object is dropped from a height of <em data-effect=\"italics\">h<\/em> feet, the time in seconds it will take to reach the ground is found by using the formula<\/p>\n<div data-type=\"equation\" id=\"fs-id1169149103897\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3490b72d80ae0f6ca49756cb4339450_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#104;&#125;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1169146628740\">For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by substituting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e48c241955bc35e28678a07830ddd788_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"52\" style=\"vertical-align: -1px;\" \/> into the formula.<\/p>\n<table class=\"unnumbered unstyled\" summary=\"Since h equals 64 we rewrite the formula, replacing h with the number 64. The formula then becomes t equals square root of 64 divided by 4. Taking the square root of 64 we get t equals 8 divided by 4. Simplifying the fraction we get t equals 2. It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.\" 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-id1169149065270\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149032044\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\">Take the square root of 64.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149342082\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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 the fraction.<\/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_06_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169149190917\">It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.<\/p>\n<div data-type=\"example\" id=\"fs-id1169149292896\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1169148889574\">\n<div data-type=\"problem\" id=\"fs-id1169149120935\">\n<p id=\"fs-id1169148973896\">Marissa dropped her sunglasses from a bridge 400 feet above a river. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61154b9c5a29ce4c47540f7df2fdc4c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#104;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"52\" style=\"vertical-align: -6px;\" \/> to find how many seconds it took for the sunglasses to reach the river.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148890736\">\n<table id=\"fs-id1169149376727\" class=\"unnumbered unstyled can-break\" summary=\"The first step in the process is to read the problem. Step 2 is to identify what we are looking for. We are looking for the time it takes the sunglasses to reach the river. Step 3 is to name what we are looking for. Let t equal the time. Step 4 is to translate into an equation by writing the appropriate formula and substitute in the given information. t equals the square root of h divided by 4 and h equals 400. So t equals the square root of 400 divided by 4. Step 5 is to solve the equation. So t equals 20 divided by 4. So t equals 5. Step 6 is to check the answer in the problem and make sure it makes sense. Does 5 equal the square root of 400 divided 4. Since 5 equals 20 divided by 4, the answer is a solution to the equation. Does 5 seconds seem like a reasonable length of time? Yes. Step 7 is to answer the question. It will take 5 seconds for the sunglasses to reach the river.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">the time it takes for the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>sunglasses to reach the river<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what we are looking.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-be6dc1d69e7f8de3302461f289c68554_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> time.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into an equation by writing the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>appropriate formula. Substitute in the given<\/p>\n<div data-type=\"newline\"><\/div>\n<p>information.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148938836\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\"><strong data-effect=\"bold\">Step 5. Solve the equation.<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169146744203\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144561652\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\"><strong data-effect=\"bold\">Step 6. Check<\/strong> the answer in the problem and make<\/p>\n<div data-type=\"newline\"><\/div>\n<p>sure it makes sense.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149157046\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\">Does 5 seconds seem like a reasonable length of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>time?<\/td>\n<td data-valign=\"top\" data-align=\"left\">Yes.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 7. Answer<\/strong> the question.<\/td>\n<td data-valign=\"top\" data-align=\"left\">It will take 5 seconds for the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>sunglasses to reach the river.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148879930\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169146645348\">\n<div data-type=\"problem\" id=\"fs-id1169149115845\">\n<p id=\"fs-id1169149089351\">A helicopter dropped a rescue package from a height of 1,296 feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61154b9c5a29ce4c47540f7df2fdc4c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#104;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"52\" style=\"vertical-align: -6px;\" \/> to find how many seconds it took for the package to reach the ground.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144450967\">\n<p id=\"fs-id1169138945430\">9 seconds<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169149028779\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169148933988\">\n<div data-type=\"problem\" id=\"fs-id1169149004636\">\n<p id=\"fs-id1169149293850\">A window washer dropped a squeegee from a platform 196 feet above the sidewalk Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61154b9c5a29ce4c47540f7df2fdc4c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#104;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"52\" style=\"vertical-align: -6px;\" \/> to find how many seconds it took for the squeegee to reach the sidewalk.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144555964\">\n<p id=\"fs-id1169144614940\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a880aa3dee267fff1a375dd7cfcaa389_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/> seconds<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169148957152\">Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the <span data-type=\"term\" class=\"no-emphasis\">speed<\/span>, in miles per hour, a car was going before applying the brakes.<\/p>\n<div data-type=\"note\" id=\"fs-id1169149012194\">\n<div data-type=\"title\">Skid Marks and Speed of a Car<\/div>\n<p id=\"fs-id1169144555978\">If the length of the skid marks is <em data-effect=\"italics\">d<\/em> feet, then the speed, <em data-effect=\"italics\">s<\/em>, of the car before the brakes were applied can be found by using the formula<\/p>\n<div data-type=\"equation\" id=\"fs-id1169149092734\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a9a9c32372c90dc6afdb389799a1856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/><\/div>\n<\/div>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1169149311099\">\n<p id=\"fs-id1169147133499\">After a car accident, the skid marks for one car measured 190 feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a9a9c32372c90dc6afdb389799a1856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/> to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148993220\">\n<table id=\"fs-id1169144381624\" class=\"unnumbered unstyled can-break\" summary=\"The first step in the process is to read the problem. Step 2 is to identify what we are looking for. We are looking for the speed of the car. Step 3 is to name what we are looking for. Let s equal the speed. Step 4 is to translate into an equation by writing the appropriate formula and substitute in the given information. s equals the square root of the quantity 24 d in parentheses, and d equals 190. So s equals the square root of the quantity 24 times 190 in parentheses. Step 5 is to solve the equation. So s equals the square root of 4560. So s is approximately equal to 67.52777. Rounding to 1 decimal place we et s equal to 67.5. Step 6 is to check the answer in the problem and make sure it makes sense. Does the square root of 4560 equal the square root of the quantity 24 times 190 in parentheses? It does. Does 67.5 mph seem like a reasonable speed? Yes. Step 7 is to answer the question. The car was traveling approximately 67.5 mph before the brakes were applied.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 1. Read<\/strong> the problem<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 2. Identify<\/strong> what we are looking for.<\/td>\n<td data-valign=\"top\" data-align=\"left\">the speed of a car<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 3. Name<\/strong> what weare looking for,<\/td>\n<td data-valign=\"top\" data-align=\"left\">Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12ac9f5dee0c0d9acdb247ec52211d05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"26\" style=\"vertical-align: 0px;\" \/> the speed.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Step 4. Translate<\/strong> into an equation by writing<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the appropriate formula. Substitute in the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>given information.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169147109998\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\"><strong data-effect=\"bold\">Step 5. Solve the equation.<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169148971507\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169149219635\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_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\">Round to 1 decimal place.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1169144558132\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_012d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\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-id1169146594591\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_012e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The speed of the car before the brakes were applied<\/p>\n<div data-type=\"newline\"><\/div>\n<p>was 67.5 miles per hour.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144382040\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169144382958\">\n<div data-type=\"problem\" id=\"fs-id1169149330072\">\n<p id=\"fs-id1169149354829\">An accident investigator measured the skid marks of the car. The length of the skid marks was 76 feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a9a9c32372c90dc6afdb389799a1856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/> to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1169146662927\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bba56402cf93c2d584d5a5151045aad9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#50;&#46;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"32\" style=\"vertical-align: -1px;\" \/> feet<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169148984286\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1169149373592\">\n<div data-type=\"problem\" id=\"fs-id1169149065549\">\n<p id=\"fs-id1169149339440\">The skid marks of a vehicle involved in an accident were 122 feet long. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a9a9c32372c90dc6afdb389799a1856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/> to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144615503\">\n<p id=\"fs-id1169148872024\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-831e6660ea3e0254c19c277d51555ed1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#52;&#46;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/> feet<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1169144566399\" class=\"media-2\">\n<p id=\"fs-id1169149329664\">Access these online resources for additional instruction and practice with solving radical equations.<\/p>\n<ul id=\"fs-id1169147089449\" data-bullet-style=\"bullet\">\n<li><a href=\"https:\/\/openstax.org\/l\/37RadEquat1\">Solving an Equation Involving a Single Radical<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37RadEquat2\">Solving Equations with Radicals and Rational Exponents<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37RadEquat3\">Solving Radical Equations<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37RadEquat4\">Solve Radical Equations<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37RadEquat5\">Radical Equation Application<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169149092589\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1169149351406\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Binomial Squares<\/strong>\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-a9a7877cc62bc33f0f2d4c3caa830623_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#92;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"184\" style=\"vertical-align: -16px;\" \/><\/li>\n<li><strong data-effect=\"bold\">Solve a Radical Equation<\/strong>\n<ol id=\"fs-id1169149356299\" type=\"1\" class=\"stepwise\">\n<li>Isolate one of the radical terms on one side of the equation.<\/li>\n<li>Raise both sides of the equation to the power of the index.<\/li>\n<li>Are there any more radicals?\n<div data-type=\"newline\"><\/div>\n<p> If yes, repeat Step 1 and Step 2 again.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> If no, solve the new equation.<\/li>\n<li>Check the answer in the original equation.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Problem Solving Strategy for Applications with Formulas<\/strong>\n<ol id=\"fs-id1169144768017\" type=\"1\" class=\"stepwise\">\n<li>Read the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.<\/li>\n<li>Identify what we are looking for.<\/li>\n<li>Name what we are looking for by choosing a variable to represent it.<\/li>\n<li>Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\n<li>Solve the equation using good algebra techniques.<\/li>\n<li>Check the answer in the problem and make sure it makes sense.<\/li>\n<li>Answer the question with a complete sentence.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Falling Objects<\/strong>\n<ul id=\"fs-id1169149361874\" data-bullet-style=\"bullet\">\n<li>On Earth, if an object is dropped from a height of <em data-effect=\"italics\">h<\/em> feet, the time in seconds it will take to reach the ground is found by using the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3490b72d80ae0f6ca49756cb4339450_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#104;&#125;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Skid Marks and Speed of a Car<\/strong>\n<ul id=\"fs-id1169144615595\" data-bullet-style=\"bullet\">\n<li>If the length of the skid marks is <em data-effect=\"italics\">d<\/em> feet, then the speed, <em data-effect=\"italics\">s<\/em>, of the car before the brakes were applied can be found by using the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80a40f8a02cb4b880fd039baf0a2c458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1169148889368\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1169148955469\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1169148967553\"><strong data-effect=\"bold\">Solve Radical Equations<\/strong><\/p>\n<p id=\"fs-id1169146645835\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169144767986\">\n<div data-type=\"problem\" id=\"fs-id1169146645006\">\n<p id=\"fs-id1169144892849\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b0a53d6824615c2ff113067529b2eb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#120;&#45;&#54;&#125;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146611138\">\n<p id=\"fs-id1169149008350\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2833243c897427f5a197619a42392ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149030412\">\n<div data-type=\"problem\" id=\"fs-id1169149375804\">\n<p id=\"fs-id1169148997760\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ca93ac22463e77328bf6e8b861313a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#120;&#45;&#51;&#125;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144544993\">\n<div data-type=\"problem\" id=\"fs-id1169146669980\">\n<p id=\"fs-id1169146660926\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2cc9884e8397d0781980cbb20d4be7a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#120;&#43;&#49;&#125;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148868701\">\n<p id=\"fs-id1169144565248\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144421298\">\n<div data-type=\"problem\" id=\"fs-id1169146609692\">\n<p id=\"fs-id1169148924664\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5fe9c4d47d30fc894b0781c75f21c4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#121;&#45;&#52;&#125;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148884439\">\n<div data-type=\"problem\" id=\"fs-id1169146745549\">\n<p id=\"fs-id1169148917729\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc6d8a1778d675195a4360d62c36956a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#120;&#125;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146669918\">\n<p id=\"fs-id1169149113769\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3dc975a98ccada6f136856736d7df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149086617\">\n<div data-type=\"problem\" id=\"fs-id1169147027817\">\n<p id=\"fs-id1169148870239\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-96137971f03e06cb5ed1bab6338a1459_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#120;&#45;&#49;&#125;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149220779\">\n<div data-type=\"problem\" id=\"fs-id1169146630929\">\n<p id=\"fs-id1169144381891\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8206ed63ec8ee0c3d31957c55b23774d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#109;&#45;&#51;&#125;&#45;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"132\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149006802\">\n<p id=\"fs-id1169144545042\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2833243c897427f5a197619a42392ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144516801\">\n<div data-type=\"problem\" id=\"fs-id1169146741588\">\n<p id=\"fs-id1169149140544\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e8a976d907dd62438dcf35ebaef944e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#110;&#45;&#49;&#125;&#45;&#51;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149039277\">\n<div data-type=\"problem\" id=\"fs-id1169146665818\">\n<p id=\"fs-id1169146660008\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d40d0167f37454ac5850dfaa894a77d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#118;&#45;&#50;&#125;&#45;&#49;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146629385\">\n<p id=\"fs-id1169149115657\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a640d8ded5227ea82fe0effcdcd23ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149102979\">\n<div data-type=\"problem\" id=\"fs-id1169149285826\">\n<p id=\"fs-id1169146642595\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30e6d52dceb745987924ebbd4acb84dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#50;&#117;&#43;&#49;&#125;&#45;&#49;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148995897\">\n<div data-type=\"problem\" id=\"fs-id1169149140501\">\n<p id=\"fs-id1169144560794\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b34bc63d86e59b9a0178da20c4272c9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#109;&#43;&#50;&#125;&#43;&#50;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"132\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1169146611236\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-360685a2e6f7cd43ffe1889ef7313667_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"48\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149108832\">\n<div data-type=\"problem\" id=\"fs-id1169149040777\">\n<p id=\"fs-id1169148992914\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-651c7c2c40fcdb4e05a83edb8da0d84d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#54;&#110;&#43;&#49;&#125;&#43;&#52;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149037013\">\n<div data-type=\"problem\" id=\"fs-id1169146610982\">\n<p id=\"fs-id1169149094596\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f12acf7467dd26fe235fe92faac61530_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#117;&#45;&#51;&#125;&#43;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149189792\">\n<p id=\"fs-id1169148958541\">no solution<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149319612\">\n<div data-type=\"problem\" id=\"fs-id1169149172684\">\n<p id=\"fs-id1169149336271\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91a5f7b5cdaf1c560c484a50987861a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#118;&#45;&#50;&#125;&#43;&#53;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169140166746\">\n<div data-type=\"problem\" id=\"fs-id1169149210689\">\n<p id=\"fs-id1169144875868\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a89bcc045487a288cda521826b55748_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#117;&#45;&#51;&#125;&#45;&#51;&#61;&#117;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169148895758\">\n<p id=\"fs-id1169149230001\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb3a77ba7b7ae0b5f67c64d485797202_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#51;&#44;&#117;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"93\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148951939\">\n<div data-type=\"problem\" id=\"fs-id1169144746311\">\n<p id=\"fs-id1169148994556\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b80e220de0c1ea97373faa77aacc9cb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#118;&#45;&#49;&#48;&#125;&#43;&#49;&#48;&#61;&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146609259\">\n<div data-type=\"problem\" id=\"fs-id1169147028026\">\n<p id=\"fs-id1169146595086\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-deb0d72691ad0a820b4f5bd67c70fb04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#45;&#49;&#125;&#61;&#114;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149328130\">\n<p id=\"fs-id1169144381096\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d8af0db1478fa3ac9d4c11a2945ce73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#49;&#44;&#114;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169140091631\">\n<div data-type=\"problem\" id=\"fs-id1169144604107\">\n<p id=\"fs-id1169144604073\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb24c45279d8c09df34c287dfe8a7bee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#115;&#45;&#56;&#125;&#61;&#115;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144744651\">\n<div data-type=\"problem\" id=\"fs-id1169144744653\">\n<p id=\"fs-id1169144416352\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5a275da7fde6dabff7a19bd455b9628_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#54;&#120;&#43;&#52;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144379926\">\n<p id=\"fs-id1169140166884\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6adca822887415d6f9a00ba1a1d97657_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149351498\">\n<div data-type=\"problem\" id=\"fs-id1169146632531\">\n<p id=\"fs-id1169146632533\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15ebc3bcb586568bdf7e07196157c40e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#49;&#49;&#120;&#43;&#52;&#125;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169148871204\">\n<div data-type=\"problem\" id=\"fs-id1169144417097\">\n<p id=\"fs-id1169144604691\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ed629ede1bf6b732f50db7282546d01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#120;&#43;&#53;&#125;&#45;&#50;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144744577\">\n<p id=\"fs-id1169149326524\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7852d0704798164bd2ce5b572ac50cd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144375789\">\n<div data-type=\"problem\" id=\"fs-id1169144744881\">\n<p id=\"fs-id1169144744883\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5342aba099653eb3fe82820f7468d558_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#57;&#120;&#45;&#49;&#125;&#45;&#49;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144453078\">\n<div data-type=\"problem\" id=\"fs-id1169146833202\">\n<p id=\"fs-id1169146833204\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfab0cfbd889cbfdb84bade5b7cb9ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#45;&#51;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149357243\">\n<p id=\"fs-id1169149357245\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1dc438da70f358c2eb1bf64a8b7ea4b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144744499\">\n<div data-type=\"problem\" id=\"fs-id1169149102314\">\n<p id=\"fs-id1169149102316\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3590ceb6567afa3b9f0a260c6f74509_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#43;&#49;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144452942\">\n<div data-type=\"problem\" id=\"fs-id1169144416963\">\n<p id=\"fs-id1169144416966\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-022a04f37db41a0b5bac7fa5b7db8bb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#43;&#50;&#61;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144417003\">\n<p id=\"fs-id1169144417005\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3dc975a98ccada6f136856736d7df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146732087\">\n<div data-type=\"problem\" id=\"fs-id1169146732089\">\n<p id=\"fs-id1169146732091\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a55b2d4ead3f084b368dd13cd7cc62bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#43;&#56;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149102331\">\n<div data-type=\"problem\" id=\"fs-id1169149102333\">\n<p id=\"fs-id1169149102335\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d01886a59e9ead65e4757aae6243e17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#45;&#53;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149102373\">\n<p id=\"fs-id1169149102375\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49af7f686347e06977fd3a18757933db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147082537\">\n<div data-type=\"problem\" id=\"fs-id1169147082540\">\n<p id=\"fs-id1169147082542\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28ed0edcaf6446e3d2fb63f98be8594e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#43;&#55;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169147082594\">\n<div data-type=\"problem\" id=\"fs-id1169147082596\">\n<p id=\"fs-id1169147082598\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84febce2d64e378701665b668b6b9515_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#43;&#49;&#125;&#45;&#120;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144604719\">\n<p id=\"fs-id1169144604721\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144604734\">\n<div data-type=\"problem\" id=\"fs-id1169144604736\">\n<p id=\"fs-id1169144604738\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb9e387c31bb9e8105e76cf6a91b3575_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#121;&#43;&#52;&#125;&#45;&#121;&#43;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"148\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144376883\">\n<div data-type=\"problem\" id=\"fs-id1169144376885\">\n<p id=\"fs-id1169144376887\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6c799fc4bf47a9d4e84f6005b0be1c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#122;&#43;&#49;&#48;&#48;&#125;&#45;&#122;&#61;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"157\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144376910\">\n<p id=\"fs-id1169144376912\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-371f4571bd719043cab4df6ab4f5c8d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"50\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144376925\">\n<div data-type=\"problem\" id=\"fs-id1169144376927\">\n<p id=\"fs-id1169144376929\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8bc382b4defb1518fc7a30f2e8ceb341_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#119;&#43;&#50;&#53;&#125;&#45;&#119;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"147\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144376966\">\n<div data-type=\"problem\" id=\"fs-id1169144376968\">\n<p id=\"fs-id1169144376970\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1b6002b56408cdc29a116ed648d4553_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#45;&#51;&#125;&#45;&#50;&#48;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"145\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149355571\">\n<p id=\"fs-id1169149355573\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-194e9289e6669c66a25cf705b61f8199_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149355586\">\n<div data-type=\"problem\" id=\"fs-id1169149355588\">\n<p id=\"fs-id1169149355590\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d61ec1cee27481f2bc86b830d87f581d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#120;&#43;&#49;&#125;&#45;&#56;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"136\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149355632\">\n<div data-type=\"problem\" id=\"fs-id1169149355634\">\n<p id=\"fs-id1169149355636\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43ad5eb9465965536858435d01d4bc31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#114;&#43;&#49;&#125;&#45;&#56;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149221178\">\n<p id=\"fs-id1169149221180\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f57695ad3ee3297417eba5e756bb36e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149221193\">\n<div data-type=\"problem\" id=\"fs-id1169149221195\">\n<p id=\"fs-id1169149221197\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2cd0599ae55a74610f91c9716bab87c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#121;&#43;&#49;&#125;&#45;&#49;&#48;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169149221240\"><strong data-effect=\"bold\">Solve Radical Equations with Two Radicals<\/strong><\/p>\n<p id=\"fs-id1169149221245\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1169149221249\">\n<div data-type=\"problem\" id=\"fs-id1169149221251\">\n<p id=\"fs-id1169149221253\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4126602240b9d677d9a410929176e22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#117;&#43;&#55;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#117;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"152\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169139890415\">\n<p id=\"fs-id1169139890417\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c9362b4d3f1e72137a095026128b158_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169139890430\">\n<div data-type=\"problem\" id=\"fs-id1169139890432\">\n<p id=\"fs-id1169139890434\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3b8e0594f2a4cd616c6ac780068a907_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#118;&#43;&#49;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#118;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169139890478\">\n<div data-type=\"problem\" id=\"fs-id1169139890480\">\n<p id=\"fs-id1169139890482\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e211909451173c4a9c78cd0a923d7e8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#56;&#43;&#50;&#114;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#114;&#43;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"157\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169139890511\">\n<p id=\"fs-id1169139890513\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d75e3ba8e2d2ab0451520a284f02520e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"54\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146651576\">\n<div data-type=\"problem\" id=\"fs-id1169146651578\">\n<p id=\"fs-id1169146651580\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b58b69734d7704894d2feed331426a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#48;&#43;&#50;&#99;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#52;&#99;&#43;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"164\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169146651624\">\n<div data-type=\"problem\" id=\"fs-id1169146651626\">\n<p id=\"fs-id1169146651628\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe1aac58842193fccdefef5dcdc5dac2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#53;&#120;&#45;&#49;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#120;&#43;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169146651659\">\n<p id=\"fs-id1169146651662\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144684269\">\n<div data-type=\"problem\" id=\"fs-id1169144684271\">\n<p id=\"fs-id1169144684273\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c130da1ab61f6891baf070c52256873_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#56;&#120;&#45;&#53;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#51;&#120;&#43;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"153\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144684321\">\n<div data-type=\"problem\" id=\"fs-id1169144684323\">\n<p id=\"fs-id1169144684325\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b006d5ceb98d3a9615e06f1a4a26fcfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#45;&#49;&#56;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#120;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"250\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144684375\">\n<p id=\"fs-id1169149172962\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-673b47a45398049b77add48d113519dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#56;&#44;&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149172984\">\n<div data-type=\"problem\" id=\"fs-id1169149172986\">\n<p id=\"fs-id1169149172988\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53a837ab8c4628592e1705c5c63876b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#120;&#43;&#49;&#56;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"240\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149173059\">\n<div data-type=\"problem\" id=\"fs-id1169149173062\">\n<p id=\"fs-id1169149173064\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-10c798ee4ad3a38e2677f8228389b9a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#125;&#43;&#50;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149359734\">\n<p id=\"fs-id1169149359736\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-072bed0ebf929f9d7c14da365c8512a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149359749\">\n<div data-type=\"problem\" id=\"fs-id1169149359751\">\n<p id=\"fs-id1169149359753\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f1a6447e80d10b4d9332c4af8a5e3ac2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#125;&#43;&#54;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#43;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149359784\">\n<div data-type=\"problem\" id=\"fs-id1169149359786\">\n<p id=\"fs-id1169149359788\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16429fe4d57b0a50ab3fe2cbca140c7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#117;&#125;&#43;&#49;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#117;&#43;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149359812\">\n<p id=\"fs-id1169149359814\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e551d43b6b3a6314c45e4e70cea5337c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144557874\">\n<div data-type=\"problem\" id=\"fs-id1169144557876\">\n<p id=\"fs-id1169144557878\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ec04e95432268aeefc0055020f9168f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#43;&#49;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144557920\">\n<div data-type=\"problem\" id=\"fs-id1169144557922\">\n<p id=\"fs-id1169144557924\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8372d2d0109aee51a7968b5bb0de7c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#43;&#53;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#97;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144557948\">\n<p id=\"fs-id1169144557950\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1735c51b99ed2469a7f2a6f728de75f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144557963\">\n<div data-type=\"problem\" id=\"fs-id1169144557965\">\n<p id=\"fs-id1169144557967\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0dfdee581bf082546d2b8f4bb5e617ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#100;&#45;&#50;&#48;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"155\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144744956\">\n<div data-type=\"problem\" id=\"fs-id1169144744958\">\n<p id=\"fs-id1169144744960\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b3dacaae1060c92b27d388ed84d01f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#43;&#49;&#125;&#61;&#49;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169144744986\">\n<p id=\"fs-id1169144744988\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88b35914324b9351eb5c681558266251_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"89\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169144745009\">\n<div data-type=\"problem\" id=\"fs-id1169144745011\">\n<p id=\"fs-id1169144745013\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39d323b7efc343333b94a41942b7b2e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#120;&#43;&#49;&#125;&#61;&#49;&#43;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"182\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149327613\">\n<div data-type=\"problem\" id=\"fs-id1169149327615\">\n<p id=\"fs-id1169149327617\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ff4c95dd1cf4dc5ba02c905ee554875_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#120;&#45;&#49;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#45;&#49;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"172\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149327648\">\n<p id=\"fs-id1169149327650\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-363ecc19f056d9c0a9ec9199bb1ca471_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"88\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149327671\">\n<div data-type=\"problem\" id=\"fs-id1169149327674\">\n<p id=\"fs-id1169149327676\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8c4a5c979b4dfbe19844122eecbc044_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#43;&#49;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#45;&#50;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"163\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149335518\">\n<div data-type=\"problem\" id=\"fs-id1169149335520\">\n<p id=\"fs-id1169149335522\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a52c473aa3e9cc6a5be65cbef86dcc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#43;&#55;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#45;&#53;&#125;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"163\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149335551\">\n<p id=\"fs-id1169149335553\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-300a345ef7b973d34879ac8e90555390_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149335566\">\n<div data-type=\"problem\" id=\"fs-id1169149335568\">\n<p id=\"fs-id1169149335570\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f4677eecb5edfe313e80e0997b4ca55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#43;&#53;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#45;&#51;&#125;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"163\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169140418470\"><strong data-effect=\"bold\">Use Radicals in Applications<\/strong><\/p>\n<p id=\"fs-id1169140418475\">In the following exercises, solve. Round approximations to one decimal place.<\/p>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1169140418482\"><strong data-effect=\"bold\">Landscaping<\/strong> Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-710aedf0b9cfbdd26e3c31b132a4e7a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -2px;\" \/> to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169140418502\">\n<p id=\"fs-id1169140418504\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c3502f62f80d6ebc29154bf5cd04bc5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#46;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"23\" style=\"vertical-align: 0px;\" \/> feet<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169140418514\">\n<div data-type=\"problem\" id=\"fs-id1169140418516\">\n<p id=\"fs-id1169140418518\"><strong data-effect=\"bold\">Landscaping<\/strong> Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130 square feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-710aedf0b9cfbdd26e3c31b132a4e7a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -2px;\" \/> to find the length of each side of his patio. Round your answer to the nearest tenth of a foot.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169140418549\">\n<div data-type=\"problem\" id=\"fs-id1169140418551\">\n<p id=\"fs-id1169140418553\"><strong data-effect=\"bold\">Gravity<\/strong> A hang glider dropped his cell phone from a height of 350 feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61154b9c5a29ce4c47540f7df2fdc4c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#104;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"52\" style=\"vertical-align: -6px;\" \/> to find how many seconds it took for the cell phone to reach the ground.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149235355\">\n<p id=\"fs-id1169149235357\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa8e39683e761eafc77f9d9648a9e402_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"23\" style=\"vertical-align: -1px;\" \/> seconds<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149235366\">\n<div data-type=\"problem\" id=\"fs-id1169149235368\">\n<p id=\"fs-id1169149235371\"><strong data-effect=\"bold\">Gravity<\/strong> A construction worker dropped a hammer while building the Grand Canyon skywalk, 4000 feet above the Colorado River. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61154b9c5a29ce4c47540f7df2fdc4c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#104;&#125;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"52\" style=\"vertical-align: -6px;\" \/> to find how many seconds it took for the hammer to reach the river.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149235406\">\n<div data-type=\"problem\" id=\"fs-id1169149235408\">\n<p id=\"fs-id1169149235410\"><strong data-effect=\"bold\">Accident investigation<\/strong> The skid marks for a car involved in an accident measured 216 feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a9a9c32372c90dc6afdb389799a1856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/> to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149235432\">\n<p id=\"fs-id1169149235434\">72 feet<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149235439\">\n<div data-type=\"problem\" id=\"fs-id1169149235441\">\n<p id=\"fs-id1169149235443\"><strong data-effect=\"bold\">Accident investigation<\/strong> An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. Use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a9a9c32372c90dc6afdb389799a1856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#52;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -2px;\" \/> to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1169149335189\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1169149335196\">\n<div data-type=\"problem\" id=\"fs-id1169149335198\">\n<p id=\"fs-id1169149335200\">Explain why an equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccdeffed00e0024e8f9731efd7403e33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#120;&#125;&#43;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"87\" style=\"vertical-align: -4px;\" \/> has no solution.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1169149335219\">\n<p id=\"fs-id1169149335221\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1169149335226\">\n<div data-type=\"problem\" id=\"fs-id1169149335229\">\n<p id=\"fs-id1169149335231\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> Solve the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de5762b213341fbe4e1d316c98e5010e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#114;&#43;&#52;&#125;&#45;&#114;&#43;&#50;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Explain why one of the \u201csolutions\u201d that was found was not actually a solution to the equation.<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1169149335281\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1169144746564\"><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-id1169144746579\" data-alt=\"The table has 4 columns and 4 rows. The first row is a header row with the headers \u201cI can\u2026\u201d, \u201cConfidently\u201d, \u201cWith some help.\u201d, and \u201cNo \u2013 I don\u2019t get it!\u201d. The first column contains the phrases \u201cSolve radical equations\u201d, \u201csolve radical equations with two radicals\u201d, and \u201cuse radicals in applications\u201d. The other columns are left blank so the learner can indicate their level of understanding.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_08_06_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The table has 4 columns and 4 rows. The first row is a header row with the headers \u201cI can\u2026\u201d, \u201cConfidently\u201d, \u201cWith some help.\u201d, and \u201cNo \u2013 I don\u2019t get it!\u201d. The first column contains the phrases \u201cSolve radical equations\u201d, \u201csolve radical equations with two radicals\u201d, and \u201cuse radicals in applications\u201d. The other columns are left blank so the learner can indicate their level of understanding.\" \/><\/span><\/p>\n<p id=\"fs-id1169144746585\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/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-id1169144746596\">\n<dt>radical equation<\/dt>\n<dd id=\"fs-id1169144746602\">An equation in which a variable is in the radicand of a radical expression is called a radical equation.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3633","chapter","type-chapter","status-publish","hentry"],"part":3472,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3633","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\/3633\/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\/3633\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3633"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3633"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3633"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3633"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}