{"id":70,"date":"2023-06-21T13:22:30","date_gmt":"2023-06-21T13:22:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/multiplying-and-dividing-rational-expressions\/"},"modified":"2023-07-04T04:58:07","modified_gmt":"2023-07-04T04:58:07","slug":"multiplying-and-dividing-rational-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/multiplying-and-dividing-rational-expressions\/","title":{"raw":"\u25aa   Multiplying and Dividing Rational Expressions","rendered":"\u25aa   Multiplying and Dividing Rational Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify rational expressions.<\/li>\r\n \t<li>Multiply and divide rational expressions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Simplifying Rational Expressions<\/h2>\r\nThe quotient of two polynomial expressions is called a <strong>rational expression<\/strong>. We can apply the properties of fractions to rational expressions such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let\u2019s start with the rational expression shown.\r\n<p style=\"text-align: center;\">[latex]\\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[\/latex]<\/p>\r\n[latex]\\\\[\/latex]We can factor the numerator and denominator to rewrite the expression as [latex]\\frac{{\\left(x+4\\right)}^{2}}{\\left(x+4\\right)\\left(x+7\\right)}[\/latex]\r\n<div><\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<div>Then we can simplify the expression by canceling the common factor [latex]\\left(x+4\\right)[\/latex] to get [latex]\\frac{x+4}{x+7}[\/latex]<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a rational expression, simplify it<\/h3>\r\n<ol>\r\n \t<li>Factor the numerator and denominator.<\/li>\r\n \t<li>Cancel any common factors.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Simplifying Rational Expressions<\/h3>\r\nSimplify [latex]\\frac{{x}^{2}-9}{{x}^{2}+4x+3}[\/latex].\r\n\r\n[reveal-answer q=\"568949\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"568949\"]\r\n\r\n[latex]\\begin{array}{lllllllll}\\frac{\\left(x+3\\right)\\left(x - 3\\right)}{\\left(x+3\\right)\\left(x+1\\right)}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Factor the numerator and the denominator}.\\hfill \\\\ \\frac{x - 3}{x+1}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Cancel common factor }\\left(x+3\\right).\\hfill \\end{array}[\/latex]\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can cancel the common factor because any expression divided by itself is equal to 1.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Can the [latex]{x}^{2}[\/latex] term be cancelled in the above example?<\/strong>\r\n\r\n<em>No. A factor is an expression that is multiplied by another expression. The [latex]{x}^{2}[\/latex] term is not a factor of the numerator or the denominator.<\/em>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSimplify [latex]\\frac{x - 6}{{x}^{2}-36}[\/latex].\r\n\r\n[reveal-answer q=\"17752\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"17752\"]\r\n\r\n[latex]\\frac{1}{x+6}[\/latex][\/hidden-answer]\r\n\r\n[ohm_question]110917-110916[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Multiplying Rational Expressions<\/h2>\r\n<div class=\"textbox examples\">\r\n<h3>Recall Multiplying fractions<\/h3>\r\n<p style=\"text-align: left;\">To multiply fractions, multiply the numerators and place them over the product of the denominators.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{a}{b}\\cdot\\dfrac{c}{d} = \\dfrac {ac}{bd}[\/latex]<\/p>\r\nIt is helpful to factor the numerator and denominator and cancel common factors before multiplying terms together.\r\n\r\n<\/div>\r\nMultiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.\r\n<div class=\"textbox\">\r\n<h3>How To: Given two rational expressions, multiply them<\/h3>\r\n<ol>\r\n \t<li>Factor the numerator and denominator.<\/li>\r\n \t<li>Multiply the numerators.<\/li>\r\n \t<li>Multiply the denominators.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Multiplying Rational Expressions<\/h3>\r\nMultiply the rational expressions and show the product in simplest form:\r\n<div style=\"text-align: center;\">[latex]\\frac{{x}^{2}+4x-5}{3x+18}\\cdot \\frac{2x - 1}{x+5}[\/latex]<\/div>\r\n<div><\/div>\r\n[reveal-answer q=\"820400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"820400\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}\\frac{\\left(x+5\\right)\\left(x - 1\\right)}{3\\left(x+6\\right)}\\cdot \\frac{\\left(2x - 1\\right)}{\\left(x+5\\right)}\\hfill &amp; \\text{Factor the numerator and denominator}.\\hfill \\\\ \\frac{\\left(x+5\\right)\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)\\left(x+5\\right)}\\hfill &amp; \\text{Multiply numerators and denominators}.\\hfill \\\\ \\frac{\\cancel{\\left(x+5\\right)}\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)\\cancel{\\left(x+5\\right)}}\\hfill &amp; \\text{Cancel common factors to simplify}.\\hfill \\\\ \\frac{\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)}\\hfill &amp; \\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nMultiply the rational expressions and show the product in simplest form:\r\n<div style=\"text-align: center;\">[latex]\\frac{{x}^{2}+11x+30}{{x}^{2}+5x+6}\\cdot \\frac{{x}^{2}+7x+12}{{x}^{2}+8x+16}[\/latex]<\/div>\r\n[reveal-answer q=\"165135\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"165135\"]\r\n\r\n[latex]\\frac{\\left(x+5\\right)\\left(x+6\\right)}{\\left(x+2\\right)\\left(x+4\\right)}[\/latex][\/hidden-answer]\r\n\r\n[ohm_question]93841-93844[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Dividing Rational Expressions<\/h2>\r\n<div class=\"textbox examples\">\r\n<h3>recall dividing fractions<\/h3>\r\n<p style=\"text-align: left;\">To divide fractions, multiply the first by the reciprocal of the second.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\dfrac{a}{b}\\div\\dfrac{c}{d}=\\dfrac{a}{b}\\cdot\\dfrac{d}{c}=\\dfrac{ad}{bc}[\/latex]<\/p>\r\nRemember to factor first and cancel common factors in the numerator and denominator before multiplying individual terms together.\r\n\r\n<\/div>\r\nDivision of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite [latex]\\frac{1}{x}\\div \\frac{{x}^{2}}{3}[\/latex] as the product [latex]\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}[\/latex]. Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.\r\n<div style=\"text-align: center;\">[latex]\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}=\\frac{3}{{x}^{3}}[\/latex]<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given two rational expressions, divide them<\/h3>\r\n<ol>\r\n \t<li>Rewrite as the first rational expression multiplied by the reciprocal of the second.<\/li>\r\n \t<li>Factor the numerators and denominators.<\/li>\r\n \t<li>Multiply the numerators.<\/li>\r\n \t<li>Multiply the denominators.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Dividing Rational Expressions<\/h3>\r\nDivide the rational expressions and express the quotient in simplest form:\r\n<div style=\"text-align: center;\">[latex]\\frac{2{x}^{2}+x - 6}{{x}^{2}-1}\\div \\frac{{x}^{2}-4}{{x}^{2}+2x+1}[\/latex]<\/div>\r\n[reveal-answer q=\"266408\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"266408\"]\r\n\r\n[latex]\\begin{array}\\text{ }\\frac{2x^{2}+x-6}{x^{2}-1}\\cdot\\frac{x^{2}+2x+1}{x^{2}-4} \\hfill&amp; \\text{Rewrite as the first fraction multiplied by the reciprocal of the second fraction.} \\\\ \\frac{\\left(2x-3\\right)\\cancel{\\left(x+2\\right)}}{\\cancel{\\left(x+1\\right)}\\left(x-1\\right)}\\cdot\\frac{\\cancel{\\left(x+1\\right)}\\left(x+1\\right)}{\\cancel{\\left(x+2\\right)}\\left(x-2\\right)} \\hfill&amp; \\text{Factor and cancel common factors.} \\\\ \\frac{\\left(2x-3\\right)\\left(x+1\\right)}{\\left(x-1\\right)\\left(x-2\\right)} \\hfill&amp; \\text{Multiply numerators and denominators.} \\\\ \\frac{2x^{2}-x-3}{x^{2}-3x+2} \\hfill&amp; \\text{Simplify.}\\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDivide the rational expressions and express the quotient in simplest form:\r\n<div style=\"text-align: center;\">[latex]\\frac{9{x}^{2}-16}{3{x}^{2}+17x - 28}\\div \\frac{3{x}^{2}-2x - 8}{{x}^{2}+5x - 14}[\/latex]<\/div>\r\n[reveal-answer q=\"396693\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"396693\"]\r\n\r\n[latex]1[\/latex][\/hidden-answer]\r\n\r\n[ohm_question]93845-93847[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify rational expressions.<\/li>\n<li>Multiply and divide rational expressions.<\/li>\n<\/ul>\n<\/div>\n<h2>Simplifying Rational Expressions<\/h2>\n<p>The quotient of two polynomial expressions is called a <strong>rational expression<\/strong>. We can apply the properties of fractions to rational expressions such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let\u2019s start with the rational expression shown.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{x}^{2}+8x+16}{{x}^{2}+11x+28}[\/latex]<\/p>\n<p>[latex]\\\\[\/latex]We can factor the numerator and denominator to rewrite the expression as [latex]\\frac{{\\left(x+4\\right)}^{2}}{\\left(x+4\\right)\\left(x+7\\right)}[\/latex]<\/p>\n<div><\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<div>Then we can simplify the expression by canceling the common factor [latex]\\left(x+4\\right)[\/latex] to get [latex]\\frac{x+4}{x+7}[\/latex]<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a rational expression, simplify it<\/h3>\n<ol>\n<li>Factor the numerator and denominator.<\/li>\n<li>Cancel any common factors.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Simplifying Rational Expressions<\/h3>\n<p>Simplify [latex]\\frac{{x}^{2}-9}{{x}^{2}+4x+3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q568949\">Show Solution<\/span><\/p>\n<div id=\"q568949\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{array}{lllllllll}\\frac{\\left(x+3\\right)\\left(x - 3\\right)}{\\left(x+3\\right)\\left(x+1\\right)}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Factor the numerator and the denominator}.\\hfill \\\\ \\frac{x - 3}{x+1}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Cancel common factor }\\left(x+3\\right).\\hfill \\end{array}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can cancel the common factor because any expression divided by itself is equal to 1.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Can the [latex]{x}^{2}[\/latex] term be cancelled in the above example?<\/strong><\/p>\n<p><em>No. A factor is an expression that is multiplied by another expression. The [latex]{x}^{2}[\/latex] term is not a factor of the numerator or the denominator.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Simplify [latex]\\frac{x - 6}{{x}^{2}-36}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q17752\">Show Solution<\/span><\/p>\n<div id=\"q17752\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{1}{x+6}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm110917\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=110917-110916&theme=oea&iframe_resize_id=ohm110917&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Multiplying Rational Expressions<\/h2>\n<div class=\"textbox examples\">\n<h3>Recall Multiplying fractions<\/h3>\n<p style=\"text-align: left;\">To multiply fractions, multiply the numerators and place them over the product of the denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{a}{b}\\cdot\\dfrac{c}{d} = \\dfrac {ac}{bd}[\/latex]<\/p>\n<p>It is helpful to factor the numerator and denominator and cancel common factors before multiplying terms together.<\/p>\n<\/div>\n<p>Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given two rational expressions, multiply them<\/h3>\n<ol>\n<li>Factor the numerator and denominator.<\/li>\n<li>Multiply the numerators.<\/li>\n<li>Multiply the denominators.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying Rational Expressions<\/h3>\n<p>Multiply the rational expressions and show the product in simplest form:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{{x}^{2}+4x-5}{3x+18}\\cdot \\frac{2x - 1}{x+5}[\/latex]<\/div>\n<div><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q820400\">Show Solution<\/span><\/p>\n<div id=\"q820400\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}\\frac{\\left(x+5\\right)\\left(x - 1\\right)}{3\\left(x+6\\right)}\\cdot \\frac{\\left(2x - 1\\right)}{\\left(x+5\\right)}\\hfill & \\text{Factor the numerator and denominator}.\\hfill \\\\ \\frac{\\left(x+5\\right)\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)\\left(x+5\\right)}\\hfill & \\text{Multiply numerators and denominators}.\\hfill \\\\ \\frac{\\cancel{\\left(x+5\\right)}\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)\\cancel{\\left(x+5\\right)}}\\hfill & \\text{Cancel common factors to simplify}.\\hfill \\\\ \\frac{\\left(x - 1\\right)\\left(2x - 1\\right)}{3\\left(x+6\\right)}\\hfill & \\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Multiply the rational expressions and show the product in simplest form:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{{x}^{2}+11x+30}{{x}^{2}+5x+6}\\cdot \\frac{{x}^{2}+7x+12}{{x}^{2}+8x+16}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q165135\">Show Solution<\/span><\/p>\n<div id=\"q165135\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{\\left(x+5\\right)\\left(x+6\\right)}{\\left(x+2\\right)\\left(x+4\\right)}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm93841\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93841-93844&theme=oea&iframe_resize_id=ohm93841&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Dividing Rational Expressions<\/h2>\n<div class=\"textbox examples\">\n<h3>recall dividing fractions<\/h3>\n<p style=\"text-align: left;\">To divide fractions, multiply the first by the reciprocal of the second.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{a}{b}\\div\\dfrac{c}{d}=\\dfrac{a}{b}\\cdot\\dfrac{d}{c}=\\dfrac{ad}{bc}[\/latex]<\/p>\n<p>Remember to factor first and cancel common factors in the numerator and denominator before multiplying individual terms together.<\/p>\n<\/div>\n<p>Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite [latex]\\frac{1}{x}\\div \\frac{{x}^{2}}{3}[\/latex] as the product [latex]\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}[\/latex]. Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{1}{x}\\cdot \\frac{3}{{x}^{2}}=\\frac{3}{{x}^{3}}[\/latex]<\/div>\n<div class=\"textbox\">\n<h3>How To: Given two rational expressions, divide them<\/h3>\n<ol>\n<li>Rewrite as the first rational expression multiplied by the reciprocal of the second.<\/li>\n<li>Factor the numerators and denominators.<\/li>\n<li>Multiply the numerators.<\/li>\n<li>Multiply the denominators.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Dividing Rational Expressions<\/h3>\n<p>Divide the rational expressions and express the quotient in simplest form:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{2{x}^{2}+x - 6}{{x}^{2}-1}\\div \\frac{{x}^{2}-4}{{x}^{2}+2x+1}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266408\">Show Solution<\/span><\/p>\n<div id=\"q266408\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{array}\\text{ }\\frac{2x^{2}+x-6}{x^{2}-1}\\cdot\\frac{x^{2}+2x+1}{x^{2}-4} \\hfill& \\text{Rewrite as the first fraction multiplied by the reciprocal of the second fraction.} \\\\ \\frac{\\left(2x-3\\right)\\cancel{\\left(x+2\\right)}}{\\cancel{\\left(x+1\\right)}\\left(x-1\\right)}\\cdot\\frac{\\cancel{\\left(x+1\\right)}\\left(x+1\\right)}{\\cancel{\\left(x+2\\right)}\\left(x-2\\right)} \\hfill& \\text{Factor and cancel common factors.} \\\\ \\frac{\\left(2x-3\\right)\\left(x+1\\right)}{\\left(x-1\\right)\\left(x-2\\right)} \\hfill& \\text{Multiply numerators and denominators.} \\\\ \\frac{2x^{2}-x-3}{x^{2}-3x+2} \\hfill& \\text{Simplify.}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Divide the rational expressions and express the quotient in simplest form:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{9{x}^{2}-16}{3{x}^{2}+17x - 28}\\div \\frac{3{x}^{2}-2x - 8}{{x}^{2}+5x - 14}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q396693\">Show Solution<\/span><\/p>\n<div id=\"q396693\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]1[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm93845\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93845-93847&theme=oea&iframe_resize_id=ohm93845&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-70\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 110917, 110916. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC- BY + GPL<\/li><li>Question ID 93841, 93844, 93845, 93847. <strong>Authored by<\/strong>: Michael Jenck. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: 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