{"id":788,"date":"2016-06-01T20:49:09","date_gmt":"2016-06-01T20:49:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=788"},"modified":"2019-08-06T18:27:11","modified_gmt":"2019-08-06T18:27:11","slug":"read-product-and-quotient-rules-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-product-and-quotient-rules-2\/","title":{"raw":"Product and Quotient Rules","rendered":"Product and Quotient Rules"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify exponential expressions with like bases using the product, quotient, and power rules<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Use the Product Rule to Multiply Exponential Expressions<\/h2>\r\n<b>Exponential notation<\/b> was developed to write repeated multiplication more efficiently. There are times when it is easier or faster to leave the expressions in exponential notation when multiplying or dividing. Let us look at rules that will allow you to do this.\r\n\r\nFor example, the notation [latex]5^{4}[\/latex]\u00a0can be expanded and written as [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or\u00a0[latex]625[\/latex]. Do not forget, the exponent only applies to the number immediately to its left unless there are parentheses.\r\n\r\nWhat happens if you multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[\/latex]. Expanding each exponent, this can be rewritten as [latex]\\left(2\\cdot2\\cdot2\\right)\\left(2\\cdot2\\cdot2\\cdot2\\right)[\/latex] or [latex]2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2[\/latex]. In exponential form, you would write the product as [latex]2^{7}[\/latex]. Notice that [latex]7[\/latex] is the sum of the original two exponents,\u00a0[latex]3[\/latex] and\u00a0[latex]4[\/latex].\r\n\r\nWhat about [latex]{x}^{2}{x}^{6}[\/latex]? This can be written as [latex]\\left(x\\cdot{x}\\right)\\left(x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\right)=x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}[\/latex] or [latex]x^{8}[\/latex]. And, once again, 8 is the sum of the original two exponents. This concept can be generalized in the following way:\r\n<div class=\"textbox shaded\">\r\n<h3>The Product Rule for Exponents<\/h3>\r\nFor any number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>,\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].\r\n\r\nTo multiply exponential terms with the same base, add the exponents.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite each of the following products with a single base. Do not simplify further.\r\n<ol>\r\n \t<li>[latex]{t}^{5}\\cdot {t}^{3}[\/latex]<\/li>\r\n \t<li>[latex]\\left(-3\\right)^{5}\\cdot \\left(-3\\right)[\/latex]<\/li>\r\n \t<li>[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"772709\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"772709\"]\r\n\r\nUse the product rule to simplify each expression.\r\n<ol>\r\n \t<li>[latex]{t}^{5}\\cdot {t}^{3}={t}^{5+3}={t}^{8}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(-3\\right)}^{5}\\cdot \\left(-3\\right)={\\left(-3\\right)}^{5}\\cdot {\\left(-3\\right)}^{1}={\\left(-3\\right)}^{5+1}={\\left(-3\\right)}^{6}[\/latex]<\/li>\r\n \t<li>[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}[\/latex]<\/li>\r\n<\/ol>\r\nAt first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.\r\n<div style=\"text-align: center;\">[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}=\\left({x}^{2}\\cdot {x}^{5}\\right)\\cdot {x}^{3}=\\left({x}^{2+5}\\right)\\cdot {x}^{3}={x}^{7}\\cdot {x}^{3}={x}^{7+3}={x}^{10}[\/latex]<\/div>\r\nNotice we get the same result by adding the three exponents in one step.\r\n<div style=\"text-align: center;\">[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[\/latex]<\/div>\r\n<div style=\"text-align: center;\">[\/hidden-answer]<\/div>\r\n<\/div>\r\n<span style=\"color: #000000;\">In the following video, you will see more examples of using the product rule for exponents to simplify expressions.<\/span>\r\n\r\nhttps:\/\/youtu.be\/P0UVIMy2nuI\r\n\r\nIn our last product rule example, we will show that an exponent can be an algebraic expression. We can use the product rule for exponents no matter what the exponent looks like, as long as the base is the same.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMultiply.\u00a0[latex]x^{a+2}\\cdot{x^{3a-9}}[\/latex]\r\n[reveal-answer q=\"740329\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"740329\"]\r\n\r\nWe have two exponential terms with the same base. The product rule for exponents says that we can add the exponents.\r\n\r\n[latex]x^{a+2}\\cdot{x^{3a-9}}=x^{(a+2)+(3a-9)}=x^{4a-7}[\/latex]\r\n\r\nThe expression cannot be simplified any further.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Use the Quotient Rule to Divide Exponential Expressions<\/h2>\r\nLet us look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}[\/latex]<\/p>\r\nYou can rewrite the expression as: [latex] \\displaystyle \\frac{4\\cdot 4\\cdot 4\\cdot 4\\cdot 4}{4\\cdot 4}[\/latex]. Then you can cancel the common factors of 4 in the numerator and denominator: [latex] \\displaystyle [\/latex]\r\n\r\nFinally, this expression can be rewritten as [latex]4^{3}[\/latex]\u00a0using exponential notation. Notice that the exponent,\u00a0[latex]3[\/latex], is the difference between the two exponents in the original expression,\u00a0[latex]5[\/latex] and\u00a0[latex]2[\/latex].\r\n\r\nSo,\u00a0[latex] \\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[\/latex].\r\n\r\nBe careful that you subtract the exponent in the denominator from the exponent in the numerator.\r\n\r\nSo, to divide two exponential terms with the same base, subtract the exponents.\r\n<div class=\"textbox shaded\">\r\n<h3>The Quotient (Division) Rule for Exponents<\/h3>\r\nFor any non-zero number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>: [latex] \\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite each of the following products with a single base. Do not simplify further.\r\n<ol>\r\n \t<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"978732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"978732\"]\r\n\r\nUse the quotient rule to simplify each expression.\r\n<ol>\r\n \t<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}={\\left(-2\\right)}^{14 - 9}={\\left(-2\\right)}^{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}={\\left(z\\sqrt{2}\\right)}^{5 - 1}={\\left(z\\sqrt{2}\\right)}^{4}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAs we showed with the product rule, you may be given a quotient with an exponent that is an algebraic expression to simplify. \u00a0As long as the bases agree, you may use the quotient rule for exponents.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]\\dfrac{y^{x-3}}{y^{9-x}}[\/latex]\r\n[reveal-answer q=\"836863\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"836863\"]\r\n\r\nWe have a quotient whose terms have the same base so we can use the quotient rule for exponents.\r\n\r\n[latex]\\dfrac{y^{x-3}}{y^{9-x}}=y^{(x-3)-(9-x)}=y^{2x-12}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<span style=\"color: #000000;\">In the following video, you will see more examples of using the quotient rule for exponents.<\/span>\r\n\r\nhttps:\/\/youtu.be\/xy6WW7y_GcU","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify exponential expressions with like bases using the product, quotient, and power rules<\/li>\n<\/ul>\n<\/div>\n<h2>Use the Product Rule to Multiply Exponential Expressions<\/h2>\n<p><b>Exponential notation<\/b> was developed to write repeated multiplication more efficiently. There are times when it is easier or faster to leave the expressions in exponential notation when multiplying or dividing. Let us look at rules that will allow you to do this.<\/p>\n<p>For example, the notation [latex]5^{4}[\/latex]\u00a0can be expanded and written as [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or\u00a0[latex]625[\/latex]. Do not forget, the exponent only applies to the number immediately to its left unless there are parentheses.<\/p>\n<p>What happens if you multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[\/latex]. Expanding each exponent, this can be rewritten as [latex]\\left(2\\cdot2\\cdot2\\right)\\left(2\\cdot2\\cdot2\\cdot2\\right)[\/latex] or [latex]2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2[\/latex]. In exponential form, you would write the product as [latex]2^{7}[\/latex]. Notice that [latex]7[\/latex] is the sum of the original two exponents,\u00a0[latex]3[\/latex] and\u00a0[latex]4[\/latex].<\/p>\n<p>What about [latex]{x}^{2}{x}^{6}[\/latex]? This can be written as [latex]\\left(x\\cdot{x}\\right)\\left(x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\right)=x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}[\/latex] or [latex]x^{8}[\/latex]. And, once again, 8 is the sum of the original two exponents. This concept can be generalized in the following way:<\/p>\n<div class=\"textbox shaded\">\n<h3>The Product Rule for Exponents<\/h3>\n<p>For any number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>,\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<\/p>\n<p>To multiply exponential terms with the same base, add the exponents.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/p>\n<ol>\n<li>[latex]{t}^{5}\\cdot {t}^{3}[\/latex]<\/li>\n<li>[latex]\\left(-3\\right)^{5}\\cdot \\left(-3\\right)[\/latex]<\/li>\n<li>[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q772709\">Show Solution<\/span><\/p>\n<div id=\"q772709\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the product rule to simplify each expression.<\/p>\n<ol>\n<li>[latex]{t}^{5}\\cdot {t}^{3}={t}^{5+3}={t}^{8}[\/latex]<\/li>\n<li>[latex]{\\left(-3\\right)}^{5}\\cdot \\left(-3\\right)={\\left(-3\\right)}^{5}\\cdot {\\left(-3\\right)}^{1}={\\left(-3\\right)}^{5+1}={\\left(-3\\right)}^{6}[\/latex]<\/li>\n<li>[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}[\/latex]<\/li>\n<\/ol>\n<p>At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.<\/p>\n<div style=\"text-align: center;\">[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}=\\left({x}^{2}\\cdot {x}^{5}\\right)\\cdot {x}^{3}=\\left({x}^{2+5}\\right)\\cdot {x}^{3}={x}^{7}\\cdot {x}^{3}={x}^{7+3}={x}^{10}[\/latex]<\/div>\n<p>Notice we get the same result by adding the three exponents in one step.<\/p>\n<div style=\"text-align: center;\">[latex]{x}^{2}\\cdot {x}^{5}\\cdot {x}^{3}={x}^{2+5+3}={x}^{10}[\/latex]<\/div>\n<div style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"color: #000000;\">In the following video, you will see more examples of using the product rule for exponents to simplify expressions.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Expressions Using the Product Rule of Exponents (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/P0UVIMy2nuI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In our last product rule example, we will show that an exponent can be an algebraic expression. We can use the product rule for exponents no matter what the exponent looks like, as long as the base is the same.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Multiply.\u00a0[latex]x^{a+2}\\cdot{x^{3a-9}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q740329\">Show Solution<\/span><\/p>\n<div id=\"q740329\" class=\"hidden-answer\" style=\"display: none\">\n<p>We have two exponential terms with the same base. The product rule for exponents says that we can add the exponents.<\/p>\n<p>[latex]x^{a+2}\\cdot{x^{3a-9}}=x^{(a+2)+(3a-9)}=x^{4a-7}[\/latex]<\/p>\n<p>The expression cannot be simplified any further.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Use the Quotient Rule to Divide Exponential Expressions<\/h2>\n<p>Let us look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}[\/latex]<\/p>\n<p>You can rewrite the expression as: [latex]\\displaystyle \\frac{4\\cdot 4\\cdot 4\\cdot 4\\cdot 4}{4\\cdot 4}[\/latex]. Then you can cancel the common factors of 4 in the numerator and denominator: [latex]\\displaystyle[\/latex]<\/p>\n<p>Finally, this expression can be rewritten as [latex]4^{3}[\/latex]\u00a0using exponential notation. Notice that the exponent,\u00a0[latex]3[\/latex], is the difference between the two exponents in the original expression,\u00a0[latex]5[\/latex] and\u00a0[latex]2[\/latex].<\/p>\n<p>So,\u00a0[latex]\\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[\/latex].<\/p>\n<p>Be careful that you subtract the exponent in the denominator from the exponent in the numerator.<\/p>\n<p>So, to divide two exponential terms with the same base, subtract the exponents.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Quotient (Division) Rule for Exponents<\/h3>\n<p>For any non-zero number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>: [latex]\\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write each of the following products with a single base. Do not simplify further.<\/p>\n<ol>\n<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q978732\">Show Solution<\/span><\/p>\n<div id=\"q978732\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use the quotient rule to simplify each expression.<\/p>\n<ol>\n<li>[latex]\\dfrac{{\\left(-2\\right)}^{14}}{{\\left(-2\\right)}^{9}}={\\left(-2\\right)}^{14 - 9}={\\left(-2\\right)}^{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}[\/latex]<\/li>\n<li>[latex]\\dfrac{{\\left(z\\sqrt{2}\\right)}^{5}}{z\\sqrt{2}}={\\left(z\\sqrt{2}\\right)}^{5 - 1}={\\left(z\\sqrt{2}\\right)}^{4}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>As we showed with the product rule, you may be given a quotient with an exponent that is an algebraic expression to simplify. \u00a0As long as the bases agree, you may use the quotient rule for exponents.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\dfrac{y^{x-3}}{y^{9-x}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q836863\">Show Solution<\/span><\/p>\n<div id=\"q836863\" class=\"hidden-answer\" style=\"display: none\">\n<p>We have a quotient whose terms have the same base so we can use the quotient rule for exponents.<\/p>\n<p>[latex]\\dfrac{y^{x-3}}{y^{9-x}}=y^{(x-3)-(9-x)}=y^{2x-12}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"color: #000000;\">In the following video, you will see more examples of using the quotient rule for exponents.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Simplify Expressions Using the Quotient Rule of Exponents (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/xy6WW7y_GcU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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-788\">\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>Simplify Expressions Using the Quotient Rule of Exponents (Basic). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/xy6WW7y_GcU\">https:\/\/youtu.be\/xy6WW7y_GcU<\/a>. <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@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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@3.278:1\/Preface<\/li><li>Ex: Expanding and Evaluating Exponential Notation . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KOnQpKSpVRo\">https:\/\/youtu.be\/KOnQpKSpVRo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Simplify Exponential Expressions Using the Product Property of Exponents . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/hA9AT7QsXWo\">https:\/\/youtu.be\/hA9AT7QsXWo<\/a>. <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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\"},{\"type\":\"cc\",\"description\":\"Ex: 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