{"id":4684,"date":"2017-06-07T18:55:16","date_gmt":"2017-06-07T18:55:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-product-and-quotient-rules\/"},"modified":"2017-08-15T15:34:38","modified_gmt":"2017-08-15T15:34:38","slug":"read-product-and-quotient-rules","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-product-and-quotient-rules\/","title":{"raw":"Product and Quotient Rules","rendered":"Product and Quotient Rules"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use the product rule to multiply exponential expressions<\/li>\r\n \t<li>Use the quotient rule to divide exponential expressions<\/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\u2019s 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 625. And don\u2019t 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 7 is the sum of the original two exponents, 3 and 4.\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<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184952\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"88\" height=\"78\" \/>Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. \u00a0For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says \"For any number <em>x<\/em>, and any integers <em>a<\/em> and <em>b<\/em>.\"\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center\">[latex](a^{3})(a^{7})[\/latex]<\/p>\r\n[reveal-answer q=\"356596\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"356596\"]The base of both exponents is <i>a<\/i>, so the product rule applies.\r\n<p style=\"text-align: center\">[latex]\\left(a^{3}\\right)\\left(a^{7}\\right)[\/latex]<\/p>\r\nAdd the exponents with a common base.\r\n<p style=\"text-align: center\">[latex]a^{3+7}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(a^{3}\\right)\\left(a^{7}\\right) = a^{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen multiplying more complicated terms, multiply the coefficients and then multiply the variables.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center\">[latex]5a^{4}\\cdot7a^{6}[\/latex]<\/p>\r\n[reveal-answer q=\"215459\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"215459\"]Multiply the coefficients.\r\n<p style=\"text-align: center\">[latex]35\\cdot{a}^{4}\\cdot{a}^{6}[\/latex]<\/p>\r\nThe base of both exponents is <i>a<\/i>, so the product rule applies. Add the exponents.\r\n<p style=\"text-align: center\">[latex]35\\cdot{a}^{4+6}[\/latex]<\/p>\r\nAdd the exponents with a common base.\r\n<p style=\"text-align: center\">[latex]35\\cdot{a}^{10}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]5a^{4}\\cdot7a^{6}=35a^{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185514\/traffic-sign-160659.png\" alt=\"traffic-sign-160659\" width=\"96\" height=\"83\" \/><\/h3>\r\nCaution! Do not try to apply this rule to sums.\r\n\r\nThink about the expression\u00a0[latex]\\left(2+3\\right)^{2}[\/latex]\r\n<p style=\"text-align: center\">Does [latex]\\left(2+3\\right)^{2}[\/latex] equal [latex]2^{2}+3^{2}[\/latex]?<\/p>\r\nNo, it does not because of the order of operations!\r\n<p style=\"text-align: center\">[latex]\\left(2+3\\right)^{2}=5^{2}=25[\/latex]<\/p>\r\n<p style=\"text-align: center\">and<\/p>\r\n<p style=\"text-align: center\">[latex]2^{2}+3^{2}=4+9=13[\/latex]<\/p>\r\nTherefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next).\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/hA9AT7QsXWo\r\n<h2>Use the quotient rule to divide exponential expressions<\/h2>\r\nLet\u2019s 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, 3, is the difference between the two exponents in the original expression, 5 and 2.\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\nEvaluate. [latex] \\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}[\/latex]\r\n\r\n[reveal-answer q=\"96156\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"96156\"]These two exponents have the same base, 4. According to the Quotient Rule, you can subtract the power in the denominator from the power in the numerator.\r\n<p style=\"text-align: center\">[latex] \\displaystyle {{4}^{9-4}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}=4^{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]\r\n\r\n[reveal-answer q=\"23604\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"23604\"]Separate into numerical and variable factors.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\left( \\frac{12}{2} \\right)\\left( \\frac{{{x}^{4}}}{x} \\right)[\/latex]<\/p>\r\nSince the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.\r\n<p style=\"text-align: center\">[latex] \\displaystyle 6\\left( {{x}^{4-1}} \\right)[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]=[latex] \\displaystyle 6{{x}^{3}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p class=\"no-indent\" style=\"text-align: left\">In the following video we show another example of how to use the quotient rule to divide exponential expressions<\/p>\r\nhttps:\/\/youtu.be\/Jmf-CPhm3XM\r\n<h2><\/h2>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use the product rule to multiply exponential expressions<\/li>\n<li>Use the quotient rule to divide exponential expressions<\/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\u2019s 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 625. And don\u2019t 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 7 is the sum of the original two exponents, 3 and 4.<\/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<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184952\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"88\" height=\"78\" \/>Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. \u00a0For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says &#8220;For any number <em>x<\/em>, and any integers <em>a<\/em> and <em>b<\/em>.&#8221;<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center\">[latex](a^{3})(a^{7})[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q356596\">Show Solution<\/span><\/p>\n<div id=\"q356596\" class=\"hidden-answer\" style=\"display: none\">The base of both exponents is <i>a<\/i>, so the product rule applies.<\/p>\n<p style=\"text-align: center\">[latex]\\left(a^{3}\\right)\\left(a^{7}\\right)[\/latex]<\/p>\n<p>Add the exponents with a common base.<\/p>\n<p style=\"text-align: center\">[latex]a^{3+7}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(a^{3}\\right)\\left(a^{7}\\right) = a^{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When multiplying more complicated terms, multiply the coefficients and then multiply the variables.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center\">[latex]5a^{4}\\cdot7a^{6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q215459\">Show Solution<\/span><\/p>\n<div id=\"q215459\" class=\"hidden-answer\" style=\"display: none\">Multiply the coefficients.<\/p>\n<p style=\"text-align: center\">[latex]35\\cdot{a}^{4}\\cdot{a}^{6}[\/latex]<\/p>\n<p>The base of both exponents is <i>a<\/i>, so the product rule applies. Add the exponents.<\/p>\n<p style=\"text-align: center\">[latex]35\\cdot{a}^{4+6}[\/latex]<\/p>\n<p>Add the exponents with a common base.<\/p>\n<p style=\"text-align: center\">[latex]35\\cdot{a}^{10}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]5a^{4}\\cdot7a^{6}=35a^{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185514\/traffic-sign-160659.png\" alt=\"traffic-sign-160659\" width=\"96\" height=\"83\" \/><\/h3>\n<p>Caution! Do not try to apply this rule to sums.<\/p>\n<p>Think about the expression\u00a0[latex]\\left(2+3\\right)^{2}[\/latex]<\/p>\n<p style=\"text-align: center\">Does [latex]\\left(2+3\\right)^{2}[\/latex] equal [latex]2^{2}+3^{2}[\/latex]?<\/p>\n<p>No, it does not because of the order of operations!<\/p>\n<p style=\"text-align: center\">[latex]\\left(2+3\\right)^{2}=5^{2}=25[\/latex]<\/p>\n<p style=\"text-align: center\">and<\/p>\n<p style=\"text-align: center\">[latex]2^{2}+3^{2}=4+9=13[\/latex]<\/p>\n<p>Therefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next).<\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Simplify Exponential Expressions Using the Product Property of Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hA9AT7QsXWo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Use the quotient rule to divide exponential expressions<\/h2>\n<p>Let\u2019s 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, 3, is the difference between the two exponents in the original expression, 5 and 2.<\/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>Evaluate. [latex]\\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q96156\">Show Solution<\/span><\/p>\n<div id=\"q96156\" class=\"hidden-answer\" style=\"display: none\">These two exponents have the same base, 4. According to the Quotient Rule, you can subtract the power in the denominator from the power in the numerator.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle {{4}^{9-4}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}=4^{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q23604\">Show Solution<\/span><\/p>\n<div id=\"q23604\" class=\"hidden-answer\" style=\"display: none\">Separate into numerical and variable factors.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\left( \\frac{12}{2} \\right)\\left( \\frac{{{x}^{4}}}{x} \\right)[\/latex]<\/p>\n<p>Since the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle 6\\left( {{x}^{4-1}} \\right)[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]=[latex]\\displaystyle 6{{x}^{3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p class=\"no-indent\" style=\"text-align: left\">In the following video we show another example of how to use the quotient rule to divide exponential expressions<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Simplify Exponential Expressions Using the Quotient Property of Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Jmf-CPhm3XM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><\/h2>\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-4684\">\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>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><li>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/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\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Ex: Simplify Exponential Expressions Using the Quotient 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\/Jmf-CPhm3XM\">https:\/\/youtu.be\/Jmf-CPhm3XM<\/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 Power 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\/Hgu9HKDHTUA\">https:\/\/youtu.be\/Hgu9HKDHTUA<\/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 Fractions Raised to Powers (Positive Exponents Only) Version 1 . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ZbxgDRV35dE\">https:\/\/youtu.be\/ZbxgDRV35dE<\/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\">Public domain content<\/div><ul class=\"citation-list\"><li>Caution Sign. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/traffic-sign-sign-160659\/\">https:\/\/pixabay.com\/en\/traffic-sign-sign-160659\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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