{"id":4685,"date":"2017-06-07T18:55:16","date_gmt":"2017-06-07T18:55:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-the-power-rule-for-exponents\/"},"modified":"2017-08-15T15:36:51","modified_gmt":"2017-08-15T15:36:51","slug":"read-the-power-rule-for-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-the-power-rule-for-exponents\/","title":{"raw":"The Power Rule for Exponents","rendered":"The Power Rule for Exponents"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use the power rule to simplify expressions involving\u00a0products, quotients, and exponents<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<h2>Raise powers to powers<\/h2>\r\nAnother word for exponent is power. \u00a0You have likely seen or heard an example such as [latex]3^5[\/latex] can be described as 3 raised to the 5th power. In this section we will further expand our capabilities with exponents. We will learn what to do when a term with a\u00a0power\u00a0is raised to another power, and what to do when two numbers or variables are multiplied and both are raised to an exponent. \u00a0We will also learn what to do when numbers or variables that are divided are raised to a power. \u00a0We will begin by raising powers to powers.\r\n\r\nLet\u2019s simplify [latex]\\left(5^{2}\\right)^{4}[\/latex]. In this case, the base is [latex]5^2[\/latex]<sup>\u00a0<\/sup>and the exponent is 4, so you multiply [latex]5^{2}[\/latex]<sup>\u00a0<\/sup>four times: [latex]\\left(5^{2}\\right)^{4}=5^{2}\\cdot5^{2}\\cdot5^{2}\\cdot5^{2}=5^{8}[\/latex]<sup>\u00a0<\/sup>(using the Product Rule\u2014add the exponents).\r\n\r\n[latex]\\left(5^{2}\\right)^{4}[\/latex]<sup>\u00a0<\/sup>is a power of a power. It is the fourth power of 5 to the second power. And we saw above that the answer is [latex]5^{8}[\/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\\cdot4=8[\/latex].\r\n\r\nSo, [latex]\\left(5^{2}\\right)^{4}=5^{2\\cdot4}=5^{8}[\/latex]\u00a0(which equals 390,625, if you do the multiplication).\r\n\r\nLikewise, [latex]\\left(x^{4}\\right)^{3}=x^{4\\cdot3}=x^{12}[\/latex]\r\n\r\nThis leads to another rule for exponents\u2014the <b>Power Rule for Exponents<\/b>. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\\left(2^{3}\\right)^{5}=2^{15}[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>The Power Rule for Exponents<\/h3>\r\nFor any positive number <i>x<\/i> and integers <i>a<\/i> and <i>b<\/i>: [latex]\\left(x^{a}\\right)^{b}=x^{a\\cdot{b}}[\/latex].\r\n\r\nTake a moment to contrast how this is different from the product rule for exponents found on the previous page.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]6\\left(c^{4}\\right)^{2}[\/latex].\r\n\r\n[reveal-answer q=\"841688\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"841688\"]Since you are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses.\r\n<p style=\"text-align: center\">[latex]6\\left(c^{4}\\right)^{2}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]6\\left(c^{4}\\right)^{2}=6c^{8}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><span style=\"color: #3b86a8\">Raise a product to a power<\/span><\/h2>\r\nSimplify this expression.\r\n<p style=\"text-align: center\">[latex]\\left(2a\\right)^{4}=\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)=\\left(2\\cdot2\\cdot2\\cdot2\\right)\\left(a\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}\\right)=\\left(2^{4}\\right)\\left(a^{4}\\right)=16a^{4}[\/latex]<\/p>\r\nNotice that the exponent is applied to each factor of 2<i>a<\/i>. So, we can eliminate the middle steps.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\left(2a\\right)^{4} = \\left(2^{4}\\right)\\left(a^{4}\\right)\\text{, applying the }4\\text{ to each factor, }2\\text{ and }a\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=16a^{4}\\end{array}[\/latex]<\/p>\r\nThe product of two or more numbers raised to a power is equal to the product of each number raised to the same power.\r\n<div class=\"textbox shaded\">\r\n<h3>A Product Raised to a Power<\/h3>\r\nFor any nonzero numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>, [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].\r\n\r\nHow is this rule different from the power raised to a power rule? How is it different from the product rule for exponents on the previous page?\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify.\u00a0[latex]\\left(2yz\\right)^{6}[\/latex]\r\n\r\n[reveal-answer q=\"368657\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"368657\"]\r\n\r\nApply the exponent to each number in the product.\r\n\r\n[latex]2^{6}y^{6}z^{6}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]\\left(2yz\\right)^{6}=64y^{6}z^{6}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIf the variable has an exponent with it, use the Power Rule: multiply the exponents.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]\\left(\u22127a^{4}b\\right)^{2}[\/latex]\r\n\r\n[reveal-answer q=\"136794\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"136794\"]Apply the exponent 2 to each factor within the parentheses.\r\n\r\n[latex]\\left(\u22127\\right)^{2}\\left(a^{4}\\right)^{2}\\left(b\\right)^{2}[\/latex]\r\n\r\nSquare the coefficient and use the Power Rule to square\u00a0[latex]\\left(a^{4}\\right)^{2}[\/latex].\r\n<p style=\"text-align: center\">[latex]49a^{4\\cdot2}b^{2}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center\">[latex]49a^{8}b^{2}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(-7a^{4}b\\right)^{2}=49a^{8}b^{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/Hgu9HKDHTUA\r\n<h2><span style=\"color: #3b86a8\">Raise a quotient to a power<\/span><\/h2>\r\nNow let\u2019s look at what happens if you raise a quotient to a power. Remember that quotient means divide. Suppose you have [latex] \\displaystyle \\frac{3}{4}[\/latex] and raise it to the 3<sup>rd<\/sup> power.\r\n<p style=\"text-align: center\">[latex] \\displaystyle {{\\left( \\frac{3}{4} \\right)}^{3}}=\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)=\\frac{3\\cdot 3\\cdot 3}{4\\cdot 4\\cdot 4}=\\frac{{{3}^{3}}}{{{4}^{3}}}[\/latex]<\/p>\r\nYou can see that raising the quotient to the power of 3 can also be written as the numerator (3) to the power of 3, and the denominator (4) to the power of 3.\r\n\r\nSimilarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to power.\r\n<p style=\"text-align: center\">[latex] \\displaystyle {{\\left( \\frac{a}{b} \\right)}^{4}}=\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)=\\frac{a\\cdot a\\cdot a\\cdot a}{b\\cdot b\\cdot b\\cdot b}=\\frac{{{a}^{4}}}{{{b}^{4}}}[\/latex]<\/p>\r\nWhen a quotient is raised to a power, you can apply the power to the numerator and denominator individually, as shown below.\r\n<p style=\"text-align: center\">[latex] \\displaystyle {{\\left( \\frac{a}{b} \\right)}^{4}}=\\frac{{{a}^{4}}}{{{b}^{4}}}[\/latex]<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>A Quotient Raised to a Power<\/h3>\r\nFor any number <i>a<\/i>, any non-zero number <i>b<\/i>, and any integer <i>x<\/i>, [latex] \\displaystyle {\\left(\\frac{a}{b}\\right)}^{x}=\\frac{a^{x}}{b^{x}}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}[\/latex]\r\n\r\n[reveal-answer q=\"875425\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"875425\"]Apply the power to each factor individually.\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\frac{{{2}^{3}{\\left({x}^{2}\\right)}^{3}{y}^{3}}}{{{x}^{3}}}[\/latex]<\/p>\r\nSeparate into numerical and variable factors.\r\n<p style=\"text-align: center\">[latex] \\displaystyle {{2}^{3}}\\cdot \\frac{{{x}^{3\\cdot2}}}{{{x}^{3}}}\\cdot \\frac{{{y}^{3}}}{1}[\/latex]<\/p>\r\nSimplify by taking 2 to the third power and applying the Power and Quotient Rules for exponents\u2014multiply\u00a0and subtract the exponents of matching variables.\r\n<p style=\"text-align: center\">[latex] \\displaystyle 8\\cdot {{x}^{(6-3)}}\\cdot {{y}^{3}}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center\">[latex] \\displaystyle 8{{x}^{3}}{{y}^{3}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}=8{{x}^{3}}{{y}^{3}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you will be shown examples of simplifying quotients that are raised to a power.\r\n\r\nhttps:\/\/youtu.be\/ZbxgDRV35dE\r\n<h2>Summary<\/h2>\r\n<ul>\r\n \t<li>Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.<\/li>\r\n \t<li>The product rule for exponents:\u00a0For 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].<\/li>\r\n \t<li>The quotient rule for exponents:\u00a0For 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]<\/li>\r\n \t<li>The power rule for exponents:\r\n<ol>\r\n \t<li>For any nonzero numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>, [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].<\/li>\r\n \t<li>For any number <i>a<\/i>, any non-zero number <i>b<\/i>, and any integer <i>x<\/i>, [latex] \\displaystyle {\\left(\\frac{a}{b}\\right)}^{x}=\\frac{a^{x}}{b^{x}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use the power rule to simplify expressions involving\u00a0products, quotients, and exponents<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Raise powers to powers<\/h2>\n<p>Another word for exponent is power. \u00a0You have likely seen or heard an example such as [latex]3^5[\/latex] can be described as 3 raised to the 5th power. In this section we will further expand our capabilities with exponents. We will learn what to do when a term with a\u00a0power\u00a0is raised to another power, and what to do when two numbers or variables are multiplied and both are raised to an exponent. \u00a0We will also learn what to do when numbers or variables that are divided are raised to a power. \u00a0We will begin by raising powers to powers.<\/p>\n<p>Let\u2019s simplify [latex]\\left(5^{2}\\right)^{4}[\/latex]. In this case, the base is [latex]5^2[\/latex]<sup>\u00a0<\/sup>and the exponent is 4, so you multiply [latex]5^{2}[\/latex]<sup>\u00a0<\/sup>four times: [latex]\\left(5^{2}\\right)^{4}=5^{2}\\cdot5^{2}\\cdot5^{2}\\cdot5^{2}=5^{8}[\/latex]<sup>\u00a0<\/sup>(using the Product Rule\u2014add the exponents).<\/p>\n<p>[latex]\\left(5^{2}\\right)^{4}[\/latex]<sup>\u00a0<\/sup>is a power of a power. It is the fourth power of 5 to the second power. And we saw above that the answer is [latex]5^{8}[\/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\\cdot4=8[\/latex].<\/p>\n<p>So, [latex]\\left(5^{2}\\right)^{4}=5^{2\\cdot4}=5^{8}[\/latex]\u00a0(which equals 390,625, if you do the multiplication).<\/p>\n<p>Likewise, [latex]\\left(x^{4}\\right)^{3}=x^{4\\cdot3}=x^{12}[\/latex]<\/p>\n<p>This leads to another rule for exponents\u2014the <b>Power Rule for Exponents<\/b>. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\\left(2^{3}\\right)^{5}=2^{15}[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>The Power Rule for Exponents<\/h3>\n<p>For any positive number <i>x<\/i> and integers <i>a<\/i> and <i>b<\/i>: [latex]\\left(x^{a}\\right)^{b}=x^{a\\cdot{b}}[\/latex].<\/p>\n<p>Take a moment to contrast how this is different from the product rule for exponents found on the previous page.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]6\\left(c^{4}\\right)^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q841688\">Show Solution<\/span><\/p>\n<div id=\"q841688\" class=\"hidden-answer\" style=\"display: none\">Since you are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses.<\/p>\n<p style=\"text-align: center\">[latex]6\\left(c^{4}\\right)^{2}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]6\\left(c^{4}\\right)^{2}=6c^{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><span style=\"color: #3b86a8\">Raise a product to a power<\/span><\/h2>\n<p>Simplify this expression.<\/p>\n<p style=\"text-align: center\">[latex]\\left(2a\\right)^{4}=\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)=\\left(2\\cdot2\\cdot2\\cdot2\\right)\\left(a\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}\\right)=\\left(2^{4}\\right)\\left(a^{4}\\right)=16a^{4}[\/latex]<\/p>\n<p>Notice that the exponent is applied to each factor of 2<i>a<\/i>. So, we can eliminate the middle steps.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}\\left(2a\\right)^{4} = \\left(2^{4}\\right)\\left(a^{4}\\right)\\text{, applying the }4\\text{ to each factor, }2\\text{ and }a\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=16a^{4}\\end{array}[\/latex]<\/p>\n<p>The product of two or more numbers raised to a power is equal to the product of each number raised to the same power.<\/p>\n<div class=\"textbox shaded\">\n<h3>A Product Raised to a Power<\/h3>\n<p>For any nonzero numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>, [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].<\/p>\n<p>How is this rule different from the power raised to a power rule? How is it different from the product rule for exponents on the previous page?<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify.\u00a0[latex]\\left(2yz\\right)^{6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q368657\">Show Solution<\/span><\/p>\n<div id=\"q368657\" class=\"hidden-answer\" style=\"display: none\">\n<p>Apply the exponent to each number in the product.<\/p>\n<p>[latex]2^{6}y^{6}z^{6}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(2yz\\right)^{6}=64y^{6}z^{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>If the variable has an exponent with it, use the Power Rule: multiply the exponents.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\left(\u22127a^{4}b\\right)^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q136794\">Show Solution<\/span><\/p>\n<div id=\"q136794\" class=\"hidden-answer\" style=\"display: none\">Apply the exponent 2 to each factor within the parentheses.<\/p>\n<p>[latex]\\left(\u22127\\right)^{2}\\left(a^{4}\\right)^{2}\\left(b\\right)^{2}[\/latex]<\/p>\n<p>Square the coefficient and use the Power Rule to square\u00a0[latex]\\left(a^{4}\\right)^{2}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]49a^{4\\cdot2}b^{2}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center\">[latex]49a^{8}b^{2}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(-7a^{4}b\\right)^{2}=49a^{8}b^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Simplify Exponential Expressions Using the Power Property of Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Hgu9HKDHTUA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><span style=\"color: #3b86a8\">Raise a quotient to a power<\/span><\/h2>\n<p>Now let\u2019s look at what happens if you raise a quotient to a power. Remember that quotient means divide. Suppose you have [latex]\\displaystyle \\frac{3}{4}[\/latex] and raise it to the 3<sup>rd<\/sup> power.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle {{\\left( \\frac{3}{4} \\right)}^{3}}=\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)=\\frac{3\\cdot 3\\cdot 3}{4\\cdot 4\\cdot 4}=\\frac{{{3}^{3}}}{{{4}^{3}}}[\/latex]<\/p>\n<p>You can see that raising the quotient to the power of 3 can also be written as the numerator (3) to the power of 3, and the denominator (4) to the power of 3.<\/p>\n<p>Similarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to power.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle {{\\left( \\frac{a}{b} \\right)}^{4}}=\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)=\\frac{a\\cdot a\\cdot a\\cdot a}{b\\cdot b\\cdot b\\cdot b}=\\frac{{{a}^{4}}}{{{b}^{4}}}[\/latex]<\/p>\n<p>When a quotient is raised to a power, you can apply the power to the numerator and denominator individually, as shown below.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle {{\\left( \\frac{a}{b} \\right)}^{4}}=\\frac{{{a}^{4}}}{{{b}^{4}}}[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>A Quotient Raised to a Power<\/h3>\n<p>For any number <i>a<\/i>, any non-zero number <i>b<\/i>, and any integer <i>x<\/i>, [latex]\\displaystyle {\\left(\\frac{a}{b}\\right)}^{x}=\\frac{a^{x}}{b^{x}}[\/latex].<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q875425\">Show Solution<\/span><\/p>\n<div id=\"q875425\" class=\"hidden-answer\" style=\"display: none\">Apply the power to each factor individually.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\frac{{{2}^{3}{\\left({x}^{2}\\right)}^{3}{y}^{3}}}{{{x}^{3}}}[\/latex]<\/p>\n<p>Separate into numerical and variable factors.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle {{2}^{3}}\\cdot \\frac{{{x}^{3\\cdot2}}}{{{x}^{3}}}\\cdot \\frac{{{y}^{3}}}{1}[\/latex]<\/p>\n<p>Simplify by taking 2 to the third power and applying the Power and Quotient Rules for exponents\u2014multiply\u00a0and subtract the exponents of matching variables.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle 8\\cdot {{x}^{(6-3)}}\\cdot {{y}^{3}}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle 8{{x}^{3}}{{y}^{3}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}=8{{x}^{3}}{{y}^{3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you will be shown examples of simplifying quotients that are raised to a power.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Simplify Fractions Raised to Powers (Positive Exponents Only) Version 1\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ZbxgDRV35dE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<ul>\n<li>Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.<\/li>\n<li>The product rule for exponents:\u00a0For 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].<\/li>\n<li>The quotient rule for exponents:\u00a0For 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]<\/li>\n<li>The power rule for exponents:\n<ol>\n<li>For any nonzero numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>, [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].<\/li>\n<li>For any number <i>a<\/i>, any non-zero number <i>b<\/i>, and any integer <i>x<\/i>, [latex]\\displaystyle {\\left(\\frac{a}{b}\\right)}^{x}=\\frac{a^{x}}{b^{x}}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ul>\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-4685\">\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: 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><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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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