{"id":4075,"date":"2020-04-05T18:59:20","date_gmt":"2020-04-05T18:59:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=4075"},"modified":"2021-02-05T23:50:15","modified_gmt":"2021-02-05T23:50:15","slug":"using-the-quotient-property","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/using-the-quotient-property\/","title":{"raw":"Simplifying Variable Expressions Using the Quotient Property of Exponents","rendered":"Simplifying Variable Expressions Using the Quotient Property of Exponents"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify a polynomial expression using the quotient property of exponents<\/li>\r\n \t<li>Simplify expressions with exponents equal to zero<\/li>\r\n \t<li>Simplify quotients raised to a power<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Simplify Expressions Using the Quotient Property of Exponents<\/h2>\r\nEarlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties here.\r\n<div class=\"textbox shaded\">\r\n<h3>Summary of Exponent Properties for Multiplication<\/h3>\r\nIf [latex]a\\text{ and }b[\/latex] are real numbers and [latex]m\\text{ and }n[\/latex] are whole numbers, then\r\n\r\n[latex]\\begin{array}{cccc}\\text{Product Property}\\hfill &amp; &amp; &amp; \\hfill {a}^{m}\\cdot {a}^{n}={a}^{m+n}\\hfill \\\\ \\text{Power Property}\\hfill &amp; &amp; &amp; \\hfill {\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}\\hfill \\\\ \\text{Product to a Power}\\hfill &amp; &amp; &amp; \\hfill {\\left(ab\\right)}^{m}={a}^{m}{b}^{m}\\hfill \\end{array}[\/latex]\r\n\r\n<\/div>\r\nNow we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Fractions you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions\u2014which are also quotients.\r\n<div class=\"textbox shaded\">\r\n<h3>Equivalent Fractions Property<\/h3>\r\nIf [latex]a,b,c[\/latex] are whole numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then\r\n\r\n[latex]{\\Large\\frac{a}{b}}={\\Large\\frac{a\\cdot c}{b\\cdot c}}\\text{ and }{\\Large\\frac{a\\cdot c}{b\\cdot c}}={\\Large\\frac{a}{b}}[\/latex]\r\n\r\n<\/div>\r\nAs before, we'll try to discover a property by looking at some examples.\r\n\r\n[latex]\\begin{array}{cccccccccc}\\text{Consider}\\hfill &amp; &amp; &amp; \\hfill {\\Large\\frac{{x}^{5}}{{x}^{2}}}\\hfill &amp; &amp; &amp; \\text{and}\\hfill &amp; &amp; &amp; \\hfill {\\Large\\frac{{x}^{2}}{{x}^{3}}}\\hfill \\\\ \\text{What do they mean?}\\hfill &amp; &amp; &amp; \\hfill {\\Large\\frac{x\\cdot x\\cdot x\\cdot x\\cdot x}{x\\cdot x}}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\hfill {\\Large\\frac{x\\cdot x}{x\\cdot x\\cdot x}}\\hfill \\\\ \\text{Use the Equivalent Fractions Property.}\\hfill &amp; &amp; &amp; \\hfill \\frac{\\overline{)x}\\cdot \\overline{)x}\\cdot x\\cdot x\\cdot x}{\\overline{)x}\\cdot \\overline{)x}\\cdot 1}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{\\overline{)x}\\cdot \\overline{)x}\\cdot 1}{\\overline{)x}\\cdot \\overline{)x}\\cdot x}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill {x}^{3}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\hfill {\\Large\\frac{1}{x}}\\hfill \\end{array}[\/latex]\r\n\r\nNotice that in each case the bases were the same and we subtracted the exponents.\r\n<ul id=\"fs-id916657\">\r\n \t<li>When the larger exponent was in the numerator, we were left with factors in the numerator and [latex]1[\/latex] in the denominator, which we simplified.<\/li>\r\n \t<li>When the larger exponent was in the denominator, we were left with factors in the denominator, and [latex]1[\/latex] in the numerator, which could not be simplified.<\/li>\r\n<\/ul>\r\nWe write:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{ccccc}\\frac{{x}^{5}}{{x}^{2}}\\hfill &amp; &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{{x}^{3}}\\hfill \\\\ {x}^{5 - 2}\\hfill &amp; &amp; &amp; &amp; \\hfill \\frac{1}{{x}^{3 - 2}}\\hfill \\\\ {x}^{3}\\hfill &amp; &amp; &amp; &amp; \\hfill \\frac{1}{x}\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Quotient Property of Exponents<\/h3>\r\nIf [latex]a[\/latex] is a real number, [latex]a\\ne 0[\/latex], and [latex]m,n[\/latex] are whole numbers, then\r\n\r\n[latex]{\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m&gt;n\\text{ and }{\\Large\\frac{{a}^{m}}{{a}^{n}}}={\\Large\\frac{1}{{a}^{n-m}}},n&gt;m[\/latex]\r\n\r\n<\/div>\r\nA couple of examples with numbers may help to verify this property.\r\n\r\n[latex]\\begin{array}{cccc}\\frac{{3}^{4}}{{3}^{2}}\\stackrel{?}{=}{3}^{4 - 2}\\hfill &amp; &amp; &amp; \\hfill \\frac{{5}^{2}}{{5}^{3}}\\stackrel{?}{=}\\frac{1}{{5}^{3 - 2}}\\hfill \\\\ \\frac{81}{9}\\stackrel{?}{=}{3}^{2}\\hfill &amp; &amp; &amp; \\hfill \\frac{25}{125}\\stackrel{?}{=}\\frac{1}{{5}^{1}}\\hfill \\\\ 9=9 \\hfill &amp; &amp; &amp; \\hfill \\frac{1}{5}=\\frac{1}{5}\\hfill \\end{array}[\/latex]\r\n\r\nWhen we work with numbers and the exponent is less than or equal to [latex]3[\/latex], we will apply the exponent. When the exponent is greater than [latex]3[\/latex] , we leave the answer in exponential form.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\Large\\frac{{x}^{10}}{{x}^{8}}[\/latex]\r\n2. [latex]\\Large\\frac{{2}^{9}}{{2}^{2}}[\/latex]\r\n\r\nSolution\r\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.\r\n<table id=\"eip-id1168469856047\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 10th over x to the 8th. Beside that is written \">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since 10 &gt; 8, there are more factors of [latex]x[\/latex] in the numerator.<\/td>\r\n<td>[latex]\\Large\\frac{{x}^{10}}{{x}^{8}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the quotient property with [latex]m&gt;n,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize ={a}^{m-n}[\/latex] .<\/td>\r\n<td>[latex]{x}^{\\color{red}{10-8}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467300585\" class=\"unnumbered unstyled\" summary=\"The first line shows 2 to the 9th over 2 squared. Beside that is written \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since 9 &gt; 2, there are more factors of 2 in the numerator.<\/td>\r\n<td>[latex]\\Large\\frac{{2}^{9}}{{2}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the quotient property with [latex]m&gt;n,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize ={a}^{m-n}[\/latex].<\/td>\r\n<td>[latex]{2}^{\\color{red}{9-2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{2}^{7}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNotice that when the larger exponent is in the numerator, we are left with factors in the numerator.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146219[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\Large\\frac{{b}^{10}}{{b}^{15}}[\/latex]\r\n2. [latex]\\Large\\frac{{3}^{3}}{{3}^{5}}[\/latex]\r\n[reveal-answer q=\"738923\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"738923\"]\r\n\r\nSolution\r\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.\r\n<table id=\"eip-id1168469768330\" class=\"unnumbered unstyled\" summary=\"The first line shows b to the 10th over b to the 15th. Beside that is written \">\r\n<tbody>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">1.<\/td>\r\n<td style=\"height: 15px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.34375px\">\r\n<td style=\"height: 15.34375px\">Since [latex]15&gt;10[\/latex], there are more factors of [latex]b[\/latex] in the denominator.<\/td>\r\n<td style=\"height: 15.34375px\">[latex]\\Large\\frac{{b}^{10}}{{b}^{15}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Use the quotient property with [latex]n&gt;m,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize =\\Large\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\r\n<td style=\"height: 15px\">[latex]\\Large\\frac{\\color{red}{1}}{{b}^{\\color{red}{15-10}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Simplify.<\/td>\r\n<td style=\"height: 15px\">[latex]\\Large\\frac{1}{{b}^{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469585119\" class=\"unnumbered unstyled\" summary=\"The first line shows 3 to the 3rd over 3 to the 5th. Beside that is written \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since [latex]5&gt;3[\/latex], there are more factors of [latex]3[\/latex] in the denominator.<\/td>\r\n<td>[latex]\\Large\\frac{{3}^{3}}{{3}^{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the quotient property with [latex]n&gt;m,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize =\\Large\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\r\n<td>[latex]\\Large\\frac{\\color{red}{1}}{{3}^{\\color{red}{5-3}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{1}{{3}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Apply the exponent.<\/td>\r\n<td>[latex]\\Large\\frac{1}{9}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that when the larger exponent is in the denominator, we are left with factors in the denominator and [latex]1[\/latex] in the numerator.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146220[\/ohm_question]\r\n\r\n<\/div>\r\nNow let's see if you can determine when you will end up with factors in the denominator, and when you will end up with factors in the numerator.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\Large\\frac{{a}^{5}}{{a}^{9}}[\/latex]\r\n2. [latex]\\Large\\frac{{x}^{11}}{{x}^{7}}[\/latex]\r\n[reveal-answer q=\"903400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"903400\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468311860\" class=\"unnumbered unstyled\" summary=\"The first line shows a to the 5th over a to the 9th. Beside that is written \">\r\n<tbody>\r\n<tr style=\"height: 15.2344px\">\r\n<td style=\"height: 15.2344px\">1.<\/td>\r\n<td style=\"height: 15.2344px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px\">\r\n<td style=\"height: 30px\">Since [latex]9&gt;5[\/latex], there are more [latex]a[\/latex] 's in the denominator and so we will end up with factors in the denominator.<\/td>\r\n<td style=\"height: 30px\">[latex]\\Large\\frac{{a}^{5}}{{a}^{9}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px\">\r\n<td style=\"height: 30px\">Use the Quotient Property for [latex]n&gt;m,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize =\\Large\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\r\n<td style=\"height: 30px\">[latex]\\Large\\frac{\\color{red}{1}}{{a}^{\\color{red}{9-5}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Simplify.<\/td>\r\n<td style=\"height: 15px\">[latex]\\Large\\frac{1}{{a}^{4}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468257607\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 11th over x to the 7th. Beside that is written \">\r\n<tbody>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">2.<\/td>\r\n<td style=\"height: 15px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px\">\r\n<td style=\"height: 30px\">Notice there are more factors of [latex]x[\/latex] in the numerator, since 11 &gt; 7. So we will end up with factors in the numerator.<\/td>\r\n<td style=\"height: 30px\">[latex]\\Large\\frac{{x}^{11}}{{x}^{7}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.28125px\">\r\n<td style=\"height: 15.28125px\">Use the Quotient Property for [latex]m&gt;n,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize ={a}^{n-m}[\/latex].<\/td>\r\n<td style=\"height: 15.28125px\">[latex]{x}^{\\color{red}{11-7}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Simplify.<\/td>\r\n<td style=\"height: 15px\">[latex]{x}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146889[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video for more examples of how to simplify quotients that contain exponents. Pay attention to the last example where we demonstrate the difference between subtracting terms with exponents, and subtracting exponents to simplify a quotient.\r\n\r\nhttps:\/\/youtu.be\/Jmf-CPhm3XM\r\n<h2>Simplify Expressions with Zero Exponents<\/h2>\r\nA special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like [latex]\\Large\\frac{{a}^{m}}{{a}^{m}}[\/latex]. From earlier work with fractions, we know that\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{2}{2}\\normalsize =1\\Large\\frac{17}{17}\\normalsize =1\\Large\\frac{-43}{-43}\\normalsize =1[\/latex]<\/p>\r\nIn words, a number divided by itself is [latex]1[\/latex]. So [latex]\\Large\\frac{x}{x}\\normalsize =1[\/latex], for any [latex]x[\/latex] ( [latex]x\\ne 0[\/latex] ), since any number divided by itself is [latex]1[\/latex].\r\n\r\nThe Quotient Property of Exponents shows us how to simplify [latex]\\Large\\frac{{a}^{m}}{{a}^{n}}[\/latex] when [latex]m&gt;n[\/latex] and when [latex]n&lt;m[\/latex] by subtracting exponents. What if [latex]m=n[\/latex] ?\r\n\r\nNow we will simplify [latex]\\Large\\frac{{a}^{m}}{{a}^{m}}[\/latex] in two ways to lead us to the definition of the zero exponent.\r\nConsider first [latex]\\Large\\frac{8}{8}[\/latex], which we know is [latex]1[\/latex].\r\n<table id=\"eip-id1168469654144\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 11th over x to the 7th. Beside that is written \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large\\frac{8}{8}\\normalsize =1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]8[\/latex] as [latex]{2}^{3}[\/latex] .<\/td>\r\n<td>[latex]\\Large\\frac{{2}^{3}}{{2}^{3}}\\normalsize =1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract exponents.<\/td>\r\n<td>[latex]{2}^{3 - 3}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{2}^{0}=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224544\/CNX_BMath_Figure_10_04_019_img.png\" alt=\".\" \/>\r\nWe see [latex]\\Large\\frac{{a}^{m}}{{a}^{n}}[\/latex] simplifies to a [latex]{a}^{0}[\/latex] and to [latex]1[\/latex] . So [latex]{a}^{0}=1[\/latex] .\r\n<div class=\"textbox shaded\">\r\n<h3>Zero Exponent<\/h3>\r\nIf [latex]a[\/latex] is a non-zero number, then [latex]{a}^{0}=1[\/latex].\r\nAny nonzero number raised to the zero power is [latex]1[\/latex].\r\n\r\n<\/div>\r\nIn this text, we assume any variable that we raise to the zero power is not zero.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{12}^{0}[\/latex]\r\n2. [latex]{y}^{0}[\/latex]\r\n[reveal-answer q=\"363472\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"363472\"]\r\n\r\nSolution\r\nThe definition says any non-zero number raised to the zero power is [latex]1[\/latex].\r\n<table id=\"eip-id1168468469984\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{12}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of the zero exponent.<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466072251\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{y}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of the zero exponent.<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146221[\/ohm_question]\r\n\r\n[ohm_question]146890[\/ohm_question]\r\n\r\n<\/div>\r\nNow that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.\r\nWhat about raising an expression to the zero power? Let's look at [latex]{\\left(2x\\right)}^{0}[\/latex]. We can use the product to a power rule to rewrite this expression.\r\n<table id=\"eip-id1168469871755\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\"><\/td>\r\n<td style=\"height: 15px\">[latex]{\\left(2x\\right)}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Use the Product to a Power Rule.<\/td>\r\n<td style=\"height: 15px\">[latex]{2}^{0}{x}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Use the Zero Exponent Property.<\/td>\r\n<td style=\"height: 15px\">[latex]1\\cdot 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.53125px\">\r\n<td style=\"height: 15.53125px\">Simplify.<\/td>\r\n<td style=\"height: 15.53125px\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis tells us that any non-zero expression raised to the zero power is one.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left(7z\\right)}^{0}[\/latex].\r\n[reveal-answer q=\"685861\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"685861\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467118106\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(7z\\right)}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of the zero exponent.<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146222[\/ohm_question]\r\n\r\n<\/div>\r\nNow let's compare the difference between the previous example, where the entire expression was raised to a zero exponent, and what happens when only one factor is raised to a zero exponent.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{\\left(-3{x}^{2}y\\right)}^{0}[\/latex]\r\n2. [latex]-3{x}^{2}{y}^{0}[\/latex]\r\n[reveal-answer q=\"80948\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"80948\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468244600\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product is raised to the zero power.<\/td>\r\n<td>[latex]{\\left(-3{x}^{2}y\\right)}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of the zero exponent.<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168047201113\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that only the variable [latex]y[\/latex] is being raised to the zero power.<\/td>\r\n<td>[latex]{-3{x}^{2}y}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of the zero exponent.<\/td>\r\n<td>[latex]-3{x}^{2}\\cdot 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-3{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try a similar problem to make sure you see the difference between raising an entire expression to a zero power and having only one factor raised to a zero power.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146223[\/ohm_question]\r\n\r\n[ohm_question]146222[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next video we show some different examples of how you can apply the zero exponent rule.\r\n\r\nhttps:\/\/youtu.be\/zQJy1aBm1dQ\r\n<h2>Simplify Quotients Raised to a Power<\/h2>\r\nNow we will look at an example that will lead us to the Quotient to a Power Property.\r\n<table id=\"eip-id1168469592930\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(\\Large\\frac{x}{y}\\normalsize\\right)}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>This means<\/td>\r\n<td>[latex]\\Large\\frac{x}{y}\\normalsize\\cdot\\Large\\frac{x}{y}\\normalsize\\cdot\r\n\r\n\\Large\\frac{x}{y}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the fractions.<\/td>\r\n<td>[latex]\\Large\\frac{x\\cdot x\\cdot x}{y\\cdot y\\cdot y}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write with exponents.<\/td>\r\n<td>[latex]\\Large\\frac{{x}^{3}}{{y}^{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that the exponent applies to both the numerator and the denominator.\r\n\r\nWe see that [latex]{\\left(\\frac{x}{y}\\normalsize\\right)}^{3}[\/latex] is [latex]\\Large\\frac{{x}^{3}}{{y}^{3}}[\/latex].\r\n\r\n[latex]\\begin{array}{ccccc}\\text{We write:}\\hfill &amp; &amp; &amp; &amp; {\\left(\\frac{x}{y}\\right)}^{3}\\hfill \\\\ &amp; &amp; &amp; &amp; \\frac{{x}^{3}}{{y}^{3}}\\hfill \\end{array}[\/latex]\r\n\r\nThis leads to the Quotient to a Power Property for Exponents.\r\n<div class=\"textbox shaded\">\r\n<h3>Quotient to a Power Property of Exponents<\/h3>\r\nIf [latex]a[\/latex] and [latex]b[\/latex] are real numbers, [latex]b\\ne 0[\/latex], and [latex]m[\/latex] is a counting number, then\r\n\r\n[latex]{\\left(\\Large\\frac{a}{b}\\normalsize\\right)}^{m}=\\Large\\frac{{a}^{m}}{{b}^{m}}[\/latex]\r\nTo raise a fraction to a power, raise the numerator and denominator to that power.\r\n\r\n<\/div>\r\nAn example with numbers may help you understand this property:\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{2}{3}^3[\/latex] =\u00a0[latex]\\Large\\frac{{2}^{3}}{{3}^{3}}[\/latex] = [latex]\\Large\\frac{8}{27}[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{\\left(\\Large\\frac{5}{8}\\normalsize\\right)}^{2}[\/latex]\r\n2. [latex]{\\left(\\Large\\frac{x}{3}\\normalsize\\right)}^{4}[\/latex]\r\n3. [latex]{\\left(\\Large\\frac{y}{m}\\normalsize\\right)}^{3}[\/latex]\r\n\r\n[reveal-answer q=\"803388\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"803388\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467475453\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 5 over 8 raised to the second power. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large(\\frac{5}{8})^2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Quotient to a Power Property, [latex]{\\Large\\left(\\frac{a}{b}\\right)}^{m}\\normalsize =\\Large\\frac{{a}^{m}}{{b}^{m}}[\/latex] .<\/td>\r\n<td>[latex]\\Large\\frac{5^{\\color{red}{2}}}{8^{\\color{red}{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{25}{64}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469801450\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x over 3 raised to the fourth power. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large(\\frac{x}{3})^4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Quotient to a Power Property, [latex]{\\Large\\left(\\frac{a}{b}\\right)}^{m}\\normalsize =\\Large\\frac{{a}^{m}}{{b}^{m}}[\/latex] .<\/td>\r\n<td>[latex]\\Large\\frac{x^{\\color{red}{4}}}{3^{\\color{red}{4}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{x^4}{81}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468503320\" class=\"unnumbered unstyled\" summary=\"Parentheses y over m raised to the third power is shown. Below that, it says, \">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large(\\frac{y}{m})^3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Raise the numerator and denominator to the third power.<\/td>\r\n<td>[latex]\\Large\\frac{y^{3}}{m^{3}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146227[\/ohm_question]\r\n\r\n[ohm_question]146891[\/ohm_question]\r\n\r\n[ohm_question]146892[\/ohm_question]\r\n\r\n<\/div>\r\nFor more examples of how to simplify a quotient raised to a power, watch the following video.\r\n\r\nhttps:\/\/youtu.be\/BoBe31pRxFM","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify a polynomial expression using the quotient property of exponents<\/li>\n<li>Simplify expressions with exponents equal to zero<\/li>\n<li>Simplify quotients raised to a power<\/li>\n<\/ul>\n<\/div>\n<h2>Simplify Expressions Using the Quotient Property of Exponents<\/h2>\n<p>Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties here.<\/p>\n<div class=\"textbox shaded\">\n<h3>Summary of Exponent Properties for Multiplication<\/h3>\n<p>If [latex]a\\text{ and }b[\/latex] are real numbers and [latex]m\\text{ and }n[\/latex] are whole numbers, then<\/p>\n<p>[latex]\\begin{array}{cccc}\\text{Product Property}\\hfill & & & \\hfill {a}^{m}\\cdot {a}^{n}={a}^{m+n}\\hfill \\\\ \\text{Power Property}\\hfill & & & \\hfill {\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}\\hfill \\\\ \\text{Product to a Power}\\hfill & & & \\hfill {\\left(ab\\right)}^{m}={a}^{m}{b}^{m}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<p>Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. In Fractions you learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help us work with algebraic fractions\u2014which are also quotients.<\/p>\n<div class=\"textbox shaded\">\n<h3>Equivalent Fractions Property<\/h3>\n<p>If [latex]a,b,c[\/latex] are whole numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then<\/p>\n<p>[latex]{\\Large\\frac{a}{b}}={\\Large\\frac{a\\cdot c}{b\\cdot c}}\\text{ and }{\\Large\\frac{a\\cdot c}{b\\cdot c}}={\\Large\\frac{a}{b}}[\/latex]<\/p>\n<\/div>\n<p>As before, we&#8217;ll try to discover a property by looking at some examples.<\/p>\n<p>[latex]\\begin{array}{cccccccccc}\\text{Consider}\\hfill & & & \\hfill {\\Large\\frac{{x}^{5}}{{x}^{2}}}\\hfill & & & \\text{and}\\hfill & & & \\hfill {\\Large\\frac{{x}^{2}}{{x}^{3}}}\\hfill \\\\ \\text{What do they mean?}\\hfill & & & \\hfill {\\Large\\frac{x\\cdot x\\cdot x\\cdot x\\cdot x}{x\\cdot x}}\\hfill & & & & & & \\hfill {\\Large\\frac{x\\cdot x}{x\\cdot x\\cdot x}}\\hfill \\\\ \\text{Use the Equivalent Fractions Property.}\\hfill & & & \\hfill \\frac{\\overline{)x}\\cdot \\overline{)x}\\cdot x\\cdot x\\cdot x}{\\overline{)x}\\cdot \\overline{)x}\\cdot 1}\\hfill & & & & & & \\hfill \\frac{\\overline{)x}\\cdot \\overline{)x}\\cdot 1}{\\overline{)x}\\cdot \\overline{)x}\\cdot x}\\hfill \\\\ \\text{Simplify.}\\hfill & & & \\hfill {x}^{3}\\hfill & & & & & & \\hfill {\\Large\\frac{1}{x}}\\hfill \\end{array}[\/latex]<\/p>\n<p>Notice that in each case the bases were the same and we subtracted the exponents.<\/p>\n<ul id=\"fs-id916657\">\n<li>When the larger exponent was in the numerator, we were left with factors in the numerator and [latex]1[\/latex] in the denominator, which we simplified.<\/li>\n<li>When the larger exponent was in the denominator, we were left with factors in the denominator, and [latex]1[\/latex] in the numerator, which could not be simplified.<\/li>\n<\/ul>\n<p>We write:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ccccc}\\frac{{x}^{5}}{{x}^{2}}\\hfill & & & & \\hfill \\frac{{x}^{2}}{{x}^{3}}\\hfill \\\\ {x}^{5 - 2}\\hfill & & & & \\hfill \\frac{1}{{x}^{3 - 2}}\\hfill \\\\ {x}^{3}\\hfill & & & & \\hfill \\frac{1}{x}\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Quotient Property of Exponents<\/h3>\n<p>If [latex]a[\/latex] is a real number, [latex]a\\ne 0[\/latex], and [latex]m,n[\/latex] are whole numbers, then<\/p>\n<p>[latex]{\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m>n\\text{ and }{\\Large\\frac{{a}^{m}}{{a}^{n}}}={\\Large\\frac{1}{{a}^{n-m}}},n>m[\/latex]<\/p>\n<\/div>\n<p>A couple of examples with numbers may help to verify this property.<\/p>\n<p>[latex]\\begin{array}{cccc}\\frac{{3}^{4}}{{3}^{2}}\\stackrel{?}{=}{3}^{4 - 2}\\hfill & & & \\hfill \\frac{{5}^{2}}{{5}^{3}}\\stackrel{?}{=}\\frac{1}{{5}^{3 - 2}}\\hfill \\\\ \\frac{81}{9}\\stackrel{?}{=}{3}^{2}\\hfill & & & \\hfill \\frac{25}{125}\\stackrel{?}{=}\\frac{1}{{5}^{1}}\\hfill \\\\ 9=9 \\hfill & & & \\hfill \\frac{1}{5}=\\frac{1}{5}\\hfill \\end{array}[\/latex]<\/p>\n<p>When we work with numbers and the exponent is less than or equal to [latex]3[\/latex], we will apply the exponent. When the exponent is greater than [latex]3[\/latex] , we leave the answer in exponential form.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\Large\\frac{{x}^{10}}{{x}^{8}}[\/latex]<br \/>\n2. [latex]\\Large\\frac{{2}^{9}}{{2}^{2}}[\/latex]<\/p>\n<p>Solution<br \/>\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.<\/p>\n<table id=\"eip-id1168469856047\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 10th over x to the 8th. Beside that is written\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Since 10 &gt; 8, there are more factors of [latex]x[\/latex] in the numerator.<\/td>\n<td>[latex]\\Large\\frac{{x}^{10}}{{x}^{8}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the quotient property with [latex]m>n,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize ={a}^{m-n}[\/latex] .<\/td>\n<td>[latex]{x}^{\\color{red}{10-8}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{x}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467300585\" class=\"unnumbered unstyled\" summary=\"The first line shows 2 to the 9th over 2 squared. Beside that is written\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Since 9 &gt; 2, there are more factors of 2 in the numerator.<\/td>\n<td>[latex]\\Large\\frac{{2}^{9}}{{2}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the quotient property with [latex]m>n,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize ={a}^{m-n}[\/latex].<\/td>\n<td>[latex]{2}^{\\color{red}{9-2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{2}^{7}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146219\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146219&theme=oea&iframe_resize_id=ohm146219&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\Large\\frac{{b}^{10}}{{b}^{15}}[\/latex]<br \/>\n2. [latex]\\Large\\frac{{3}^{3}}{{3}^{5}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q738923\">Show Solution<\/span><\/p>\n<div id=\"q738923\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.<\/p>\n<table id=\"eip-id1168469768330\" class=\"unnumbered unstyled\" summary=\"The first line shows b to the 10th over b to the 15th. Beside that is written\">\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">1.<\/td>\n<td style=\"height: 15px\"><\/td>\n<\/tr>\n<tr style=\"height: 15.34375px\">\n<td style=\"height: 15.34375px\">Since [latex]15>10[\/latex], there are more factors of [latex]b[\/latex] in the denominator.<\/td>\n<td style=\"height: 15.34375px\">[latex]\\Large\\frac{{b}^{10}}{{b}^{15}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Use the quotient property with [latex]n>m,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize =\\Large\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\n<td style=\"height: 15px\">[latex]\\Large\\frac{\\color{red}{1}}{{b}^{\\color{red}{15-10}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Simplify.<\/td>\n<td style=\"height: 15px\">[latex]\\Large\\frac{1}{{b}^{5}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469585119\" class=\"unnumbered unstyled\" summary=\"The first line shows 3 to the 3rd over 3 to the 5th. Beside that is written\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Since [latex]5>3[\/latex], there are more factors of [latex]3[\/latex] in the denominator.<\/td>\n<td>[latex]\\Large\\frac{{3}^{3}}{{3}^{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the quotient property with [latex]n>m,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize =\\Large\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\n<td>[latex]\\Large\\frac{\\color{red}{1}}{{3}^{\\color{red}{5-3}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{1}{{3}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Apply the exponent.<\/td>\n<td>[latex]\\Large\\frac{1}{9}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and [latex]1[\/latex] in the numerator.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146220\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146220&theme=oea&iframe_resize_id=ohm146220&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Now let&#8217;s see if you can determine when you will end up with factors in the denominator, and when you will end up with factors in the numerator.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\Large\\frac{{a}^{5}}{{a}^{9}}[\/latex]<br \/>\n2. [latex]\\Large\\frac{{x}^{11}}{{x}^{7}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q903400\">Show Solution<\/span><\/p>\n<div id=\"q903400\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468311860\" class=\"unnumbered unstyled\" summary=\"The first line shows a to the 5th over a to the 9th. Beside that is written\">\n<tbody>\n<tr style=\"height: 15.2344px\">\n<td style=\"height: 15.2344px\">1.<\/td>\n<td style=\"height: 15.2344px\"><\/td>\n<\/tr>\n<tr style=\"height: 30px\">\n<td style=\"height: 30px\">Since [latex]9>5[\/latex], there are more [latex]a[\/latex] &#8216;s in the denominator and so we will end up with factors in the denominator.<\/td>\n<td style=\"height: 30px\">[latex]\\Large\\frac{{a}^{5}}{{a}^{9}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 30px\">\n<td style=\"height: 30px\">Use the Quotient Property for [latex]n>m,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize =\\Large\\frac{1}{{a}^{n-m}}[\/latex].<\/td>\n<td style=\"height: 30px\">[latex]\\Large\\frac{\\color{red}{1}}{{a}^{\\color{red}{9-5}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Simplify.<\/td>\n<td style=\"height: 15px\">[latex]\\Large\\frac{1}{{a}^{4}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468257607\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 11th over x to the 7th. Beside that is written\">\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">2.<\/td>\n<td style=\"height: 15px\"><\/td>\n<\/tr>\n<tr style=\"height: 30px\">\n<td style=\"height: 30px\">Notice there are more factors of [latex]x[\/latex] in the numerator, since 11 &gt; 7. So we will end up with factors in the numerator.<\/td>\n<td style=\"height: 30px\">[latex]\\Large\\frac{{x}^{11}}{{x}^{7}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.28125px\">\n<td style=\"height: 15.28125px\">Use the Quotient Property for [latex]m>n,\\Large\\frac{{a}^{m}}{{a}^{n}}\\normalsize ={a}^{n-m}[\/latex].<\/td>\n<td style=\"height: 15.28125px\">[latex]{x}^{\\color{red}{11-7}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Simplify.<\/td>\n<td style=\"height: 15px\">[latex]{x}^{4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146889\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146889&theme=oea&iframe_resize_id=ohm146889&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video for more examples of how to simplify quotients that contain exponents. Pay attention to the last example where we demonstrate the difference between subtracting terms with exponents, and subtracting exponents to simplify a quotient.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" 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>Simplify Expressions with Zero Exponents<\/h2>\n<p>A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like [latex]\\Large\\frac{{a}^{m}}{{a}^{m}}[\/latex]. From earlier work with fractions, we know that<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{2}{2}\\normalsize =1\\Large\\frac{17}{17}\\normalsize =1\\Large\\frac{-43}{-43}\\normalsize =1[\/latex]<\/p>\n<p>In words, a number divided by itself is [latex]1[\/latex]. So [latex]\\Large\\frac{x}{x}\\normalsize =1[\/latex], for any [latex]x[\/latex] ( [latex]x\\ne 0[\/latex] ), since any number divided by itself is [latex]1[\/latex].<\/p>\n<p>The Quotient Property of Exponents shows us how to simplify [latex]\\Large\\frac{{a}^{m}}{{a}^{n}}[\/latex] when [latex]m>n[\/latex] and when [latex]n<m[\/latex] by subtracting exponents. What if [latex]m=n[\/latex] ?\n\nNow we will simplify [latex]\\Large\\frac{{a}^{m}}{{a}^{m}}[\/latex] in two ways to lead us to the definition of the zero exponent.\nConsider first [latex]\\Large\\frac{8}{8}[\/latex], which we know is [latex]1[\/latex].\n\n\n<table id=\"eip-id1168469654144\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 11th over x to the 7th. Beside that is written\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{8}{8}\\normalsize =1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write [latex]8[\/latex] as [latex]{2}^{3}[\/latex] .<\/td>\n<td>[latex]\\Large\\frac{{2}^{3}}{{2}^{3}}\\normalsize =1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract exponents.<\/td>\n<td>[latex]{2}^{3 - 3}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{2}^{0}=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224544\/CNX_BMath_Figure_10_04_019_img.png\" alt=\".\" \/><br \/>\nWe see [latex]\\Large\\frac{{a}^{m}}{{a}^{n}}[\/latex] simplifies to a [latex]{a}^{0}[\/latex] and to [latex]1[\/latex] . So [latex]{a}^{0}=1[\/latex] .<\/p>\n<div class=\"textbox shaded\">\n<h3>Zero Exponent<\/h3>\n<p>If [latex]a[\/latex] is a non-zero number, then [latex]{a}^{0}=1[\/latex].<br \/>\nAny nonzero number raised to the zero power is [latex]1[\/latex].<\/p>\n<\/div>\n<p>In this text, we assume any variable that we raise to the zero power is not zero.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{12}^{0}[\/latex]<br \/>\n2. [latex]{y}^{0}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q363472\">Show Solution<\/span><\/p>\n<div id=\"q363472\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nThe definition says any non-zero number raised to the zero power is [latex]1[\/latex].<\/p>\n<table id=\"eip-id1168468469984\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{12}^{0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of the zero exponent.<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466072251\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{y}^{0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of the zero exponent.<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146221\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146221&theme=oea&iframe_resize_id=ohm146221&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146890\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146890&theme=oea&iframe_resize_id=ohm146890&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.<br \/>\nWhat about raising an expression to the zero power? Let&#8217;s look at [latex]{\\left(2x\\right)}^{0}[\/latex]. We can use the product to a power rule to rewrite this expression.<\/p>\n<table id=\"eip-id1168469871755\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\"><\/td>\n<td style=\"height: 15px\">[latex]{\\left(2x\\right)}^{0}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Use the Product to a Power Rule.<\/td>\n<td style=\"height: 15px\">[latex]{2}^{0}{x}^{0}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Use the Zero Exponent Property.<\/td>\n<td style=\"height: 15px\">[latex]1\\cdot 1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.53125px\">\n<td style=\"height: 15.53125px\">Simplify.<\/td>\n<td style=\"height: 15.53125px\">[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This tells us that any non-zero expression raised to the zero power is one.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{\\left(7z\\right)}^{0}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q685861\">Show Solution<\/span><\/p>\n<div id=\"q685861\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467118106\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(7z\\right)}^{0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of the zero exponent.<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146222\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146222&theme=oea&iframe_resize_id=ohm146222&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Now let&#8217;s compare the difference between the previous example, where the entire expression was raised to a zero exponent, and what happens when only one factor is raised to a zero exponent.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{\\left(-3{x}^{2}y\\right)}^{0}[\/latex]<br \/>\n2. [latex]-3{x}^{2}{y}^{0}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q80948\">Show Solution<\/span><\/p>\n<div id=\"q80948\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468244600\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>The product is raised to the zero power.<\/td>\n<td>[latex]{\\left(-3{x}^{2}y\\right)}^{0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of the zero exponent.<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168047201113\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Notice that only the variable [latex]y[\/latex] is being raised to the zero power.<\/td>\n<td>[latex]{-3{x}^{2}y}^{0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of the zero exponent.<\/td>\n<td>[latex]-3{x}^{2}\\cdot 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-3{x}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a similar problem to make sure you see the difference between raising an entire expression to a zero power and having only one factor raised to a zero power.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146223\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146223&theme=oea&iframe_resize_id=ohm146223&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146222\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146222&theme=oea&iframe_resize_id=ohm146222&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next video we show some different examples of how you can apply the zero exponent rule.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 3: Exponent Properties (Zero Exponent)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/zQJy1aBm1dQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Simplify Quotients Raised to a Power<\/h2>\n<p>Now we will look at an example that will lead us to the Quotient to a Power Property.<\/p>\n<table id=\"eip-id1168469592930\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(\\Large\\frac{x}{y}\\normalsize\\right)}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>This means<\/td>\n<td>[latex]\\Large\\frac{x}{y}\\normalsize\\cdot\\Large\\frac{x}{y}\\normalsize\\cdot    \\Large\\frac{x}{y}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the fractions.<\/td>\n<td>[latex]\\Large\\frac{x\\cdot x\\cdot x}{y\\cdot y\\cdot y}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write with exponents.<\/td>\n<td>[latex]\\Large\\frac{{x}^{3}}{{y}^{3}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that the exponent applies to both the numerator and the denominator.<\/p>\n<p>We see that [latex]{\\left(\\frac{x}{y}\\normalsize\\right)}^{3}[\/latex] is [latex]\\Large\\frac{{x}^{3}}{{y}^{3}}[\/latex].<\/p>\n<p>[latex]\\begin{array}{ccccc}\\text{We write:}\\hfill & & & & {\\left(\\frac{x}{y}\\right)}^{3}\\hfill \\\\ & & & & \\frac{{x}^{3}}{{y}^{3}}\\hfill \\end{array}[\/latex]<\/p>\n<p>This leads to the Quotient to a Power Property for Exponents.<\/p>\n<div class=\"textbox shaded\">\n<h3>Quotient to a Power Property of Exponents<\/h3>\n<p>If [latex]a[\/latex] and [latex]b[\/latex] are real numbers, [latex]b\\ne 0[\/latex], and [latex]m[\/latex] is a counting number, then<\/p>\n<p>[latex]{\\left(\\Large\\frac{a}{b}\\normalsize\\right)}^{m}=\\Large\\frac{{a}^{m}}{{b}^{m}}[\/latex]<br \/>\nTo raise a fraction to a power, raise the numerator and denominator to that power.<\/p>\n<\/div>\n<p>An example with numbers may help you understand this property:<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{2}{3}^3[\/latex] =\u00a0[latex]\\Large\\frac{{2}^{3}}{{3}^{3}}[\/latex] = [latex]\\Large\\frac{8}{27}[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{\\left(\\Large\\frac{5}{8}\\normalsize\\right)}^{2}[\/latex]<br \/>\n2. [latex]{\\left(\\Large\\frac{x}{3}\\normalsize\\right)}^{4}[\/latex]<br \/>\n3. [latex]{\\left(\\Large\\frac{y}{m}\\normalsize\\right)}^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q803388\">Show Solution<\/span><\/p>\n<div id=\"q803388\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467475453\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 5 over 8 raised to the second power. The next line says,\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\Large(\\frac{5}{8})^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Quotient to a Power Property, [latex]{\\Large\\left(\\frac{a}{b}\\right)}^{m}\\normalsize =\\Large\\frac{{a}^{m}}{{b}^{m}}[\/latex] .<\/td>\n<td>[latex]\\Large\\frac{5^{\\color{red}{2}}}{8^{\\color{red}{2}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{25}{64}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469801450\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x over 3 raised to the fourth power. The next line says,\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\Large(\\frac{x}{3})^4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Quotient to a Power Property, [latex]{\\Large\\left(\\frac{a}{b}\\right)}^{m}\\normalsize =\\Large\\frac{{a}^{m}}{{b}^{m}}[\/latex] .<\/td>\n<td>[latex]\\Large\\frac{x^{\\color{red}{4}}}{3^{\\color{red}{4}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{x^4}{81}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468503320\" class=\"unnumbered unstyled\" summary=\"Parentheses y over m raised to the third power is shown. Below that, it says,\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\Large(\\frac{y}{m})^3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Raise the numerator and denominator to the third power.<\/td>\n<td>[latex]\\Large\\frac{y^{3}}{m^{3}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146227\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146227&theme=oea&iframe_resize_id=ohm146227&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146891\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146891&theme=oea&iframe_resize_id=ohm146891&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146892\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146892&theme=oea&iframe_resize_id=ohm146892&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>For more examples of how to simplify a quotient raised to a power, watch the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Simplify Expressions Using Exponent Rules (Power of a Quotient)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/BoBe31pRxFM?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-4075\">\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>Question ID: 146892, 146891, 146227, 146222, 146223, 146890, 146221, 146889, 146220, 146219 . <strong>Authored by<\/strong>: Alyson Day. <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, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Simplify Exponential Expressions Using the Quotient Property of Exponents. <strong>Authored by<\/strong>: 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 3: Exponent Properties (Zero Exponent). <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/zQJy1aBm1dQ\">https:\/\/youtu.be\/zQJy1aBm1dQ<\/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>Simplify Expressions Using Exponent Rules (Power of a Quotient). <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/BoBe31pRxFM\">https:\/\/youtu.be\/BoBe31pRxFM<\/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, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <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\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":25777,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Ex: Simplify Exponential Expressions Using the Quotient Property of Exponents\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Jmf-CPhm3XM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 3: Exponent Properties (Zero Exponent)\",\"author\":\"Lumen 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