{"id":10852,"date":"2017-06-05T21:31:27","date_gmt":"2017-06-05T21:31:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10852"},"modified":"2020-10-22T09:25:56","modified_gmt":"2020-10-22T09:25:56","slug":"combining-properties-to-simplify-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/combining-properties-to-simplify-expressions\/","title":{"raw":"11.1.e - Simplifying Complex Expressions I","rendered":"11.1.e &#8211; Simplifying Complex Expressions I"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify complex expressions using a combination of exponent rules<\/li>\r\n \t<li>Simplify quotients that require a combination of the properties of exponents<\/li>\r\n<\/ul>\r\n<\/div>\r\nAll the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference as we will be using them here to simplify various exponential expressions.\r\n<div class=\"textbox shaded\">\r\n<h3>Summary of Exponent Properties<\/h3>\r\nIf [latex]a,b[\/latex] are real numbers and [latex]m,n[\/latex] are integers, then\r\n\r\n[latex]\\begin{array}{cccc}\\mathbf{\\text{Product Property}}\\hfill &amp; &amp; &amp; {a}^{m}\\cdot {a}^{n}={a}^{m+n}\\hfill \\\\ \\mathbf{\\text{Power Property}}\\hfill &amp; &amp; &amp; {\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}\\hfill \\\\ \\mathbf{\\text{Product to a Power Property}}\\hfill &amp; &amp; &amp; {\\left(ab\\right)}^{m}={a}^{m}{b}^{m}\\hfill \\\\ \\mathbf{\\text{Quotient Property}}\\hfill &amp; &amp; &amp; {\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},a\\ne 0\\hfill \\\\ \\mathbf{\\text{Zero Exponent Property}}\\hfill &amp; &amp; &amp; {a}^{0}=1,a\\ne 0\\hfill \\\\ \\mathbf{\\text{Quotient to a Power Property}}\\hfill &amp; &amp; &amp; {\\left({\\Large\\frac{a}{b}}\\right)}^{m}={\\Large\\frac{{a}^{m}}{{b}^{m}}},b\\ne 0\\hfill \\\\ \\mathbf{\\text{Definition of Negative Exponent}}\\hfill &amp; &amp; &amp; {a}^{-n}={\\Large\\frac{1}{{a}^{n}}}\\hfill \\end{array}[\/latex]\r\n\r\n<\/div>\r\n<h3>Expressions with negative exponents<\/h3>\r\nThe following examples involve simplifying expressions with negative exponents.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{x}^{-4}\\cdot {x}^{6}[\/latex]\r\n2. [latex]{y}^{-6}\\cdot {y}^{4}[\/latex]\r\n3. [latex]{z}^{-5}\\cdot {z}^{-3}[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168469494417\" 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]{x}^{-4}\\cdot {x}^{6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Product Property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\r\n<td>[latex]{x}^{-4+6}[\/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-id1168467250235\" 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}^{-6}\\cdot {y}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The bases are the same, so add the exponents.<\/td>\r\n<td>[latex]{y}^{-6+4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{y}^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of a negative exponent, [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/td>\r\n<td>[latex]{\\Large\\frac{1}{{y}^{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468607449\" class=\"unnumbered unstyled\" summary=\".\">\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]{z}^{-5}\\cdot {z}^{-3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The bases are the same, so add the exponents.<\/td>\r\n<td>[latex]{z}^{-5 - 3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{z}^{-8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of a negative exponent, [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/td>\r\n<td>[latex]{\\Large\\frac{1}{{z}^{8}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146301[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next two examples, we\u2019ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\left({m}^{4}{n}^{-3}\\right)\\left({m}^{-5}{n}^{-2}\\right)[\/latex]\r\n[reveal-answer q=\"198141\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"198141\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468721741\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left({m}^{4}{n}^{-3}\\right)\\left({m}^{-5}{n}^{-2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Commutative Property to get like bases together.<\/td>\r\n<td>[latex]{m}^{4}{m}^{-5}\\cdot {n}^{-2}{n}^{-3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the exponents for each base.<\/td>\r\n<td>[latex]{m}^{-1}\\cdot {n}^{-5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take reciprocals and change the signs of the exponents.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{{m}^{1}}\\cdot \\frac{1}{{n}^{5}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{m{n}^{5}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146303[\/ohm_question]\r\n\r\n<\/div>\r\nIf we multipy two expressions with numerical coefficients, we multiply the coefficients together.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]\\left(2{x}^{-6}{y}^{8}\\right)\\left(-5{x}^{5}{y}^{-3}\\right)[\/latex]\r\n[reveal-answer q=\"989732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"989732\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466697901\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(2{x}^{-6}{y}^{8}\\right)\\left(-5{x}^{5}{y}^{-3}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite with the like bases together.<\/td>\r\n<td>[latex]2\\left(-5\\right)\\cdot \\left({x}^{-6}{x}^{5}\\right)\\cdot \\left({y}^{8}{y}^{-3}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-10\\cdot {x}^{-1}\\cdot {y}^{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of a negative exponent, [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/td>\r\n<td>[latex]-10\\cdot {\\Large\\frac{1}{{x}^{1}}}\\cdot {y}^{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\Large\\frac{-10{y}^{5}}{x}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146304[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next two examples, we\u2019ll use the Power Property and the Product to a Power Property to simplify expressions with negative exponents.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left({k}^{3}\\right)}^{-2}[\/latex].\r\n[reveal-answer q=\"769374\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"769374\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168464917598\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left({k}^{3}\\right)}^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Product to a Power Property, [latex]{\\left(ab\\right)}^{m}={a}^{m}{b}^{m}[\/latex].<\/td>\r\n<td>[latex]{k}^{3\\left(-2\\right)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{k}^{-6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite with a positive exponent.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{{k}^{6}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146306[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left(5{x}^{-3}\\right)}^{2}[\/latex]\r\n[reveal-answer q=\"40374\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"40374\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466013404\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(5{x}^{-3}\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Product to a Power Property, [latex]{\\left(ab\\right)}^{m}={a}^{m}{b}^{m}[\/latex].<\/td>\r\n<td>[latex]{5}^{2}{\\left({x}^{-3}\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify [latex]{5}^{2}[\/latex] and multiply the exponents of [latex]x[\/latex] using the\r\n\r\nPower Property, [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/td>\r\n<td>[latex]25{x}^{-6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite [latex]{x}^{-6}[\/latex] by using the definition of a negative\r\n\r\nexponent, [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/td>\r\n<td>[latex]25\\cdot {\\Large\\frac{1}{{x}^{6}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify<\/td>\r\n<td>[latex]{\\Large\\frac{25}{{x}^{6}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146307[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show another example of how to simplify a product that contains negative exponents.\r\n\r\nhttps:\/\/youtu.be\/J9A-JlTXnsQ\r\n\r\nThe following examples involve solving exponential expressions with quotients.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\Large\\frac{{\\left({x}^{2}\\right)}^{3}}{{x}^{5}}}[\/latex].\r\n\r\nSolution\r\n<table id=\"eip-id1168468505651\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\Large\\frac{{\\left({x}^{2}\\right)}^{3}}{{x}^{5}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the exponents in the numerator, using the\r\n\r\nPower Property.<\/td>\r\n<td>[latex]{\\Large\\frac{{x}^{6}}{{x}^{5}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract the exponents.<\/td>\r\n<td>[latex]x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146230[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\Large\\frac{{m}^{8}}{{\\left({m}^{2}\\right)}^{4}}}[\/latex]\r\n[reveal-answer q=\"680453\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"680453\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468260532\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\Large\\frac{{m}^{8}}{{\\left({m}^{2}\\right)}^{4}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the exponents in the numerator, using the\r\n\r\nPower Property.<\/td>\r\n<td>[latex]{\\Large\\frac{{m}^{8}}{{m}^{8}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract the exponents.<\/td>\r\n<td>[latex]{m}^{0}=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146231[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left({\\Large\\frac{{x}^{7}}{{x}^{3}}}\\right)}^{2}[\/latex]\r\n[reveal-answer q=\"903996\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"903996\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468297399\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(\\frac{{x}^{7}}{{x}^{3}}\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remember parentheses come before exponents, and the\r\n\r\nbases are the same so we can simplify inside the\r\n\r\nparentheses. Subtract the exponents.<\/td>\r\n<td>[latex]{\\left({x}^{7 - 3}\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\left({x}^{4}\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the exponents.<\/td>\r\n<td>[latex]{x}^{8}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146233[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left({\\Large\\frac{{p}^{2}}{{q}^{5}}}\\right)}^{3}[\/latex]\r\n[reveal-answer q=\"867763\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"867763\"]\r\n\r\nSolution\r\nHere we cannot simplify inside the parentheses first, since the bases are not the same.\r\n<table id=\"eip-id1168467258528\" class=\"unnumbered unstyled\" style=\"width: 859px\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 477.117px\"><\/td>\r\n<td style=\"width: 357.883px\">[latex]{\\Large{\\left(\\frac{{p}^{2}}{{q}^{5}}\\right)}}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 477.117px\">Raise the numerator and denominator to the third power\u00a0using the Quotient to a Power Property, [latex]{\\Large{\\left(\\frac{a}{b}\\right)}}^{m}={\\Large\\frac{{a}^{m}}{{b}^{m}}}[\/latex]<\/td>\r\n<td style=\"width: 357.883px\">[latex]{\\Large\\frac{(p^2)^{3}}{(q^5)^{3}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 477.117px\">Use the Power Property, [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/td>\r\n<td style=\"width: 357.883px\">[latex]{\\Large\\frac{{p}^{6}}{{q}^{15}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146234[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\Large{\\left(\\frac{2{x}^{3}}{3y}\\right)}}^{4}[\/latex]\r\n[reveal-answer q=\"521774\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"521774\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468606012\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\Large{\\left(\\frac{2{x}^{3}}{3y}\\right)}}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Raise the numerator and denominator to the fourth\r\n\r\npower using the Quotient to a Power Property.<\/td>\r\n<td>[latex]{\\Large\\frac{{\\left(2{x}^{3}\\right)}^{4}}{{\\left(3y\\right)}^{4}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Raise each factor to the fourth power, using the Power\r\n\r\nto a Power Property.<\/td>\r\n<td>[latex]{\\Large\\frac{{2}^{4}{\\left({x}^{3}\\right)}^{4}}{{3}^{4}{y}^{4}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Power Property and simplify.<\/td>\r\n<td>[latex]{\\Large\\frac{16{x}^{12}}{81{y}^{4}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146235[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\Large\\frac{{\\left({y}^{2}\\right)}^{3}{\\left({y}^{2}\\right)}^{4}}{{\\left({y}^{5}\\right)}^{4}}}[\/latex]\r\n[reveal-answer q=\"189952\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"189952\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466312387\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\Large\\frac{{\\left({y}^{2}\\right)}^{3}{\\left({y}^{2}\\right)}^{4}}{{\\left({y}^{5}\\right)}^{4}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Power Property.<\/td>\r\n<td>[latex]{\\Large\\frac{\\left({y}^{6}\\right)\\left({y}^{8}\\right)}{{y}^{20}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the exponents in the numerator, using the Product Property.<\/td>\r\n<td>[latex]{\\Large\\frac{{y}^{14}}{{y}^{20}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Quotient Property.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{{y}^{6}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146893[\/ohm_question]\r\n\r\n[ohm_question]146241[\/ohm_question]\r\n\r\n<\/div>\r\nFor more similar examples, watch the following video.\r\n\r\nhttps:\/\/youtu.be\/Mqx8AXl75UY\r\n\r\nTo conclude this section, we will simplify quotient expressions with a negative exponent.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\Large\\frac{{r}^{5}}{{r}^{-4}}}[\/latex].\r\n[reveal-answer q=\"556096\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"556096\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467267504\" class=\"unnumbered unstyled\" summary=\"r to the 5th over r to the negative 4 is shown. The first step says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\Large\\frac{r^5}{r^{-4}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Quotient Property, [latex]{\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n}[\/latex] .<\/td>\r\n<td>[latex]{r}^{5-(\\color{red}{-4})}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Be careful to subtract [latex]5-(\\color{red}{-4})[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]r^9[\/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]146308[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the next video we share more examples of simplifying a quotient with negative exponents.\r\n\r\nhttps:\/\/youtu.be\/J5MrZbpaAGc","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify complex expressions using a combination of exponent rules<\/li>\n<li>Simplify quotients that require a combination of the properties of exponents<\/li>\n<\/ul>\n<\/div>\n<p>All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference as we will be using them here to simplify various exponential expressions.<\/p>\n<div class=\"textbox shaded\">\n<h3>Summary of Exponent Properties<\/h3>\n<p>If [latex]a,b[\/latex] are real numbers and [latex]m,n[\/latex] are integers, then<\/p>\n<p>[latex]\\begin{array}{cccc}\\mathbf{\\text{Product Property}}\\hfill & & & {a}^{m}\\cdot {a}^{n}={a}^{m+n}\\hfill \\\\ \\mathbf{\\text{Power Property}}\\hfill & & & {\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}\\hfill \\\\ \\mathbf{\\text{Product to a Power Property}}\\hfill & & & {\\left(ab\\right)}^{m}={a}^{m}{b}^{m}\\hfill \\\\ \\mathbf{\\text{Quotient Property}}\\hfill & & & {\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},a\\ne 0\\hfill \\\\ \\mathbf{\\text{Zero Exponent Property}}\\hfill & & & {a}^{0}=1,a\\ne 0\\hfill \\\\ \\mathbf{\\text{Quotient to a Power Property}}\\hfill & & & {\\left({\\Large\\frac{a}{b}}\\right)}^{m}={\\Large\\frac{{a}^{m}}{{b}^{m}}},b\\ne 0\\hfill \\\\ \\mathbf{\\text{Definition of Negative Exponent}}\\hfill & & & {a}^{-n}={\\Large\\frac{1}{{a}^{n}}}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<h3>Expressions with negative exponents<\/h3>\n<p>The following examples involve simplifying expressions with negative exponents.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{x}^{-4}\\cdot {x}^{6}[\/latex]<br \/>\n2. [latex]{y}^{-6}\\cdot {y}^{4}[\/latex]<br \/>\n3. [latex]{z}^{-5}\\cdot {z}^{-3}[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168469494417\" 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]{x}^{-4}\\cdot {x}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Product Property, [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex].<\/td>\n<td>[latex]{x}^{-4+6}[\/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-id1168467250235\" 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}^{-6}\\cdot {y}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The bases are the same, so add the exponents.<\/td>\n<td>[latex]{y}^{-6+4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{y}^{-2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of a negative exponent, [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/td>\n<td>[latex]{\\Large\\frac{1}{{y}^{2}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468607449\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{z}^{-5}\\cdot {z}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The bases are the same, so add the exponents.<\/td>\n<td>[latex]{z}^{-5 - 3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{z}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of a negative exponent, [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/td>\n<td>[latex]{\\Large\\frac{1}{{z}^{8}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146301\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146301&theme=oea&iframe_resize_id=ohm146301&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next two examples, we\u2019ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\left({m}^{4}{n}^{-3}\\right)\\left({m}^{-5}{n}^{-2}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q198141\">Show Solution<\/span><\/p>\n<div id=\"q198141\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468721741\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left({m}^{4}{n}^{-3}\\right)\\left({m}^{-5}{n}^{-2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Commutative Property to get like bases together.<\/td>\n<td>[latex]{m}^{4}{m}^{-5}\\cdot {n}^{-2}{n}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the exponents for each base.<\/td>\n<td>[latex]{m}^{-1}\\cdot {n}^{-5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take reciprocals and change the signs of the exponents.<\/td>\n<td>[latex]{\\Large\\frac{1}{{m}^{1}}\\cdot \\frac{1}{{n}^{5}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\Large\\frac{1}{m{n}^{5}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146303\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146303&theme=oea&iframe_resize_id=ohm146303&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>If we multipy two expressions with numerical coefficients, we multiply the coefficients together.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]\\left(2{x}^{-6}{y}^{8}\\right)\\left(-5{x}^{5}{y}^{-3}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q989732\">Show Solution<\/span><\/p>\n<div id=\"q989732\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466697901\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(2{x}^{-6}{y}^{8}\\right)\\left(-5{x}^{5}{y}^{-3}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite with the like bases together.<\/td>\n<td>[latex]2\\left(-5\\right)\\cdot \\left({x}^{-6}{x}^{5}\\right)\\cdot \\left({y}^{8}{y}^{-3}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-10\\cdot {x}^{-1}\\cdot {y}^{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of a negative exponent, [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/td>\n<td>[latex]-10\\cdot {\\Large\\frac{1}{{x}^{1}}}\\cdot {y}^{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\Large\\frac{-10{y}^{5}}{x}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146304\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146304&theme=oea&iframe_resize_id=ohm146304&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next two examples, we\u2019ll use the Power Property and the Product to a Power Property to simplify expressions with negative exponents.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{\\left({k}^{3}\\right)}^{-2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q769374\">Show Solution<\/span><\/p>\n<div id=\"q769374\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168464917598\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left({k}^{3}\\right)}^{-2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Product to a Power Property, [latex]{\\left(ab\\right)}^{m}={a}^{m}{b}^{m}[\/latex].<\/td>\n<td>[latex]{k}^{3\\left(-2\\right)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{k}^{-6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite with a positive exponent.<\/td>\n<td>[latex]{\\Large\\frac{1}{{k}^{6}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146306\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146306&theme=oea&iframe_resize_id=ohm146306&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: [latex]{\\left(5{x}^{-3}\\right)}^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q40374\">Show Solution<\/span><\/p>\n<div id=\"q40374\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466013404\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(5{x}^{-3}\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Product to a Power Property, [latex]{\\left(ab\\right)}^{m}={a}^{m}{b}^{m}[\/latex].<\/td>\n<td>[latex]{5}^{2}{\\left({x}^{-3}\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify [latex]{5}^{2}[\/latex] and multiply the exponents of [latex]x[\/latex] using the<\/p>\n<p>Power Property, [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/td>\n<td>[latex]25{x}^{-6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite [latex]{x}^{-6}[\/latex] by using the definition of a negative<\/p>\n<p>exponent, [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/td>\n<td>[latex]25\\cdot {\\Large\\frac{1}{{x}^{6}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify<\/td>\n<td>[latex]{\\Large\\frac{25}{{x}^{6}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146307\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146307&theme=oea&iframe_resize_id=ohm146307&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show another example of how to simplify a product that contains negative exponents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify A Product of Expressions with Neg Exponents (2 Methods)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/J9A-JlTXnsQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The following examples involve solving exponential expressions with quotients.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{\\Large\\frac{{\\left({x}^{2}\\right)}^{3}}{{x}^{5}}}[\/latex].<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468505651\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\Large\\frac{{\\left({x}^{2}\\right)}^{3}}{{x}^{5}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the exponents in the numerator, using the<\/p>\n<p>Power Property.<\/td>\n<td>[latex]{\\Large\\frac{{x}^{6}}{{x}^{5}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract the exponents.<\/td>\n<td>[latex]x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146230\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146230&theme=oea&iframe_resize_id=ohm146230&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: [latex]{\\Large\\frac{{m}^{8}}{{\\left({m}^{2}\\right)}^{4}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q680453\">Show Solution<\/span><\/p>\n<div id=\"q680453\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468260532\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\Large\\frac{{m}^{8}}{{\\left({m}^{2}\\right)}^{4}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the exponents in the numerator, using the<\/p>\n<p>Power Property.<\/td>\n<td>[latex]{\\Large\\frac{{m}^{8}}{{m}^{8}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract the exponents.<\/td>\n<td>[latex]{m}^{0}=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146231\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146231&theme=oea&iframe_resize_id=ohm146231&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: [latex]{\\left({\\Large\\frac{{x}^{7}}{{x}^{3}}}\\right)}^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q903996\">Show Solution<\/span><\/p>\n<div id=\"q903996\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468297399\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(\\frac{{x}^{7}}{{x}^{3}}\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remember parentheses come before exponents, and the<\/p>\n<p>bases are the same so we can simplify inside the<\/p>\n<p>parentheses. Subtract the exponents.<\/td>\n<td>[latex]{\\left({x}^{7 - 3}\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\left({x}^{4}\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the exponents.<\/td>\n<td>[latex]{x}^{8}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146233\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146233&theme=oea&iframe_resize_id=ohm146233&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: [latex]{\\left({\\Large\\frac{{p}^{2}}{{q}^{5}}}\\right)}^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q867763\">Show Solution<\/span><\/p>\n<div id=\"q867763\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nHere we cannot simplify inside the parentheses first, since the bases are not the same.<\/p>\n<table id=\"eip-id1168467258528\" class=\"unnumbered unstyled\" style=\"width: 859px\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 477.117px\"><\/td>\n<td style=\"width: 357.883px\">[latex]{\\Large{\\left(\\frac{{p}^{2}}{{q}^{5}}\\right)}}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 477.117px\">Raise the numerator and denominator to the third power\u00a0using the Quotient to a Power Property, [latex]{\\Large{\\left(\\frac{a}{b}\\right)}}^{m}={\\Large\\frac{{a}^{m}}{{b}^{m}}}[\/latex]<\/td>\n<td style=\"width: 357.883px\">[latex]{\\Large\\frac{(p^2)^{3}}{(q^5)^{3}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 477.117px\">Use the Power Property, [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex].<\/td>\n<td style=\"width: 357.883px\">[latex]{\\Large\\frac{{p}^{6}}{{q}^{15}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146234\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146234&theme=oea&iframe_resize_id=ohm146234&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: [latex]{\\Large{\\left(\\frac{2{x}^{3}}{3y}\\right)}}^{4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q521774\">Show Solution<\/span><\/p>\n<div id=\"q521774\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468606012\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\Large{\\left(\\frac{2{x}^{3}}{3y}\\right)}}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Raise the numerator and denominator to the fourth<\/p>\n<p>power using the Quotient to a Power Property.<\/td>\n<td>[latex]{\\Large\\frac{{\\left(2{x}^{3}\\right)}^{4}}{{\\left(3y\\right)}^{4}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Raise each factor to the fourth power, using the Power<\/p>\n<p>to a Power Property.<\/td>\n<td>[latex]{\\Large\\frac{{2}^{4}{\\left({x}^{3}\\right)}^{4}}{{3}^{4}{y}^{4}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Power Property and simplify.<\/td>\n<td>[latex]{\\Large\\frac{16{x}^{12}}{81{y}^{4}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146235\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146235&theme=oea&iframe_resize_id=ohm146235&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: [latex]{\\Large\\frac{{\\left({y}^{2}\\right)}^{3}{\\left({y}^{2}\\right)}^{4}}{{\\left({y}^{5}\\right)}^{4}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q189952\">Show Solution<\/span><\/p>\n<div id=\"q189952\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466312387\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\Large\\frac{{\\left({y}^{2}\\right)}^{3}{\\left({y}^{2}\\right)}^{4}}{{\\left({y}^{5}\\right)}^{4}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Power Property.<\/td>\n<td>[latex]{\\Large\\frac{\\left({y}^{6}\\right)\\left({y}^{8}\\right)}{{y}^{20}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the exponents in the numerator, using the Product Property.<\/td>\n<td>[latex]{\\Large\\frac{{y}^{14}}{{y}^{20}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Quotient Property.<\/td>\n<td>[latex]{\\Large\\frac{1}{{y}^{6}}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146893\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146893&theme=oea&iframe_resize_id=ohm146893&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146241\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146241&theme=oea&iframe_resize_id=ohm146241&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>For more similar examples, watch the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1:  Simplify Expressions using Exponent Properties (Quotient \/ Power Properties)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/Mqx8AXl75UY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>To conclude this section, we will simplify quotient expressions with a negative exponent.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{\\Large\\frac{{r}^{5}}{{r}^{-4}}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q556096\">Show Solution<\/span><\/p>\n<div id=\"q556096\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467267504\" class=\"unnumbered unstyled\" summary=\"r to the 5th over r to the negative 4 is shown. The first step says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\Large\\frac{r^5}{r^{-4}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Quotient Property, [latex]{\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n}[\/latex] .<\/td>\n<td>[latex]{r}^{5-(\\color{red}{-4})}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Be careful to subtract [latex]5-(\\color{red}{-4})[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]r^9[\/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=\"ohm146308\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146308&theme=oea&iframe_resize_id=ohm146308&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the next video we share more examples of simplifying a quotient with negative exponents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 2:  Simplify Exponential Expressions With Negative Exponents - Basic\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/J5MrZbpaAGc?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-10852\">\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: 146230, 146231, 146233, 146234, 146235, 146893, 146241. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Simplify Expressions using Exponent Properties (Quotient \/ Power Properties). <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Provided by<\/strong>: `. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Mqx8AXl75UY\">https:\/\/youtu.be\/Mqx8AXl75UY<\/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":21046,"menu_order":7,"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 1: Simplify Expressions using Exponent Properties (Quotient \/ Power Properties)\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"`\",\"url\":\"https:\/\/youtu.be\/Mqx8AXl75UY\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID: 146230, 146231, 146233, 146234, 146235, 146893, 146241\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"0180f73e52424109bcb3c78fcfbbbc1f, 326ccb3fb2bd46e587586c8a2784b189, 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