{"id":4432,"date":"2020-04-13T13:14:15","date_gmt":"2020-04-13T13:14:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=4432"},"modified":"2021-02-06T00:05:03","modified_gmt":"2021-02-06T00:05:03","slug":"simplifying-expressions-with-negative-exponents","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/simplifying-expressions-with-negative-exponents\/","title":{"raw":"Simplifying Expressions With Negative Exponents","rendered":"Simplifying Expressions With Negative Exponents"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the properties of exponents to simplify products and quotients that contain negative exponents and variables<\/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.\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&nbsp;\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&nbsp;\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&nbsp;\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 the monomials have numerical coefficients, we multiply the coefficients, just as we did in Integer Exponents and Scientific Notation.\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&nbsp;\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.\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&nbsp;\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&nbsp;\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\nTo simplify a fraction, we use the Quotient Property.\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\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>Use the properties of exponents to simplify products and quotients that contain negative exponents and variables<\/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.<\/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<p>&nbsp;<\/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<p>&nbsp;<\/p>\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<p>&nbsp;<\/p>\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 the monomials have numerical coefficients, we multiply the coefficients, just as we did in Integer Exponents and Scientific Notation.<\/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<p>&nbsp;<\/p>\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.<\/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<p>&nbsp;<\/p>\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<p>&nbsp;<\/p>\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>To simplify a fraction, we use the Quotient Property.<\/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>In the next video we share more examples of simplifying a quotient with negative exponents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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-4432\">\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 146301, 146303, 146304, 146306, 146307, 146308. <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><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Simplify A Product of Expressions with Neg Exponents (2 Methods). <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/J9A-JlTXnsQ\">https:\/\/youtu.be\/J9A-JlTXnsQ<\/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 2: Simplify Exponential Expressions With Negative Exponents - Basic. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/J5MrZbpaAGc\">https:\/\/youtu.be\/J5MrZbpaAGc<\/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":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\":\"original\",\"description\":\"Question ID 146301, 146303, 146304, 146306, 146307, 146308\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Simplify A Product of Expressions with Neg Exponents (2 Methods)\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/J9A-JlTXnsQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 2: Simplify Exponential Expressions With Negative Exponents - Basic\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/J5MrZbpaAGc\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4432","chapter","type-chapter","status-web-only","hentry"],"part":4412,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4432","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4432\/revisions"}],"predecessor-version":[{"id":4433,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4432\/revisions\/4433"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/4412"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4432\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=4432"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=4432"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=4432"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=4432"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}