{"id":4426,"date":"2020-04-13T13:05:17","date_gmt":"2020-04-13T13:05:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=4426"},"modified":"2021-02-06T00:05:02","modified_gmt":"2021-02-06T00:05:02","slug":"writing-negative-exponents-as-positive-exponents","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/writing-negative-exponents-as-positive-exponents\/","title":{"raw":"Writing Negative Exponents as Positive Exponents","rendered":"Writing Negative Exponents as Positive Exponents"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify expressions with negative exponents<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe Quotient Property of Exponents has two forms depending on whether the exponent in the numerator or denominator was larger.\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\nWhat if we just subtract exponents, regardless of which is larger? Let\u2019s consider [latex]{\\Large\\frac{{x}^{2}}{{x}^{5}}}[\/latex]\r\nWe subtract the exponent in the denominator from the exponent in the numerator.\r\n<p style=\"text-align: center\">[latex]{\\Large\\frac{{x}^{2}}{{x}^{5}}}[\/latex]\u00a0\u00a0 [latex]=[\/latex] \u00a0 [latex]{x}^{2 - 5}[\/latex]\u00a0\u00a0 [latex]=[\/latex] \u00a0 [latex]{x}^{-3}[\/latex]<\/p>\r\nWe can also simplify [latex]{\\Large\\frac{{x}^{2}}{{x}^{5}}}[\/latex] by dividing out common factors:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224553\/CNX_BMath_Figure_10_05_014_img.png\" alt=\"A fraction is shown. The numerator is x times x, the denominator is x times x times x times x times x. Two x's are crossed out in red on the top and on the bottom. Below that, the fraction 1 over x cubed is shown.\" \/>\r\nThis implies that [latex]{x}^{-3}={\\Large\\frac{1}{{x}^{3}}}[\/latex] and it leads us to the definition of a negative exponent.\r\n<div class=\"textbox shaded\">\r\n<h3>Negative Exponent<\/h3>\r\nIf [latex]n[\/latex] is a positive integer and [latex]a\\ne 0[\/latex], then [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].\r\n\r\n<\/div>\r\nThe negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{4}^{-2}[\/latex]\r\n2. [latex]{10}^{-3}[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168469720473\" 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]{4}^{-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}{{4}^{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{16}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467437540\" 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]{10}^{-3}[\/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}{{10}^{3}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{1000}}[\/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]146245[\/ohm_question]\r\n\r\n<\/div>\r\nWhen simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{\\left(-3\\right)}^{-2}[\/latex]\r\n2 [latex]{-3}^{-2}[\/latex]\r\n[reveal-answer q=\"719497\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"719497\"]\r\n\r\nSolution\r\nThe negative in the exponent does not affect the sign of the base.\r\n<table id=\"eip-id1168465147148\" 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 exponent applies to the base, [latex]-3[\/latex] .<\/td>\r\n<td>[latex]{\\left(-3\\right)}^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the reciprocal of the base and change the sign of the exponent.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{{\\left(-3\\right)}^{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{9}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466253794\" 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>The expression [latex]-{3}^{-2}[\/latex] means: find the opposite of [latex]{3}^{-2}[\/latex]\r\n\r\nThe exponent applies only to the base, [latex]3[\/latex].<\/td>\r\n<td>[latex]-{3}^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite as a product with [latex]\u22121[\/latex].<\/td>\r\n<td>[latex]-1\\cdot {3}^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the reciprocal of the base and change the sign of the exponent.<\/td>\r\n<td>[latex]-1\\cdot {\\Large\\frac{1}{{3}^{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/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\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146247[\/ohm_question]\r\n\r\n<\/div>\r\nWe must be careful to follow the order of operations. In the next example, parts 1 and 2 look similar, but we get different results.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]4\\cdot {2}^{-1}[\/latex]\r\n2. [latex]{\\left(4\\cdot 2\\right)}^{-1}[\/latex]\r\n[reveal-answer q=\"617492\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"617492\"]\r\n\r\nSolution\r\nRemember to always follow the order of operations.\r\n<table id=\"eip-id1168468498674\" 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>Do exponents before multiplication.<\/td>\r\n<td>[latex]4\\cdot {2}^{-1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/td>\r\n<td>[latex]4\\cdot {\\Large\\frac{1}{{2}^{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466077342\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td>[latex]{\\left(4\\cdot 2\\right)}^{-1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify inside the parentheses first.<\/td>\r\n<td>[latex]{\\left(8\\right)}^{-1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/td>\r\n<td>[latex]{\\Large\\frac{1}{{8}^{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{8}}[\/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]146298[\/ohm_question]\r\n\r\n<\/div>\r\nWhen a variable is raised to a negative exponent, we apply the definition the same way we did with numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{x}^{-6}[\/latex]\r\n[reveal-answer q=\"205196\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"205196\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469876523\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{x}^{-6}[\/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}{{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]146299[\/ohm_question]\r\n\r\n<\/div>\r\nWhen there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We\u2019ll see how this works in the next example.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]5{y}^{-1}[\/latex]\r\n2. [latex]{\\left(5y\\right)}^{-1}[\/latex]\r\n3. [latex]{\\left(-5y\\right)}^{-1}[\/latex]\r\n[reveal-answer q=\"633196\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"633196\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469497194\" 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>Notice the exponent applies to just the base [latex]y[\/latex] .<\/td>\r\n<td>[latex]5{y}^{-1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the reciprocal of [latex]y[\/latex] and change the sign of the exponent.<\/td>\r\n<td>[latex]5\\cdot {\\Large\\frac{1}{{y}^{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\Large\\frac{5}{y}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469766330\" 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>Here the parentheses make the exponent apply to the base [latex]5y[\/latex] .<\/td>\r\n<td>[latex]{\\left(5y\\right)}^{-1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the reciprocal of [latex]5y[\/latex] and change the sign of the exponent.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{{\\left(5y\\right)}^{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{5y}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469859636\" 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]{\\left(-5y\\right)}^{-1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The base is [latex]-5y[\/latex] . Take the reciprocal of [latex]-5y[\/latex] and change the sign of the exponent.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{{\\left(-5y\\right)}^{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\Large\\frac{1}{-5y}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use [latex]{\\Large\\frac{a}{-b}}=-{\\Large\\frac{a}{b}}[\/latex]<\/td>\r\n<td>[latex]-{\\Large\\frac{1}{5y}}[\/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]146300[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNow that we have defined negative exponents, the Quotient Property of Exponents needs only one form, [latex]{\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n}[\/latex], where [latex]a\\ne 0[\/latex] and <em>m<\/em> and <em>n<\/em> are integers.\r\n\r\nWhen the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify expressions with negative exponents<\/li>\n<\/ul>\n<\/div>\n<p>The Quotient Property of Exponents has two forms depending on whether the exponent in the numerator or denominator was larger.<\/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>What if we just subtract exponents, regardless of which is larger? Let\u2019s consider [latex]{\\Large\\frac{{x}^{2}}{{x}^{5}}}[\/latex]<br \/>\nWe subtract the exponent in the denominator from the exponent in the numerator.<\/p>\n<p style=\"text-align: center\">[latex]{\\Large\\frac{{x}^{2}}{{x}^{5}}}[\/latex]\u00a0\u00a0 [latex]=[\/latex] \u00a0 [latex]{x}^{2 - 5}[\/latex]\u00a0\u00a0 [latex]=[\/latex] \u00a0 [latex]{x}^{-3}[\/latex]<\/p>\n<p>We can also simplify [latex]{\\Large\\frac{{x}^{2}}{{x}^{5}}}[\/latex] by dividing out common factors:<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224553\/CNX_BMath_Figure_10_05_014_img.png\" alt=\"A fraction is shown. The numerator is x times x, the denominator is x times x times x times x times x. Two x's are crossed out in red on the top and on the bottom. Below that, the fraction 1 over x cubed is shown.\" \/><br \/>\nThis implies that [latex]{x}^{-3}={\\Large\\frac{1}{{x}^{3}}}[\/latex] and it leads us to the definition of a negative exponent.<\/p>\n<div class=\"textbox shaded\">\n<h3>Negative Exponent<\/h3>\n<p>If [latex]n[\/latex] is a positive integer and [latex]a\\ne 0[\/latex], then [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/p>\n<\/div>\n<p>The negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{4}^{-2}[\/latex]<br \/>\n2. [latex]{10}^{-3}[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168469720473\" 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]{4}^{-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}{{4}^{2}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\Large\\frac{1}{16}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467437540\" 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]{10}^{-3}[\/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}{{10}^{3}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\Large\\frac{1}{1000}}[\/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=\"ohm146245\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146245&theme=oea&iframe_resize_id=ohm146245&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{\\left(-3\\right)}^{-2}[\/latex]<br \/>\n2 [latex]{-3}^{-2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q719497\">Show Solution<\/span><\/p>\n<div id=\"q719497\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nThe negative in the exponent does not affect the sign of the base.<\/p>\n<table id=\"eip-id1168465147148\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>The exponent applies to the base, [latex]-3[\/latex] .<\/td>\n<td>[latex]{\\left(-3\\right)}^{-2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take the reciprocal of the base and change the sign of the exponent.<\/td>\n<td>[latex]{\\Large\\frac{1}{{\\left(-3\\right)}^{2}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\Large\\frac{1}{9}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466253794\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>The expression [latex]-{3}^{-2}[\/latex] means: find the opposite of [latex]{3}^{-2}[\/latex]<\/p>\n<p>The exponent applies only to the base, [latex]3[\/latex].<\/td>\n<td>[latex]-{3}^{-2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite as a product with [latex]\u22121[\/latex].<\/td>\n<td>[latex]-1\\cdot {3}^{-2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take the reciprocal of the base and change the sign of the exponent.<\/td>\n<td>[latex]-1\\cdot {\\Large\\frac{1}{{3}^{2}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-{\\Large\\frac{1}{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=\"ohm146247\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146247&theme=oea&iframe_resize_id=ohm146247&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>We must be careful to follow the order of operations. In the next example, parts 1 and 2 look similar, but we get different results.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]4\\cdot {2}^{-1}[\/latex]<br \/>\n2. [latex]{\\left(4\\cdot 2\\right)}^{-1}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q617492\">Show Solution<\/span><\/p>\n<div id=\"q617492\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nRemember to always follow the order of operations.<\/p>\n<table id=\"eip-id1168468498674\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Do exponents before multiplication.<\/td>\n<td>[latex]4\\cdot {2}^{-1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/td>\n<td>[latex]4\\cdot {\\Large\\frac{1}{{2}^{1}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466077342\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td>[latex]{\\left(4\\cdot 2\\right)}^{-1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify inside the parentheses first.<\/td>\n<td>[latex]{\\left(8\\right)}^{-1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/td>\n<td>[latex]{\\Large\\frac{1}{{8}^{1}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\Large\\frac{1}{8}}[\/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=\"ohm146298\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146298&theme=oea&iframe_resize_id=ohm146298&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{x}^{-6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q205196\">Show Solution<\/span><\/p>\n<div id=\"q205196\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469876523\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{x}^{-6}[\/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}{{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=\"ohm146299\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146299&theme=oea&iframe_resize_id=ohm146299&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We\u2019ll see how this works in the next example.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]5{y}^{-1}[\/latex]<br \/>\n2. [latex]{\\left(5y\\right)}^{-1}[\/latex]<br \/>\n3. [latex]{\\left(-5y\\right)}^{-1}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q633196\">Show Solution<\/span><\/p>\n<div id=\"q633196\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469497194\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Notice the exponent applies to just the base [latex]y[\/latex] .<\/td>\n<td>[latex]5{y}^{-1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take the reciprocal of [latex]y[\/latex] and change the sign of the exponent.<\/td>\n<td>[latex]5\\cdot {\\Large\\frac{1}{{y}^{1}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\Large\\frac{5}{y}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469766330\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Here the parentheses make the exponent apply to the base [latex]5y[\/latex] .<\/td>\n<td>[latex]{\\left(5y\\right)}^{-1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take the reciprocal of [latex]5y[\/latex] and change the sign of the exponent.<\/td>\n<td>[latex]{\\Large\\frac{1}{{\\left(5y\\right)}^{1}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\Large\\frac{1}{5y}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469859636\" 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]{\\left(-5y\\right)}^{-1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The base is [latex]-5y[\/latex] . Take the reciprocal of [latex]-5y[\/latex] and change the sign of the exponent.<\/td>\n<td>[latex]{\\Large\\frac{1}{{\\left(-5y\\right)}^{1}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\Large\\frac{1}{-5y}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use [latex]{\\Large\\frac{a}{-b}}=-{\\Large\\frac{a}{b}}[\/latex]<\/td>\n<td>[latex]-{\\Large\\frac{1}{5y}}[\/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=\"ohm146300\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146300&theme=oea&iframe_resize_id=ohm146300&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, [latex]{\\Large\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n}[\/latex], where [latex]a\\ne 0[\/latex] and <em>m<\/em> and <em>n<\/em> are integers.<\/p>\n<p>When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].<\/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-4426\">\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: 146245, 146247, 146298, 146299, 146300. <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, 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":6,"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: 146245, 146247, 146298, 146299, 146300\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4426","chapter","type-chapter","status-web-only","hentry"],"part":4412,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4426","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\/4426\/revisions"}],"predecessor-version":[{"id":4427,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4426\/revisions\/4427"}],"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\/4426\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=4426"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=4426"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=4426"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=4426"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}