{"id":10868,"date":"2017-06-05T21:37:39","date_gmt":"2017-06-05T21:37:39","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10868"},"modified":"2024-04-30T21:32:15","modified_gmt":"2024-04-30T21:32:15","slug":"writing-negative-exponents-as-positive-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/writing-negative-exponents-as-positive-exponents\/","title":{"raw":"Simplifying Expressions with Negative Exponents and Exponents of 0 and 1","rendered":"Simplifying Expressions with Negative Exponents and Exponents of 0 and 1"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify exponential expressions containing negative exponents<\/li>\r\n \t<li>Simplify exponential expressions containing exponents of 0 and 1<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe previously saw that the 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\n<h2 id=\"title2\">Define and use the negative exponent rule<\/h2>\r\nWe now propose another question about exponents.\u00a0 Given a quotient like\u00a0[latex] \\displaystyle \\frac{{{2}^{m}}}{{{2}^{n}}}[\/latex] what happens when <em>n<\/em> is larger than <em>m<\/em>? We will need to use the <em>negative rule of exponents<\/em> to simplify the expression so that it is easier to understand.\r\n\r\nLet's look at an example to clarify this idea. Given the expression:\r\n<p style=\"text-align: center;\">[latex]\\frac{{h}^{3}}{{h}^{5}}[\/latex]<\/p>\r\nExpand the numerator and denominator, all the terms in the numerator will cancel to [latex]1[\/latex], leaving two <em>h<\/em>s multiplied in the denominator, and a numerator of [latex]1[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l} \\frac{{h}^{3}}{{h}^{5}}\\,\\,\\,=\\,\\,\\,\\frac{h\\cdot{h}\\cdot{h}}{h\\cdot{h}\\cdot{h}\\cdot{h}\\cdot{h}} \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\frac{\\cancel{h}\\cdot \\cancel{h}\\cdot \\cancel{h}}{\\cancel{h}\\cdot \\cancel{h}\\cdot \\cancel{h}\\cdot {h}\\cdot {h}}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\frac{1}{h\\cdot{h}}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\frac{1}{{h}^{2}} \\end{array}[\/latex]<\/div>\r\nWe could have also applied the quotient rule from the last section, to obtain the following result:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{h^{3}}{h^{5}}\\,\\,\\,=\\,\\,\\,h^{3-5}\\\\\\\\=\\,\\,\\,h^{-2}\\,\\,\\end{array}[\/latex]<\/p>\r\nPutting the answers together, we have [latex]{h}^{-2}=\\frac{1}{{h}^{2}}[\/latex]. This is true when <em>h<\/em>, or any variable, is a real number and is not zero.\r\n<div class=\"textbox shaded\">\r\n<h3>The Negative Rule of Exponents<\/h3>\r\nFor any nonzero real number [latex]a[\/latex] and natural number [latex]n[\/latex], the negative rule of exponents states that\r\n<div style=\"text-align: center;\">[latex]{a}^{-n}=\\frac{1}{{a}^{n}}[\/latex]<\/div>\r\n<\/div>\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, 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. We rewrite it by using the definition of negative exponents, [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].\u00a0 Any expression that has negative exponents is not considered to be in simplest form, so 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=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\n<p style=\"text-align: left;\">Evaluate the expression [latex]{4}^{-3}[\/latex].<\/p>\r\n<p style=\"text-align: left;\">[reveal-answer q=\"231258\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"231258\"]First, write the expression with positive exponents by putting the term with the negative exponent in the denominator.<\/p>\r\n<p style=\"text-align: center;\">[latex]{4}^{-3} = \\frac{1}{{4}^{3}} = \\frac{1}{4\\cdot4\\cdot4}[\/latex]<\/p>\r\nNow that we have an expression that looks somewhat familiar.\r\n<p style=\"text-align: center;\">[latex]\\frac{1}{4\\cdot4\\cdot4} = \\frac{1}{64}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{1}{64}[\/latex]\r\n\r\n[\/hidden-answer]\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]{4}^{-2}[\/latex]\r\n2. [latex]{x}^{-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]{x}^{-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}{{x}^{3}}}[\/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\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 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 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\nLet's looks at some examples of how this rule applies under different circumstances.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWrite [latex]\\frac{{\\left({t}^{3}\\right)}}{{\\left({t}^{8}\\right)}}[\/latex] with positive exponents.\r\n<p style=\"text-align: left;\">[reveal-answer q=\"219981\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"219981\"]<\/p>\r\n<p style=\"text-align: left;\">Use the quotient rule to subtract the exponents of terms with like bases.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{{\\left({t}^{3}\\right)}}{{\\left({t}^{8}\\right)}}={t}^{3-8}\\\\={t}^{-5}\\,\\,\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Write the expression with positive exponents by putting the term with the negative exponent in the denominator.<\/p>\r\n<p style=\"text-align: center;\">[latex]=\\frac{1}{{t}^{5}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{1}{{t}^{5}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]{\\left(\\frac{1}{3}\\right)}^{-2}[\/latex].\r\n\r\n[reveal-answer q=\"998337\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"998337\"]Apply the power property of exponents.\r\n<p style=\"text-align: center;\">[latex]\\frac{{1}^{-2}}{{3}^{-2}}[\/latex]<\/p>\r\nWrite each term with a positive exponent, the numerator will go to the denominator and the denominator will go to the numerator.\r\n<p style=\"text-align: center;\">[latex]\\frac{{3}^{2}}{{1}^{2}}{ = }\\frac{{3}\\cdot{3}}{{1}\\cdot{1}}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\frac{{3}\\cdot{3}}{{1}\\cdot{1}}{ = }\\frac{9}{1}{ = }{9}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]9[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify.[latex]\\frac{1}{4^{-2}}[\/latex] Write your answer using positive exponents.\r\n\r\n[reveal-answer q=\"629171\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"629171\"]\r\n\r\nWrite each term with a positive exponent, the denominator will go to the numerator.\r\n<p style=\"text-align: center;\">[latex]\\frac{1}{4^{-2}}=1\\cdot\\frac{4^{2}}{1}=\\frac{16}{1}=16[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]16[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you will see examples of simplifying expressions with negative exponents.\r\n\r\nhttps:\/\/youtu.be\/WvFlHjlIITg\r\n<h2>Simplify Expressions with Zero Exponents<\/h2>\r\nA special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like [latex]\\Large\\frac{{a}^{m}}{{a}^{m}}[\/latex]. From earlier work with fractions, we know that\r\n<p style=\"text-align: center;\">[latex]\\Large\\frac{2}{2}\\normalsize =\\Large\\frac{17}{17}\\normalsize =\\Large\\frac{-43}{-43}\\normalsize =1[\/latex]<\/p>\r\nIn words, a number divided by itself is [latex]1[\/latex]. So [latex]\\Large\\frac{x}{x}\\normalsize =1[\/latex], for any [latex]x[\/latex] ( [latex]x\\ne 0[\/latex] ), since any number divided by itself is [latex]1[\/latex].\r\n\r\nThe Quotient Property of Exponents shows us how to simplify [latex]\\Large\\frac{{a}^{m}}{{a}^{n}}[\/latex] by subtracting exponents. What if [latex]m=n[\/latex] ?\r\n\r\nNow we will simplify [latex]\\Large\\frac{{a}^{m}}{{a}^{m}}[\/latex] in two ways to lead us to the definition of the zero exponent.\r\nConsider first [latex]\\Large\\frac{8}{8}[\/latex], which we know is [latex]1[\/latex].\r\n<table id=\"eip-id1168469654144\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 11th over x to the 7th. Beside that is written \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large\\frac{8}{8}\\normalsize =1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]8[\/latex] as [latex]{2}^{3}[\/latex] .<\/td>\r\n<td>[latex]\\Large\\frac{{2}^{3}}{{2}^{3}}\\normalsize =1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract exponents.<\/td>\r\n<td>[latex]{2}^{3 - 3}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{2}^{0}=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224544\/CNX_BMath_Figure_10_04_019_img.png\" alt=\"The fraction \u201ca\u201d to the \u201cm\u201d power over \u201ca\u201d to the \u201cm\u201d power is equal to \u201ca\u201d to the \u201cm\u201d minus \u201cm\u201d power which is equal to \u201ca\u201d to the zero power. In a fraction with \u201ca\u201d written \u201cm\u201d number of times multiplied by itself over the same in the denominator equals 1.\" width=\"285\" height=\"213\" \/>\r\nWe see [latex]\\Large\\frac{{a}^{m}}{{a}^{m}}[\/latex] simplifies to a [latex]{a}^{0}[\/latex] and to [latex]1[\/latex] . So [latex]{a}^{0}=1[\/latex].\u00a0 At this time, we will define both the exponent of 0 and the exponent of 1.\r\n<div class=\"textbox shaded\">\r\n<h3>Exponents of 0 or 1<\/h3>\r\nAny number or variable raised to a power of [latex]1[\/latex] is the number itself.\r\n<p style=\"text-align: center;\">[latex]n^{1}=n[\/latex]<\/p>\r\nAny non-zero number or variable raised to a power of [latex]0[\/latex] is equal to [latex]1[\/latex]\r\n<p style=\"text-align: center;\">[latex]n^{0}=1[\/latex]<\/p>\r\nThe quantity [latex]0^{0}[\/latex]\u00a0is undefined.\r\n\r\n<\/div>\r\nIn this text, we assume any variable that we raise to the zero power is not zero.\u00a0\u00a0The sole exception is the expression [latex]{0}^{0}[\/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined, or DNE (Does Not Exist).\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{12}^{0}[\/latex]\r\n2. [latex]{y}^{0}[\/latex]\r\n[reveal-answer q=\"363472\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"363472\"]\r\n\r\nSolution\r\nThe definition says any non-zero number raised to the zero power is [latex]1[\/latex].\r\n<table id=\"eip-id1168468469984\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{12}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of the zero exponent.<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466072251\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{y}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of the zero exponent.<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146221[\/ohm_question]\r\n\r\n[ohm_question]146890[\/ohm_question]\r\n\r\n<\/div>\r\nNow that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.\r\nWhat about raising an expression to the zero power? Let's look at [latex]{\\left(2x\\right)}^{0}[\/latex]. We can use the product to a power rule to rewrite this expression.\r\n<table id=\"eip-id1168469871755\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]{\\left(2x\\right)}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Use the Product to a Power Rule.<\/td>\r\n<td style=\"height: 15px;\">[latex]{2}^{0}{x}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Use the Zero Exponent Property.<\/td>\r\n<td style=\"height: 15px;\">[latex]1\\cdot 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.53125px;\">\r\n<td style=\"height: 15.53125px;\">Simplify.<\/td>\r\n<td style=\"height: 15.53125px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis tells us that any non-zero expression raised to the zero power is one.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left(7z\\right)}^{0}[\/latex].\r\n[reveal-answer q=\"685861\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"685861\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467118106\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(7z\\right)}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of the zero exponent.<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146222[\/ohm_question]\r\n\r\n<\/div>\r\nNow let's compare the difference between the previous example, where the entire expression was raised to a zero exponent, and what happens when only one factor is raised to a zero exponent.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{\\left(-3{x}^{2}y\\right)}^{0}[\/latex]\r\n2. [latex]-3{x}^{2}{y}^{0}[\/latex]\r\n[reveal-answer q=\"80948\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"80948\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468244600\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product is raised to the zero power.<\/td>\r\n<td>[latex]{\\left(-3{x}^{2}y\\right)}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of the zero exponent.<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168047201113\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that only the variable [latex]y[\/latex] is being raised to the zero power.<\/td>\r\n<td>[latex]{-3{x}^{2}y}^{0}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the definition of the zero exponent.<\/td>\r\n<td>[latex]-3{x}^{2}\\cdot 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-3{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try a similar problem to make sure you see the difference between raising an entire expression to a zero power and having only one factor raised to a zero power.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146223[\/ohm_question]\r\n\r\n[ohm_question]146222[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next video we show some different examples of how you can apply the zero exponent rule.\r\n\r\nhttps:\/\/youtu.be\/zQJy1aBm1dQ\r\n<div style=\"text-align: center;\"><\/div>\r\nAs done previously, to evaluate expressions containing exponents of [latex]0[\/latex] or [latex]1[\/latex], substitute the value of the variable into the expression and simplify.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex]2x^{0}[\/latex] if [latex]x=9[\/latex]\r\n\r\n[reveal-answer q=\"324798\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"324798\"]Substitute 9 for the variable <i>x<\/i>.\r\n<p style=\"text-align: center;\">[latex]2\\cdot9^{0}[\/latex]<\/p>\r\nEvaluate [latex]9^{0}[\/latex]. Multiply.\r\n<p style=\"text-align: center;\">[latex]2\\cdot1=2[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]2x^{0}=2[\/latex], if [latex]x=9[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify\u00a0[latex]\\frac{{c}^{3}}{{c}^{3}}[\/latex].\r\n\r\n[reveal-answer q=\"769979\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"769979\"]Use the quotient and zero exponent rules to simplify the\u00a0expression.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}\\text{ }\\frac{c^{3}}{c^{3}} \\,\\,\\,= \\,\\,\\,c^{3-3} \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,c^{0} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,1\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video there is an example of evaluating an expression with an exponent of zero, as well as simplifying when you get a result of a zero exponent.\r\n\r\nhttps:\/\/youtu.be\/jKihp_DVDa0\r\n\r\n&nbsp;\r\n\r\nhttps:\/\/youtu.be\/J5MrZbpaAGc\r\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify exponential expressions containing negative exponents<\/li>\n<li>Simplify exponential expressions containing exponents of 0 and 1<\/li>\n<\/ul>\n<\/div>\n<p>We previously saw that 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<h2 id=\"title2\">Define and use the negative exponent rule<\/h2>\n<p>We now propose another question about exponents.\u00a0 Given a quotient like\u00a0[latex]\\displaystyle \\frac{{{2}^{m}}}{{{2}^{n}}}[\/latex] what happens when <em>n<\/em> is larger than <em>m<\/em>? We will need to use the <em>negative rule of exponents<\/em> to simplify the expression so that it is easier to understand.<\/p>\n<p>Let&#8217;s look at an example to clarify this idea. Given the expression:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{h}^{3}}{{h}^{5}}[\/latex]<\/p>\n<p>Expand the numerator and denominator, all the terms in the numerator will cancel to [latex]1[\/latex], leaving two <em>h<\/em>s multiplied in the denominator, and a numerator of [latex]1[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l} \\frac{{h}^{3}}{{h}^{5}}\\,\\,\\,=\\,\\,\\,\\frac{h\\cdot{h}\\cdot{h}}{h\\cdot{h}\\cdot{h}\\cdot{h}\\cdot{h}} \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\frac{\\cancel{h}\\cdot \\cancel{h}\\cdot \\cancel{h}}{\\cancel{h}\\cdot \\cancel{h}\\cdot \\cancel{h}\\cdot {h}\\cdot {h}}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\frac{1}{h\\cdot{h}}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\frac{1}{{h}^{2}} \\end{array}[\/latex]<\/div>\n<p>We could have also applied the quotient rule from the last section, to obtain the following result:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{h^{3}}{h^{5}}\\,\\,\\,=\\,\\,\\,h^{3-5}\\\\\\\\=\\,\\,\\,h^{-2}\\,\\,\\end{array}[\/latex]<\/p>\n<p>Putting the answers together, we have [latex]{h}^{-2}=\\frac{1}{{h}^{2}}[\/latex]. This is true when <em>h<\/em>, or any variable, is a real number and is not zero.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Negative Rule of Exponents<\/h3>\n<p>For any nonzero real number [latex]a[\/latex] and natural number [latex]n[\/latex], the negative rule of exponents states that<\/p>\n<div style=\"text-align: center;\">[latex]{a}^{-n}=\\frac{1}{{a}^{n}}[\/latex]<\/div>\n<\/div>\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, 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. We rewrite it by using the definition of negative exponents, [latex]{a}^{-n}={\\Large\\frac{1}{{a}^{n}}}[\/latex].\u00a0 Any expression that has negative exponents is not considered to be in simplest form, so 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=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p style=\"text-align: left;\">Evaluate the expression [latex]{4}^{-3}[\/latex].<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q231258\">Show Solution<\/span><\/p>\n<div id=\"q231258\" class=\"hidden-answer\" style=\"display: none\">First, write the expression with positive exponents by putting the term with the negative exponent in the denominator.<\/p>\n<p style=\"text-align: center;\">[latex]{4}^{-3} = \\frac{1}{{4}^{3}} = \\frac{1}{4\\cdot4\\cdot4}[\/latex]<\/p>\n<p>Now that we have an expression that looks somewhat familiar.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{4\\cdot4\\cdot4} = \\frac{1}{64}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{1}{64}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{4}^{-2}[\/latex]<br \/>\n2. [latex]{x}^{-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]{x}^{-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}{{x}^{3}}}[\/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<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 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 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>Let&#8217;s looks at some examples of how this rule applies under different circumstances.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Write [latex]\\frac{{\\left({t}^{3}\\right)}}{{\\left({t}^{8}\\right)}}[\/latex] with positive exponents.<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q219981\">Show Solution<\/span><\/p>\n<div id=\"q219981\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">Use the quotient rule to subtract the exponents of terms with like bases.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\frac{{\\left({t}^{3}\\right)}}{{\\left({t}^{8}\\right)}}={t}^{3-8}\\\\={t}^{-5}\\,\\,\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Write the expression with positive exponents by putting the term with the negative exponent in the denominator.<\/p>\n<p style=\"text-align: center;\">[latex]=\\frac{1}{{t}^{5}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{1}{{t}^{5}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]{\\left(\\frac{1}{3}\\right)}^{-2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q998337\">Show Solution<\/span><\/p>\n<div id=\"q998337\" class=\"hidden-answer\" style=\"display: none\">Apply the power property of exponents.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{1}^{-2}}{{3}^{-2}}[\/latex]<\/p>\n<p>Write each term with a positive exponent, the numerator will go to the denominator and the denominator will go to the numerator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{3}^{2}}{{1}^{2}}{ = }\\frac{{3}\\cdot{3}}{{1}\\cdot{1}}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{3}\\cdot{3}}{{1}\\cdot{1}}{ = }\\frac{9}{1}{ = }{9}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]9[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify.[latex]\\frac{1}{4^{-2}}[\/latex] Write your answer using positive exponents.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q629171\">Show Solution<\/span><\/p>\n<div id=\"q629171\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write each term with a positive exponent, the denominator will go to the numerator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{4^{-2}}=1\\cdot\\frac{4^{2}}{1}=\\frac{16}{1}=16[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]16[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you will see examples of simplifying expressions with negative exponents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Negative Exponents - Basics\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/WvFlHjlIITg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Simplify Expressions with Zero Exponents<\/h2>\n<p>A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like [latex]\\Large\\frac{{a}^{m}}{{a}^{m}}[\/latex]. From earlier work with fractions, we know that<\/p>\n<p style=\"text-align: center;\">[latex]\\Large\\frac{2}{2}\\normalsize =\\Large\\frac{17}{17}\\normalsize =\\Large\\frac{-43}{-43}\\normalsize =1[\/latex]<\/p>\n<p>In words, a number divided by itself is [latex]1[\/latex]. So [latex]\\Large\\frac{x}{x}\\normalsize =1[\/latex], for any [latex]x[\/latex] ( [latex]x\\ne 0[\/latex] ), since any number divided by itself is [latex]1[\/latex].<\/p>\n<p>The Quotient Property of Exponents shows us how to simplify [latex]\\Large\\frac{{a}^{m}}{{a}^{n}}[\/latex] by subtracting exponents. What if [latex]m=n[\/latex] ?<\/p>\n<p>Now we will simplify [latex]\\Large\\frac{{a}^{m}}{{a}^{m}}[\/latex] in two ways to lead us to the definition of the zero exponent.<br \/>\nConsider first [latex]\\Large\\frac{8}{8}[\/latex], which we know is [latex]1[\/latex].<\/p>\n<table id=\"eip-id1168469654144\" class=\"unnumbered unstyled\" summary=\"The first line shows x to the 11th over x to the 7th. Beside that is written\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{8}{8}\\normalsize =1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write [latex]8[\/latex] as [latex]{2}^{3}[\/latex] .<\/td>\n<td>[latex]\\Large\\frac{{2}^{3}}{{2}^{3}}\\normalsize =1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract exponents.<\/td>\n<td>[latex]{2}^{3 - 3}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{2}^{0}=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224544\/CNX_BMath_Figure_10_04_019_img.png\" alt=\"The fraction \u201ca\u201d to the \u201cm\u201d power over \u201ca\u201d to the \u201cm\u201d power is equal to \u201ca\u201d to the \u201cm\u201d minus \u201cm\u201d power which is equal to \u201ca\u201d to the zero power. In a fraction with \u201ca\u201d written \u201cm\u201d number of times multiplied by itself over the same in the denominator equals 1.\" width=\"285\" height=\"213\" \/><br \/>\nWe see [latex]\\Large\\frac{{a}^{m}}{{a}^{m}}[\/latex] simplifies to a [latex]{a}^{0}[\/latex] and to [latex]1[\/latex] . So [latex]{a}^{0}=1[\/latex].\u00a0 At this time, we will define both the exponent of 0 and the exponent of 1.<\/p>\n<div class=\"textbox shaded\">\n<h3>Exponents of 0 or 1<\/h3>\n<p>Any number or variable raised to a power of [latex]1[\/latex] is the number itself.<\/p>\n<p style=\"text-align: center;\">[latex]n^{1}=n[\/latex]<\/p>\n<p>Any non-zero number or variable raised to a power of [latex]0[\/latex] is equal to [latex]1[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]n^{0}=1[\/latex]<\/p>\n<p>The quantity [latex]0^{0}[\/latex]\u00a0is undefined.<\/p>\n<\/div>\n<p>In this text, we assume any variable that we raise to the zero power is not zero.\u00a0\u00a0The sole exception is the expression [latex]{0}^{0}[\/latex]. This appears later in more advanced courses, but for now, we will consider the value to be undefined, or DNE (Does Not Exist).<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{12}^{0}[\/latex]<br \/>\n2. [latex]{y}^{0}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q363472\">Show Solution<\/span><\/p>\n<div id=\"q363472\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nThe definition says any non-zero number raised to the zero power is [latex]1[\/latex].<\/p>\n<table id=\"eip-id1168468469984\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{12}^{0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of the zero exponent.<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466072251\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{y}^{0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of the zero exponent.<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146221\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146221&theme=oea&iframe_resize_id=ohm146221&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146890\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146890&theme=oea&iframe_resize_id=ohm146890&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.<br \/>\nWhat about raising an expression to the zero power? Let&#8217;s look at [latex]{\\left(2x\\right)}^{0}[\/latex]. We can use the product to a power rule to rewrite this expression.<\/p>\n<table id=\"eip-id1168469871755\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]{\\left(2x\\right)}^{0}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Use the Product to a Power Rule.<\/td>\n<td style=\"height: 15px;\">[latex]{2}^{0}{x}^{0}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Use the Zero Exponent Property.<\/td>\n<td style=\"height: 15px;\">[latex]1\\cdot 1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.53125px;\">\n<td style=\"height: 15.53125px;\">Simplify.<\/td>\n<td style=\"height: 15.53125px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This tells us that any non-zero expression raised to the zero power is one.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{\\left(7z\\right)}^{0}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q685861\">Show Solution<\/span><\/p>\n<div id=\"q685861\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467118106\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(7z\\right)}^{0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of the zero exponent.<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146222\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146222&theme=oea&iframe_resize_id=ohm146222&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Now let&#8217;s compare the difference between the previous example, where the entire expression was raised to a zero exponent, and what happens when only one factor is raised to a zero exponent.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{\\left(-3{x}^{2}y\\right)}^{0}[\/latex]<br \/>\n2. [latex]-3{x}^{2}{y}^{0}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q80948\">Show Solution<\/span><\/p>\n<div id=\"q80948\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468244600\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>The product is raised to the zero power.<\/td>\n<td>[latex]{\\left(-3{x}^{2}y\\right)}^{0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of the zero exponent.<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168047201113\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Notice that only the variable [latex]y[\/latex] is being raised to the zero power.<\/td>\n<td>[latex]{-3{x}^{2}y}^{0}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the definition of the zero exponent.<\/td>\n<td>[latex]-3{x}^{2}\\cdot 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-3{x}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a similar problem to make sure you see the difference between raising an entire expression to a zero power and having only one factor raised to a zero power.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146223\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146223&theme=oea&iframe_resize_id=ohm146223&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146222\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146222&theme=oea&iframe_resize_id=ohm146222&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next video we show some different examples of how you can apply the zero exponent rule.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 3: Exponent Properties (Zero Exponent)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/zQJy1aBm1dQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div style=\"text-align: center;\"><\/div>\n<p>As done previously, to evaluate expressions containing exponents of [latex]0[\/latex] or [latex]1[\/latex], substitute the value of the variable into the expression and simplify.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]2x^{0}[\/latex] if [latex]x=9[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q324798\">Show Solution<\/span><\/p>\n<div id=\"q324798\" class=\"hidden-answer\" style=\"display: none\">Substitute 9 for the variable <i>x<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]2\\cdot9^{0}[\/latex]<\/p>\n<p>Evaluate [latex]9^{0}[\/latex]. Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]2\\cdot1=2[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]2x^{0}=2[\/latex], if [latex]x=9[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify\u00a0[latex]\\frac{{c}^{3}}{{c}^{3}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q769979\">Show Solution<\/span><\/p>\n<div id=\"q769979\" class=\"hidden-answer\" style=\"display: none\">Use the quotient and zero exponent rules to simplify the\u00a0expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}\\text{ }\\frac{c^{3}}{c^{3}} \\,\\,\\,= \\,\\,\\,c^{3-3} \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,c^{0} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,1\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video there is an example of evaluating an expression with an exponent of zero, as well as simplifying when you get a result of a zero exponent.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Evaluate and Simplify Expressions Using the Zero Exponent Rule\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jKihp_DVDa0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" 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-10868\">\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":21046,"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":"0180f73e52424109bcb3c78fcfbbbc1f, 5a8a9518e0a5410fbd66b236f950e99e, ed64ad8993004efc8094233e6b2a022d","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-10868","chapter","type-chapter","status-publish","hentry"],"part":16184,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10868","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/21046"}],"version-history":[{"count":35,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10868\/revisions"}],"predecessor-version":[{"id":20450,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10868\/revisions\/20450"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/16184"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/10868\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=10868"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=10868"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=10868"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=10868"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}