{"id":1424,"date":"2021-11-04T16:47:57","date_gmt":"2021-11-04T16:47:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1424"},"modified":"2023-09-25T18:58:10","modified_gmt":"2023-09-25T18:58:10","slug":"1-4-2-simplifying-expressions-with-negative-exponents-and-exponents-of-0-and-1","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/1-4-2-simplifying-expressions-with-negative-exponents-and-exponents-of-0-and-1\/","title":{"raw":"1.4.2: Exponential Properties for Division","rendered":"1.4.2: Exponential Properties for Division"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Simplify Expressions Using the Quotient Property of Exponents<\/li>\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\n<div class=\"textbox key-takeaways\">\r\n<h3>Key words<\/h3>\r\n<ul>\r\n \t<li><strong>Quotient<\/strong>: the result of dividing<\/li>\r\n \t<li><strong>Exponent<\/strong>: the power that a base number is raised to<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Simplifying Expressions Using the Quotient Property of Exponents<\/h2>\r\nSo far we have developed the properties of exponents for multiplication. We summarize these properties here.\r\n<div class=\"textbox shaded\">\r\n<h3>Summary of Exponent Properties for Multiplication<\/h3>\r\nIf [latex]a[\/latex] and [latex]b[\/latex] are real numbers and [latex]m[\/latex] and [latex]n[\/latex] are whole numbers, then,\r\n\r\nProduct Property: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]\r\n\r\nPower Property: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\left (a^m\\right )^n=a^{m\\cdot n}[\/latex]\r\n\r\nProduct to a Power: \u00a0 \u00a0 \u00a0[latex]\\left (ab\\right )^{n}={a}^{n}{b}^{n}[\/latex]\r\n\r\n<\/div>\r\nNow we will look at the exponent properties for division. We previously learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the <em><strong>Equivalent Fractions Property.<\/strong><\/em>\r\n<div class=\"textbox shaded\">\r\n<h3>Equivalent Fractions Property<\/h3>\r\nIf [latex]a,b,c[\/latex] are whole numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then\u00a0[latex]{\\frac{a}{b}}={\\frac{a\\cdot c}{b\\cdot c}}\\text{ and }{\\frac{a\\cdot c}{b\\cdot c}}={\\frac{a}{b}}[\/latex].\r\n\r\n&nbsp;\r\n\r\nIf the numerator and denominator of a fraction are multiplied or divided by the same non-zero number, the resulting fraction is equivalent to the original fraction.\r\n\r\n<\/div>\r\nAs before, we'll try to discover a property by looking at some examples.\r\n\r\nLet\u2019s look at dividing terms containing exponential expressions. What happens if we divide two numbers in exponential form with the same base? Consider the following expression.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}[\/latex]<\/p>\r\nWe can rewrite the expression as: [latex] \\displaystyle \\frac{4\\cdot 4\\cdot 4\\cdot 4\\cdot 4}{4\\cdot 4}[\/latex]. Then we can cancel the common factors of [latex]4[\/latex] in the numerator and denominator: [latex] \\displaystyle \\frac{\\color{red}{4}\\cdot \\color{red}{4}\\cdot 4\\cdot 4\\cdot 4}{\\color{red}{4}\\cdot \\color{red}{4}}[\/latex]. This leaves\u00a0[latex]4^{3}[\/latex] on the numerator and [latex]1[\/latex] on the denominator, which simplifies to [latex]4^{3}[\/latex]\u00a0using exponential notation. Notice that the exponent, [latex]3[\/latex], is the difference between the two exponents in the original expression, [latex]5[\/latex] and [latex]2[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[\/latex].<\/p>\r\n<p style=\"text-align: left;\">As another example, consider [latex]\\frac{{2}^{5}}{{2}^{2}}[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{{2}^{5}}{{2}^{2}}=\\frac{\\color{red}{2\\cdot 2}\\cdot 2\\cdot 2\\cdot 2}{\\color{red}{2\\cdot 2}}=\\frac{2\\cdot 2\\cdot 2\\cdot}{1}=2^3[\/latex]<\/p>\r\n<p style=\"text-align: left;\">If we subtract the exponents and keep the common base we get\u00a0<span style=\"font-size: 1rem; text-align: center;\">[latex]\\frac{{2}^{5}}{{2}^{2}}=2^{5-2}=2^3[\/latex]. \u00a0<\/span>The same answer we got when we expanded the exponents into multiplication.<\/p>\r\nNotice that in each case the bases were the same and we subtracted the exponents.\u00a0\u00a0So, to divide two exponential terms with the same base, subtract the exponents.\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\u00a0[latex]{\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m&gt;n[\/latex].\r\n\r\n&nbsp;\r\n\r\nTo divide two exponential terms with the same base, keep the base and subtract the exponents.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\frac{7^8}{7^3}[\/latex]\r\n\r\n[latex]=7^{8-3}=7^5[\/latex]\r\n\r\n&nbsp;\r\n\r\n2.\u00a0[latex]\\frac{(-5)^9}{(-5)^4}[\/latex]\r\n\r\n[latex]=(-5)^{9-4}=(-5)^5[\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\frac{4^5}{4^3}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2.\u00a0[latex]\\frac{(-8)^7}{(-8)^5}[\/latex]\r\n\r\n[reveal-answer q=\"743901\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"743901\"]\r\n\r\n1. [latex]\\frac{4^5}{4^3}=4^2[\/latex]\r\n\r\n&nbsp;\r\n\r\n2.\u00a0[latex]\\frac{(-8)^7}{(-8)^5}=(-8)^{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>One as an Exponent<\/h3>\r\nWhat does [latex]3^1[\/latex] or\u00a0[latex]{(-5)}^1[\/latex] equal?\r\n\r\nConsider\u00a0[latex]\\frac{2^4}{2^3}[\/latex]. If we use the quotient property,\u00a0[latex]\\frac{2^4}{2^3}=2^1[\/latex].\r\n\r\nAlternatively, we could expand the exponential terms:\u00a0[latex]\\frac{2^4}{2^3}=\\frac{2\\cdot 2\\cdot 2\\cdot 2}{2\\cdot 2\\cdot 2}[\/latex]. Then by cancelling out common factors, we get\u00a0\u00a0[latex]\\frac{\\color{red}{2\\cdot 2\\cdot 2\\cdot} 2}{\\color{red}{2\\cdot 2\\cdot 2}}=2[\/latex].\r\n\r\nThis means that [latex]2^1=2[\/latex], and leads us to the property that, for any integer [latex]a\\text{, }a^1=a[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\frac{7^4}{7^3}[\/latex]\r\n\r\n[latex]=7^{4-3}=7^1=7[\/latex]\r\n\r\n&nbsp;\r\n\r\n2.\u00a0[latex]\\frac{(-5)^5}{(-5)^4}[\/latex]\r\n\r\n[latex]=(-5)^{5-4}=(-5)^1=-5[\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<h3>Zero as an Exponent<\/h3>\r\nA special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as \u00a0[latex]\\frac{{5}^{4}}{{5}^{4}}[\/latex]. From earlier work with fractions, we know that,\r\n<p style=\"text-align: center;\">[latex]\\frac{2}{2} =\\frac{17}{17} =\\frac{-43}{-43} =1[\/latex]<\/p>\r\nIn words, a non-zero integer divided by itself is [latex]1[\/latex]. Remember that [latex]\\frac{0}{0}[\/latex] is undefined.\r\n\r\nWe also know that\u00a0[latex]a\\text{, }a^1=a[\/latex], for any non-zero integer\u00a0[latex]a[\/latex].\r\n\r\nNow consider simplifying the term\u00a0[latex]\\frac{8}{8}[\/latex] in two different ways.\r\n\r\nWe know that [latex]\\frac{8}{8}=1[\/latex] by division.\r\n\r\nWe also know that\u00a0[latex]\\frac{8}{8}=\\frac{8^1}{8^1}=8^{1-1}=8^0[\/latex] using the Quotient Property..\r\n\r\nThis means that\u00a0[latex]8^0=1[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Exponents of 0 or 1<\/h3>\r\nAny integer 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\n&nbsp;\r\n\r\nAny non-zero number integer 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\n&nbsp;\r\n\r\nThe quantity [latex]0^{0}[\/latex]\u00a0is undefined.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{12}^{0}[\/latex]\r\n\r\n2. [latex]{(-7)}^{0}[\/latex]\r\n\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]{(-7)}^{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<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146221[\/ohm_question]\r\n\r\n<\/div>\r\n<h3>Negative Exponents<\/h3>\r\n<p style=\"text-align: left;\">Now let's see what happens when the denominator has a larger exponent than the numerator, so that when we subtract the exponents, we get a negative integer:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{{2}^{2}}{{2}^{3}}=\\frac{\\color{red}{2\\cdot 2}}{\\color{red}{2\\cdot 2}\\cdot 2}=\\frac{1}{2}[\/latex]<\/p>\r\nWhen we subtract the exponents and keep the common base we get:\r\n<p style=\"text-align: center;\">[latex]\\frac{{2}^{2}}{{2}^{3}}=2^{2-3}=2^{-1}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">This means that [latex]2^{-1}=\\frac{1}{2}[\/latex].<\/p>\r\nLet's consider one more example:\r\n\r\n[latex]\\frac{{7}^{2}}{{7}^{5}}=\\frac{\\color{red}{7\\cdot 7}}{\\color{red}{7\\cdot 7\\cdot} 7\\cdot 7\\cdot 7}=\\frac{1}{7^3}[\/latex].\r\n\r\nOn the other hand, subtracting the exponents and keeping the common base gives us\u00a0[latex]\\frac{{7}^{2}}{{7}^{5}}=7^{2-5}=7^{-3}[\/latex].\r\n\r\nSo, [latex]\\frac{1}{7^3}=7^{-3}[\/latex].\r\n\r\nThis leads us to the meaning of negative exponents.\r\n<div class=\"textbox shaded\">\r\n<h3>NEGATIVE Exponents<\/h3>\r\nIf [latex]a[\/latex] is a real number, [latex]a\\ne 0[\/latex], and [latex]n[\/latex] is a whole numbers, then\u00a0[latex]a^{-n}=\\frac{1}{a^n}[\/latex].\r\n\r\n&nbsp;\r\n\r\nA negative exponent is equivalent to a positive exponent on the reciprocal of the number.\r\n\r\n<\/div>\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}={\\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&nbsp;\r\n<p style=\"text-align: left;\">[reveal-answer q=\"231258\"]Show Solution[\/reveal-answer][hidden-answer a=\"231258\"]<\/p>\r\n<p style=\"text-align: left;\">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]{4}^{-3}=\\frac{1}{64}[\/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:\r\n\r\n1. [latex]{4}^{-2}[\/latex]\r\n\r\n2. [latex]\\left ( \\frac{1}{2}\\right ) ^{-3}[\/latex]\r\n\r\n<strong>Solution<\/strong>\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}={\\frac{1}{{a}^{n}}}[\/latex].<\/td>\r\n<td>[latex]{\\frac{1}{{4}^{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{\\frac{1}{16}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467437540\" class=\"unnumbered unstyled\" style=\"height: 103px;\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 417.046875px;\">2.<\/td>\r\n<td style=\"height: 12px; width: 278.28125px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 33px;\">\r\n<td style=\"height: 33px; width: 417.046875px;\"><\/td>\r\n<td style=\"height: 33px; width: 278.28125px;\">[latex]\\left ( \\frac{1}{2}\\right ) ^{-3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 58px;\">\r\n<td style=\"height: 58px; width: 417.046875px;\">Take the reciprocal and turn the exponent positive.<\/td>\r\n<td style=\"height: 58px; width: 278.28125px;\">[latex]\\left ( \\frac{2}{1}\\right )^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 417.046875px;\">Simplify.<\/td>\r\n<td style=\"width: 278.28125px;\">[latex]2^3=8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146245[\/ohm_question]\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><\/h2>\r\nWe can now update the quotient property of exponents so that it includes negative and zero exponents by removing the condition that [latex]m\\gt n[\/latex].\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 integers, then\u00a0[latex]{\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n[latex]\\frac{{2}^{9}}{{2}^{2}}[\/latex]\r\n\r\n&nbsp;\r\n\r\nSolution\r\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.\r\n<table id=\"eip-id1168467300585\" class=\"unnumbered unstyled\" style=\"height: 48px;\" summary=\"The first line shows 2 to the 9th over 2 squared. Beside that is written \">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 506.797px;\"><\/td>\r\n<td style=\"height: 12px; width: 202.359px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 506.797px;\"><\/td>\r\n<td style=\"height: 12px; width: 202.359px;\">[latex]\\frac{{2}^{9}}{{2}^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 506.797px;\">Use the quotient property with [latex],\\frac{{a}^{m}}{{a}^{n}} ={a}^{m-n}[\/latex].<\/td>\r\n<td style=\"height: 12px; width: 202.359px;\">[latex]{2}^{\\color{red}{9-2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 506.797px;\">Simplify.<\/td>\r\n<td style=\"height: 12px; width: 202.359px;\">[latex]{2}^{7}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNotice that when the larger exponent is in the numerator, we are left with factors in the numerator.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSimplify:\r\n\r\n1. [latex]\\frac{(6)^9}{(6)^7}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]\\frac{(-3)^7}{(-3)^4}[\/latex]\r\n\r\n[reveal-answer q=\"30542\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"30542\"]\r\n\r\n&nbsp;\r\n\r\n1. [latex]\\frac{(6)^9}{(6)^7}[\/latex]\r\n\r\n[latex]=(6)^{9-7}=6^2[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]\\frac{(-3)^7}{(-3)^4}[\/latex]\r\n\r\n[latex]={(-3)}^{7-4}={(-3)}^{3}[\/latex]\r\n\r\n&nbsp;\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\n[latex]\\frac{{3}^{3}}{{3}^{5}}[\/latex]\r\n\r\n&nbsp;\r\n\r\nSolution\r\n\r\nBoth bases are [latex]3[\/latex] so we can subtract the exponents:\r\n\r\n[latex]\\frac{{3}^{3}}{{3}^{5}}=3^{3-5}=3^{-2}=\\frac{1}{3^2}=\\frac{1}{9}[\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nNotice that when the larger exponent is in the denominator, we are left with factors in the denominator and [latex]1[\/latex] in the numerator.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSimplify the terms:\r\n\r\n1. [latex]\\frac{(-5)^4}{(-5)^7}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]\\frac{2^5}{2^{-3}}[\/latex]\r\n\r\n&nbsp;\r\n\r\n3. [latex]\\frac{7^{-4}}{7^{-6}}[\/latex]\r\n\r\n&nbsp;\r\n\r\n4. [latex]\\frac{(-3)^{-6}}{(-3)^{-4}}[\/latex]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"774065\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"774065\"]\r\n\r\n&nbsp;\r\n\r\n1. [latex]\\frac{(-5)^4}{(-5)^7}[\/latex]\r\n\r\n[latex]={(-5)}^{4-7}={(-5)}^{-3}=\\left ( \\frac{-1}{5}\\right )^3=\\frac{(-1)^{3}}{5^3}=-\\frac{1}{125}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]\\frac{2^5}{2^{-3}}[\/latex]\r\n\r\n[latex]=2^{5-(-3)}=2^8=256[\/latex]\r\n\r\n&nbsp;\r\n\r\n3. [latex]\\frac{7^{-4}}{7^{-6}}[\/latex]\r\n\r\n[latex]=7^{-4-(-6)}=7^2=49[\/latex]\r\n\r\n&nbsp;\r\n\r\n4. [latex]\\frac{(-3)^{-6}}{(-3)^{-4}}[\/latex]\r\n\r\n[latex]=(-3)^{-6-(-4)}=(-3)^{-2}=\\left ( \\frac{-1}{3}\\right ) ^2=\\frac{1}{9}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Simplifying Fractions Raised to a Power<\/h2>\r\nNow we will look at an example that will lead us to the Quotient to a Power Property.\r\n\r\nLet's look at what happens if you raise a fraction to a power. Remember that a fraction bar means divide. Suppose we have [latex] \\displaystyle \\frac{5}{4}[\/latex] and raise it to the\u00a0power [latex]3[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{\\left( \\frac{5}{4} \\right)}^{3}}=\\left( \\frac{5}{4} \\right)\\left( \\frac{5}{4} \\right)\\left( \\frac{5}{4} \\right)=\\frac{5\\cdot 5\\cdot 5}{4\\cdot 4\\cdot 4}=\\frac{{{5}^{3}}}{{{4}^{3}}}[\/latex]<\/p>\r\nRaising the fraction to the power of [latex]3[\/latex] can also be written as the numerator [latex]5[\/latex] to the power of [latex]3[\/latex], and the denominator [latex]4[\/latex] to the power of [latex]3[\/latex].\r\n\r\nNotice that the exponent applies to both the numerator and the denominator.\u00a0\u00a0This leads to the <em><strong>Quotient to a Power Property for Exponents.<\/strong><\/em>\r\n<div class=\"textbox shaded\">\r\n<h3>Quotient to a Power Property of Exponents<\/h3>\r\nIf [latex]a[\/latex] and [latex]b[\/latex] are real numbers, [latex]b\\ne 0[\/latex], and [latex]n[\/latex] is an integer number, then\u00a0[latex]{\\left(\\frac{a}{b}\\right)}^{m}=\\frac{{a}^{m}}{{b}^{m}}[\/latex].\r\n\r\n&nbsp;\r\n\r\nTo raise a fraction to a power, raise the numerator and denominator to that power.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify:\r\n\r\n1. [latex]{\\left(\\frac{5}{8}\\right)}^{2}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2. [latex]{\\left(\\frac{2}{3}\\right)}^{4}[\/latex]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"803388\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"803388\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467475453\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 5 over 8 raised to the second power. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex](\\frac{5}{8})^2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Quotient to a Power Property, [latex]{\\left(\\frac{a}{b}\\right)}^{m} =\\frac{{a}^{m}}{{b}^{m}}[\/latex] .<\/td>\r\n<td>[latex]\\frac{5^{\\color{red}{2}}}{8^{\\color{red}{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\frac{25}{64}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469801450\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x over 3 raised to the fourth power. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex](\\frac{2}{3})^4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Quotient to a Power Property, [latex]{\\left(\\frac{a}{b}\\right)}^{m} =\\frac{{a}^{m}}{{b}^{m}}[\/latex] .<\/td>\r\n<td>[latex]\\frac{2^{\\color{red}{4}}}{3^{\\color{red}{4}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\frac{16}{81}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146227[\/ohm_question]\r\n\r\n<\/div>\r\n<h2><\/h2>\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\nSimplify [latex]{\\left(\\frac{1}{3}\\right)}^{-2}[\/latex].\r\n\r\n&nbsp;\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}}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] = \\frac{9}{1}{ = }{9}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{{1}^{-2}}{{3}^{-2}}=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&nbsp;\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 {4}^{2}=16[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{1}{4^{-2}}=16[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSimplify:\r\n\r\n1.\u00a0 [latex]{\\left(\\frac{5}{2}\\right)}^{-2}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2.\u00a0[latex]-\\frac{3}{5^{-2}}[\/latex]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"388784\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"388784\"]\r\n\r\n&nbsp;\r\n\r\n1.\u00a0 [latex]{\\left(\\frac{5}{2}\\right)}^{-2}={\\left(\\frac{2}{5}\\right)}^{2}=\\frac{4}{25}[\/latex]\r\n\r\n&nbsp;\r\n\r\n2.\u00a0[latex]\\frac{-3}{5^{-2}}=-3\\cdot\\frac{1}{5^{-2}}=-3\\cdot\\frac{5^{2}}{1}=-75[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2><\/h2>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Simplify Expressions Using the Quotient Property of Exponents<\/li>\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<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li><strong>Quotient<\/strong>: the result of dividing<\/li>\n<li><strong>Exponent<\/strong>: the power that a base number is raised to<\/li>\n<\/ul>\n<\/div>\n<h2>Simplifying Expressions Using the Quotient Property of Exponents<\/h2>\n<p>So far we have developed the properties of exponents for multiplication. We summarize these properties here.<\/p>\n<div class=\"textbox shaded\">\n<h3>Summary of Exponent Properties for Multiplication<\/h3>\n<p>If [latex]a[\/latex] and [latex]b[\/latex] are real numbers and [latex]m[\/latex] and [latex]n[\/latex] are whole numbers, then,<\/p>\n<p>Product Property: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/p>\n<p>Power Property: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\left (a^m\\right )^n=a^{m\\cdot n}[\/latex]<\/p>\n<p>Product to a Power: \u00a0 \u00a0 \u00a0[latex]\\left (ab\\right )^{n}={a}^{n}{b}^{n}[\/latex]<\/p>\n<\/div>\n<p>Now we will look at the exponent properties for division. We previously learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the <em><strong>Equivalent Fractions Property.<\/strong><\/em><\/p>\n<div class=\"textbox shaded\">\n<h3>Equivalent Fractions Property<\/h3>\n<p>If [latex]a,b,c[\/latex] are whole numbers where [latex]b\\ne 0,c\\ne 0[\/latex], then\u00a0[latex]{\\frac{a}{b}}={\\frac{a\\cdot c}{b\\cdot c}}\\text{ and }{\\frac{a\\cdot c}{b\\cdot c}}={\\frac{a}{b}}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>If the numerator and denominator of a fraction are multiplied or divided by the same non-zero number, the resulting fraction is equivalent to the original fraction.<\/p>\n<\/div>\n<p>As before, we&#8217;ll try to discover a property by looking at some examples.<\/p>\n<p>Let\u2019s look at dividing terms containing exponential expressions. What happens if we divide two numbers in exponential form with the same base? Consider the following expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}[\/latex]<\/p>\n<p>We can rewrite the expression as: [latex]\\displaystyle \\frac{4\\cdot 4\\cdot 4\\cdot 4\\cdot 4}{4\\cdot 4}[\/latex]. Then we can cancel the common factors of [latex]4[\/latex] in the numerator and denominator: [latex]\\displaystyle \\frac{\\color{red}{4}\\cdot \\color{red}{4}\\cdot 4\\cdot 4\\cdot 4}{\\color{red}{4}\\cdot \\color{red}{4}}[\/latex]. This leaves\u00a0[latex]4^{3}[\/latex] on the numerator and [latex]1[\/latex] on the denominator, which simplifies to [latex]4^{3}[\/latex]\u00a0using exponential notation. Notice that the exponent, [latex]3[\/latex], is the difference between the two exponents in the original expression, [latex]5[\/latex] and [latex]2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[\/latex].<\/p>\n<p style=\"text-align: left;\">As another example, consider [latex]\\frac{{2}^{5}}{{2}^{2}}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{2}^{5}}{{2}^{2}}=\\frac{\\color{red}{2\\cdot 2}\\cdot 2\\cdot 2\\cdot 2}{\\color{red}{2\\cdot 2}}=\\frac{2\\cdot 2\\cdot 2\\cdot}{1}=2^3[\/latex]<\/p>\n<p style=\"text-align: left;\">If we subtract the exponents and keep the common base we get\u00a0<span style=\"font-size: 1rem; text-align: center;\">[latex]\\frac{{2}^{5}}{{2}^{2}}=2^{5-2}=2^3[\/latex]. \u00a0<\/span>The same answer we got when we expanded the exponents into multiplication.<\/p>\n<p>Notice that in each case the bases were the same and we subtracted the exponents.\u00a0\u00a0So, to divide two exponential terms with the same base, subtract the exponents.<\/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\u00a0[latex]{\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m>n[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>To divide two exponential terms with the same base, keep the base and subtract the exponents.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\frac{7^8}{7^3}[\/latex]<\/p>\n<p>[latex]=7^{8-3}=7^5[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2.\u00a0[latex]\\frac{(-5)^9}{(-5)^4}[\/latex]<\/p>\n<p>[latex]=(-5)^{9-4}=(-5)^5[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\frac{4^5}{4^3}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2.\u00a0[latex]\\frac{(-8)^7}{(-8)^5}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q743901\">Show Answer<\/span><\/p>\n<div id=\"q743901\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]\\frac{4^5}{4^3}=4^2[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2.\u00a0[latex]\\frac{(-8)^7}{(-8)^5}=(-8)^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>One as an Exponent<\/h3>\n<p>What does [latex]3^1[\/latex] or\u00a0[latex]{(-5)}^1[\/latex] equal?<\/p>\n<p>Consider\u00a0[latex]\\frac{2^4}{2^3}[\/latex]. If we use the quotient property,\u00a0[latex]\\frac{2^4}{2^3}=2^1[\/latex].<\/p>\n<p>Alternatively, we could expand the exponential terms:\u00a0[latex]\\frac{2^4}{2^3}=\\frac{2\\cdot 2\\cdot 2\\cdot 2}{2\\cdot 2\\cdot 2}[\/latex]. Then by cancelling out common factors, we get\u00a0\u00a0[latex]\\frac{\\color{red}{2\\cdot 2\\cdot 2\\cdot} 2}{\\color{red}{2\\cdot 2\\cdot 2}}=2[\/latex].<\/p>\n<p>This means that [latex]2^1=2[\/latex], and leads us to the property that, for any integer [latex]a\\text{, }a^1=a[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\frac{7^4}{7^3}[\/latex]<\/p>\n<p>[latex]=7^{4-3}=7^1=7[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2.\u00a0[latex]\\frac{(-5)^5}{(-5)^4}[\/latex]<\/p>\n<p>[latex]=(-5)^{5-4}=(-5)^1=-5[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<h3>Zero as an Exponent<\/h3>\n<p>A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as \u00a0[latex]\\frac{{5}^{4}}{{5}^{4}}[\/latex]. From earlier work with fractions, we know that,<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2}{2} =\\frac{17}{17} =\\frac{-43}{-43} =1[\/latex]<\/p>\n<p>In words, a non-zero integer divided by itself is [latex]1[\/latex]. Remember that [latex]\\frac{0}{0}[\/latex] is undefined.<\/p>\n<p>We also know that\u00a0[latex]a\\text{, }a^1=a[\/latex], for any non-zero integer\u00a0[latex]a[\/latex].<\/p>\n<p>Now consider simplifying the term\u00a0[latex]\\frac{8}{8}[\/latex] in two different ways.<\/p>\n<p>We know that [latex]\\frac{8}{8}=1[\/latex] by division.<\/p>\n<p>We also know that\u00a0[latex]\\frac{8}{8}=\\frac{8^1}{8^1}=8^{1-1}=8^0[\/latex] using the Quotient Property..<\/p>\n<p>This means that\u00a0[latex]8^0=1[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Exponents of 0 or 1<\/h3>\n<p>Any integer 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>&nbsp;<\/p>\n<p>Any non-zero number integer 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>&nbsp;<\/p>\n<p>The quantity [latex]0^{0}[\/latex]\u00a0is undefined.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{12}^{0}[\/latex]<\/p>\n<p>2. [latex]{(-7)}^{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]{(-7)}^{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<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<\/div>\n<h3>Negative Exponents<\/h3>\n<p style=\"text-align: left;\">Now let&#8217;s see what happens when the denominator has a larger exponent than the numerator, so that when we subtract the exponents, we get a negative integer:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{2}^{2}}{{2}^{3}}=\\frac{\\color{red}{2\\cdot 2}}{\\color{red}{2\\cdot 2}\\cdot 2}=\\frac{1}{2}[\/latex]<\/p>\n<p>When we subtract the exponents and keep the common base we get:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{2}^{2}}{{2}^{3}}=2^{2-3}=2^{-1}[\/latex]<\/p>\n<p style=\"text-align: center;\">This means that [latex]2^{-1}=\\frac{1}{2}[\/latex].<\/p>\n<p>Let&#8217;s consider one more example:<\/p>\n<p>[latex]\\frac{{7}^{2}}{{7}^{5}}=\\frac{\\color{red}{7\\cdot 7}}{\\color{red}{7\\cdot 7\\cdot} 7\\cdot 7\\cdot 7}=\\frac{1}{7^3}[\/latex].<\/p>\n<p>On the other hand, subtracting the exponents and keeping the common base gives us\u00a0[latex]\\frac{{7}^{2}}{{7}^{5}}=7^{2-5}=7^{-3}[\/latex].<\/p>\n<p>So, [latex]\\frac{1}{7^3}=7^{-3}[\/latex].<\/p>\n<p>This leads us to the meaning of negative exponents.<\/p>\n<div class=\"textbox shaded\">\n<h3>NEGATIVE Exponents<\/h3>\n<p>If [latex]a[\/latex] is a real number, [latex]a\\ne 0[\/latex], and [latex]n[\/latex] is a whole numbers, then\u00a0[latex]a^{-n}=\\frac{1}{a^n}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>A negative exponent is equivalent to a positive exponent on the reciprocal of the number.<\/p>\n<\/div>\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}={\\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>&nbsp;<\/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\">\n<p style=\"text-align: left;\">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]{4}^{-3}=\\frac{1}{64}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{4}^{-2}[\/latex]<\/p>\n<p>2. [latex]\\left ( \\frac{1}{2}\\right ) ^{-3}[\/latex]<\/p>\n<p><strong>Solution<\/strong><\/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}={\\frac{1}{{a}^{n}}}[\/latex].<\/td>\n<td>[latex]{\\frac{1}{{4}^{2}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{\\frac{1}{16}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467437540\" class=\"unnumbered unstyled\" style=\"height: 103px;\" summary=\".\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 417.046875px;\">2.<\/td>\n<td style=\"height: 12px; width: 278.28125px;\"><\/td>\n<\/tr>\n<tr style=\"height: 33px;\">\n<td style=\"height: 33px; width: 417.046875px;\"><\/td>\n<td style=\"height: 33px; width: 278.28125px;\">[latex]\\left ( \\frac{1}{2}\\right ) ^{-3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 58px;\">\n<td style=\"height: 58px; width: 417.046875px;\">Take the reciprocal and turn the exponent positive.<\/td>\n<td style=\"height: 58px; width: 278.28125px;\">[latex]\\left ( \\frac{2}{1}\\right )^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 417.046875px;\">Simplify.<\/td>\n<td style=\"width: 278.28125px;\">[latex]2^3=8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"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>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><\/h2>\n<p>We can now update the quotient property of exponents so that it includes negative and zero exponents by removing the condition that [latex]m\\gt n[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Quotient Property of Exponents<\/h3>\n<p>If [latex]a[\/latex] is a real number, [latex]a\\ne 0[\/latex], and [latex]m,n[\/latex] are integers, then\u00a0[latex]{\\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>[latex]\\frac{{2}^{9}}{{2}^{2}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Solution<br \/>\nTo simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.<\/p>\n<table id=\"eip-id1168467300585\" class=\"unnumbered unstyled\" style=\"height: 48px;\" summary=\"The first line shows 2 to the 9th over 2 squared. Beside that is written\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 506.797px;\"><\/td>\n<td style=\"height: 12px; width: 202.359px;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 506.797px;\"><\/td>\n<td style=\"height: 12px; width: 202.359px;\">[latex]\\frac{{2}^{9}}{{2}^{2}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 506.797px;\">Use the quotient property with [latex],\\frac{{a}^{m}}{{a}^{n}} ={a}^{m-n}[\/latex].<\/td>\n<td style=\"height: 12px; width: 202.359px;\">[latex]{2}^{\\color{red}{9-2}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 506.797px;\">Simplify.<\/td>\n<td style=\"height: 12px; width: 202.359px;\">[latex]{2}^{7}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]\\frac{(6)^9}{(6)^7}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]\\frac{(-3)^7}{(-3)^4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q30542\">Show Answer<\/span><\/p>\n<div id=\"q30542\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<p>1. [latex]\\frac{(6)^9}{(6)^7}[\/latex]<\/p>\n<p>[latex]=(6)^{9-7}=6^2[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]\\frac{(-3)^7}{(-3)^4}[\/latex]<\/p>\n<p>[latex]={(-3)}^{7-4}={(-3)}^{3}[\/latex]<\/p>\n<p>&nbsp;<\/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>[latex]\\frac{{3}^{3}}{{3}^{5}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Solution<\/p>\n<p>Both bases are [latex]3[\/latex] so we can subtract the exponents:<\/p>\n<p>[latex]\\frac{{3}^{3}}{{3}^{5}}=3^{3-5}=3^{-2}=\\frac{1}{3^2}=\\frac{1}{9}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and [latex]1[\/latex] in the numerator.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Simplify the terms:<\/p>\n<p>1. [latex]\\frac{(-5)^4}{(-5)^7}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]\\frac{2^5}{2^{-3}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>3. [latex]\\frac{7^{-4}}{7^{-6}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>4. [latex]\\frac{(-3)^{-6}}{(-3)^{-4}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q774065\">Show Answer<\/span><\/p>\n<div id=\"q774065\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<p>1. [latex]\\frac{(-5)^4}{(-5)^7}[\/latex]<\/p>\n<p>[latex]={(-5)}^{4-7}={(-5)}^{-3}=\\left ( \\frac{-1}{5}\\right )^3=\\frac{(-1)^{3}}{5^3}=-\\frac{1}{125}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]\\frac{2^5}{2^{-3}}[\/latex]<\/p>\n<p>[latex]=2^{5-(-3)}=2^8=256[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>3. [latex]\\frac{7^{-4}}{7^{-6}}[\/latex]<\/p>\n<p>[latex]=7^{-4-(-6)}=7^2=49[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>4. [latex]\\frac{(-3)^{-6}}{(-3)^{-4}}[\/latex]<\/p>\n<p>[latex]=(-3)^{-6-(-4)}=(-3)^{-2}=\\left ( \\frac{-1}{3}\\right ) ^2=\\frac{1}{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Simplifying Fractions Raised to a Power<\/h2>\n<p>Now we will look at an example that will lead us to the Quotient to a Power Property.<\/p>\n<p>Let&#8217;s look at what happens if you raise a fraction to a power. Remember that a fraction bar means divide. Suppose we have [latex]\\displaystyle \\frac{5}{4}[\/latex] and raise it to the\u00a0power [latex]3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{\\left( \\frac{5}{4} \\right)}^{3}}=\\left( \\frac{5}{4} \\right)\\left( \\frac{5}{4} \\right)\\left( \\frac{5}{4} \\right)=\\frac{5\\cdot 5\\cdot 5}{4\\cdot 4\\cdot 4}=\\frac{{{5}^{3}}}{{{4}^{3}}}[\/latex]<\/p>\n<p>Raising the fraction to the power of [latex]3[\/latex] can also be written as the numerator [latex]5[\/latex] to the power of [latex]3[\/latex], and the denominator [latex]4[\/latex] to the power of [latex]3[\/latex].<\/p>\n<p>Notice that the exponent applies to both the numerator and the denominator.\u00a0\u00a0This leads to the <em><strong>Quotient to a Power Property for Exponents.<\/strong><\/em><\/p>\n<div class=\"textbox shaded\">\n<h3>Quotient to a Power Property of Exponents<\/h3>\n<p>If [latex]a[\/latex] and [latex]b[\/latex] are real numbers, [latex]b\\ne 0[\/latex], and [latex]n[\/latex] is an integer number, then\u00a0[latex]{\\left(\\frac{a}{b}\\right)}^{m}=\\frac{{a}^{m}}{{b}^{m}}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>To raise a fraction to a power, raise the numerator and denominator to that power.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]{\\left(\\frac{5}{8}\\right)}^{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2. [latex]{\\left(\\frac{2}{3}\\right)}^{4}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q803388\">Show Solution<\/span><\/p>\n<div id=\"q803388\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467475453\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 5 over 8 raised to the second power. The next line says,\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex](\\frac{5}{8})^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Quotient to a Power Property, [latex]{\\left(\\frac{a}{b}\\right)}^{m} =\\frac{{a}^{m}}{{b}^{m}}[\/latex] .<\/td>\n<td>[latex]\\frac{5^{\\color{red}{2}}}{8^{\\color{red}{2}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\frac{25}{64}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469801450\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x over 3 raised to the fourth power. The next line says,\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex](\\frac{2}{3})^4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Quotient to a Power Property, [latex]{\\left(\\frac{a}{b}\\right)}^{m} =\\frac{{a}^{m}}{{b}^{m}}[\/latex] .<\/td>\n<td>[latex]\\frac{2^{\\color{red}{4}}}{3^{\\color{red}{4}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\frac{16}{81}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146227\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146227&theme=oea&iframe_resize_id=ohm146227&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2><\/h2>\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>Simplify [latex]{\\left(\\frac{1}{3}\\right)}^{-2}[\/latex].<\/p>\n<p>&nbsp;<\/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}}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]= \\frac{9}{1}{ = }{9}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{{1}^{-2}}{{3}^{-2}}=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<p>&nbsp;<\/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 {4}^{2}=16[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{1}{4^{-2}}=16[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Simplify:<\/p>\n<p>1.\u00a0 [latex]{\\left(\\frac{5}{2}\\right)}^{-2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2.\u00a0[latex]-\\frac{3}{5^{-2}}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q388784\">Show Answer<\/span><\/p>\n<div id=\"q388784\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<p>1.\u00a0 [latex]{\\left(\\frac{5}{2}\\right)}^{-2}={\\left(\\frac{2}{5}\\right)}^{2}=\\frac{4}{25}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>2.\u00a0[latex]\\frac{-3}{5^{-2}}=-3\\cdot\\frac{1}{5^{-2}}=-3\\cdot\\frac{5^{2}}{1}=-75[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2><\/h2>\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-1424\">\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: 146227, 146245, 146221. <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>Adaptation and revision Prealgebra. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>Project<\/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":422605,"menu_order":17,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Adaptation and revision Prealgebra\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley 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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-1424","chapter","type-chapter","status-publish","hentry"],"part":587,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1424","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/422605"}],"version-history":[{"count":17,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1424\/revisions"}],"predecessor-version":[{"id":2743,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1424\/revisions\/2743"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/587"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1424\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=1424"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1424"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=1424"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=1424"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}