{"id":4294,"date":"2016-05-24T19:23:21","date_gmt":"2016-05-24T19:23:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/nrocarithmetic\/?post_type=chapter&#038;p=4294"},"modified":"2018-01-03T23:57:22","modified_gmt":"2018-01-03T23:57:22","slug":"read-terms-and-expressions-with-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-beginalgebra\/chapter\/read-terms-and-expressions-with-exponents\/","title":{"raw":"Rules for Exponents","rendered":"Rules for Exponents"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Product and Quotient Rules\r\n<ul>\r\n \t<li>Use the product rule to multiply exponential expressions<\/li>\r\n \t<li>Use the quotient rule to divide exponential expressions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The Power Rule for Exponents\r\n<ul>\r\n \t<li>Use the power rule to simplify expressions involving\u00a0products, quotients, and exponents<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Negative and Zero Exponents\r\n<ul>\r\n \t<li>Define and use the zero exponent rule<\/li>\r\n \t<li>Define and use the negative exponent rule<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Simplify Expressions Using the Exponent Rules\r\n<ul>\r\n \t<li>Simplify expressions using a combination of the exponent rules<\/li>\r\n \t<li>Simplify compound exponential expressions with negative exponents<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n<\/div>\r\n\r\n[caption id=\"attachment_4445\" align=\"aligncenter\" width=\"282\"]<img class=\"wp-image-4445 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/31173815\/Screen-Shot-2016-05-31-at-10.38.21-AM-150x150.png\" alt=\"Image of a woman taking a picture with a camera repeated five times in different colors.\" width=\"282\" height=\"282\" \/> Repeated Image[\/caption]\r\n<h2>Anatomy of exponential\u00a0terms<\/h2>\r\nWe use exponential notation to write repeated multiplication. For example [latex]10\\cdot10\\cdot10[\/latex] can be written more succinctly as [latex]10^{3}[\/latex]. The 10 in [latex]10^{3}[\/latex]<sup>\u00a0<\/sup>is called the <b>base<\/b>. The 3 in [latex]10^{3}[\/latex]<sup>\u00a0<\/sup>is called the <b>exponent<\/b>. The expression [latex]10^{3}[\/latex] is called the exponential expression. Knowing the names for the parts of an exponential expression or term will help you learn how to perform mathematical operations on them.\r\n<p style=\"text-align: center;\">[latex]\\text{base}\\rightarrow10^{3\\leftarrow\\text{exponent}}[\/latex]<\/p>\r\n[latex]10^{3}[\/latex] is read as \u201c10 to the third power\u201d or \u201c10 cubed.\u201d It means [latex]10\\cdot10\\cdot10[\/latex], or 1,000.\r\n\r\n[latex]8^{2}[\/latex]\u00a0is read as \u201c8 to the second power\u201d or \u201c8 squared.\u201d It means [latex]8\\cdot8[\/latex], or 64.\r\n\r\n[latex]5^{4}[\/latex]\u00a0is read as \u201c5 to the fourth power.\u201d It means [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or 625.\r\n\r\n[latex]b^{5}[\/latex]\u00a0is read as \u201c<i>b<\/i> to the fifth power.\u201d It means [latex]{b}\\cdot{b}\\cdot{b}\\cdot{b}\\cdot{b}[\/latex]. Its value will depend on the value of <i>b<\/i>.\r\n\r\nThe exponent applies only to the number that it is next to. Therefore, in the expression [latex]xy^{4}[\/latex],\u00a0only the <i>y<\/i> is affected by the 4. [latex]xy^{4}[\/latex]\u00a0means [latex]{x}\\cdot{y}\\cdot{y}\\cdot{y}\\cdot{y}[\/latex]. The <em>x<\/em> in this term is a <strong>coefficient<\/strong> of <em>y<\/em>.\r\n\r\nIf the exponential expression is negative, such as [latex]\u22123^{4}[\/latex], it means [latex]\u2013\\left(3\\cdot3\\cdot3\\cdot3\\right)[\/latex] or [latex]\u221281[\/latex].\r\n\r\nIf [latex]\u22123[\/latex] is to be the base, it must be written as [latex]\\left(\u22123\\right)^{4}[\/latex], which means [latex]\u22123\\cdot\u22123\\cdot\u22123\\cdot\u22123[\/latex], or 81.\r\n\r\nLikewise,\u00a0[latex]\\left(\u2212x\\right)^{4}=\\left(\u2212x\\right)\\cdot\\left(\u2212x\\right)\\cdot\\left(\u2212x\\right)\\cdot\\left(\u2212x\\right)=x^{4}[\/latex], while [latex]\u2212x^{4}=\u2013\\left(x\\cdot x\\cdot x\\cdot x\\right)[\/latex].\r\n\r\nYou can see that there is quite a difference, so you have to be very careful! The following examples show how to identify the base and the exponent, as well as how to identify the expanded and exponential format of writing repeated multiplication.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIdentify the exponent and the base in the following terms, then simplify:\r\n<ol>\r\n \t<li>[latex]7^{2}[\/latex]<\/li>\r\n \t<li>[latex]{\\left(\\frac{1}{2}\\right)}^{3}[\/latex]<\/li>\r\n \t<li>[latex]2x^{3}[\/latex]<\/li>\r\n \t<li>[latex]\\left(-5\\right)^{2}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"211363\"]Show Solution[\/reveal-answer]\r\n<p style=\"text-align: left;\">[hidden-answer a=\"211363\"]<\/p>\r\n1) [latex]7^{2}[\/latex]\r\n\r\nThe exponent in this term is 2 and the base is 7. To simplify, expand the term: [latex]7^{2}=7\\cdot{7}=49[\/latex]\r\n\r\n2) [latex]{\\left(\\frac{1}{2}\\right)}^{3}[\/latex]\r\n\r\nThe exponent on this term is 3, and the base is [latex]\\frac{1}{2}[\/latex]. To simplify, expand the multiplication and remember how to multiply fractions: [latex]{\\left(\\frac{1}{2}\\right)}^{3}=\\frac{1}{2}\\cdot{\\frac{1}{2}}\\cdot{\\frac{1}{2}}=\\frac{1}{16}[\/latex]\r\n\r\n3) \u00a0[latex]2x^{3}[\/latex]\r\n\r\nThe exponent on this term is 3, and the base is x, the 2 is not getting the exponent because there are no parentheses that tell us it is. \u00a0This term is in its most simplified form.\r\n\r\n4)\u00a0[latex]\\left(-5\\right)^{2}[\/latex]\r\n\r\nThe exponent on this terms is 2 and the base is [latex]-5[\/latex]. To simplify, expand the multiplication: [latex]\\left(-5\\right)^{2}=-5\\cdot{-5}=25[\/latex]\r\n\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p class=\"no-indent\">In the following video you are provided more examples of applying exponents to various bases.<\/p>\r\nhttps:\/\/youtu.be\/ocedY91LHKU\r\n<h3>Evaluate expressions<\/h3>\r\nEvaluating expressions containing exponents is the same as evaluating the linear expressions from earlier in the course. You substitute the value of the variable into the expression and simplify.\r\n\r\nYou can use the order of operations\u00a0to evaluate the expressions containing exponents. First, evaluate anything in Parentheses or grouping symbols. Next, look for Exponents, followed by Multiplication and Division (reading from left to right), and lastly, Addition and Subtraction (again, reading from left to right).\r\n\r\nSo, when you evaluate the expression [latex]5x^{3}[\/latex]\u00a0if [latex]x=4[\/latex], first substitute the value 4 for the variable <i>x<\/i>. Then evaluate, using order of operations.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate\u00a0[latex]5x^{3}[\/latex]<i>\u00a0<\/i>if [latex]x=4[\/latex].\r\n\r\n[reveal-answer q=\"411363\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"411363\"]\r\n\r\nSubstitute 4 for the variable <i>x<\/i>.\r\n<p style=\"text-align: center;\">[latex]5\\cdot4^{3}[\/latex]<\/p>\r\nEvaluate [latex]4^{3}[\/latex]. Multiply.\r\n<p style=\"text-align: center;\">[latex]5\\left(4\\cdot4\\cdot4\\right)=5\\cdot64=320[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]5x^{3}=320[\/latex]\u00a0when [latex]x=4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the example below, notice the how adding parentheses can change the outcome when you are simplifying terms with exponents.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex]\\left(5x\\right)^{3}[\/latex]\u00a0if [latex]x=4[\/latex].\r\n\r\n[reveal-answer q=\"362021\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"362021\"]Substitute 4 for the variable <i>x<\/i>.\r\n<p style=\"text-align: center;\">[latex]\\left(5\\cdot4\\right)3[\/latex]<\/p>\r\nMultiply inside the parentheses, then apply the exponent\u2014following the rules of PEMDAS.\r\n<p style=\"text-align: center;\">[latex]20^{3}[\/latex]<\/p>\r\nEvaluate [latex]20^{3}[\/latex].\r\n<p style=\"text-align: center;\">[latex]20\\cdot20\\cdot20=8,000[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(5x\\right)3=8,000[\/latex] when [latex]x=4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe addition of parentheses made quite a difference!\u00a0Parentheses allow you to apply an exponent to variables or numbers that are multiplied, divided, added, or subtracted to each other.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex]x^{3}[\/latex] if [latex]x=\u22124[\/latex].\r\n\r\n[reveal-answer q=\"86290\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"86290\"]Substitute [latex]\u22124[\/latex] for the variable <i>x<\/i>.\r\n<p style=\"text-align: center;\">[latex]\\left(\u22124\\right)^{3}[\/latex]<\/p>\r\nEvaluate. Note how placing parentheses around the [latex]\u22124[\/latex] means the negative sign also gets multiplied.\r\n<p style=\"text-align: center;\">[latex]\u22124\\cdot\u22124\\cdot\u22124[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex]\u22124\\cdot\u22124\\cdot\u22124=\u221264[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]x^{3}=\u221264[\/latex] when [latex]x=\u22124[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"75\" height=\"66\" \/>\r\n\r\nCaution! Whether to include a negative sign as part of a base or not often leads to confusion. To clarify\u00a0whether a negative sign is applied before or after the exponent, here is an example.\r\n\r\n&nbsp;\r\n\r\nWhat is the difference in the way you would evaluate these two terms?\r\n<ol>\r\n \t<li style=\"text-align: left;\">[latex]-{3}^{2}[\/latex]<\/li>\r\n \t<li style=\"text-align: left;\">[latex]{\\left(-3\\right)}^{2}[\/latex]<\/li>\r\n<\/ol>\r\nTo evaluate 1), you would apply the exponent to the three first, then apply the negative sign last, like this:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}-\\left({3}^{2}\\right)\\\\=-\\left(9\\right) = -9\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">To evaluate 2), you would apply the exponent to the 3 and the negative sign:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}{\\left(-3\\right)}^{2}\\\\=\\left(-3\\right)\\cdot\\left(-3\\right)\\\\={ 9}\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">The key to remembering this is to follow the order of operations. The first expression does not include parentheses so you would apply the exponent to the integer 3 first, then apply the negative sign. The second expression includes parentheses, so hopefully you will remember that the negative sign also gets squared.<\/p>\r\n\r\n<\/div>\r\n<p id=\"video0\" class=\"no-indent\" style=\"text-align: left;\">In the next sections, you will learn how to simplify expressions that contain exponents. Come back to this page if you forget how to apply the order of operations to a term with exponents, or forget which is the base and which is the exponent!<\/p>\r\n<p class=\"no-indent\" style=\"text-align: left;\">In the following video you are provided with examples of evaluating exponential expressions for a given number.<\/p>\r\nhttps:\/\/youtu.be\/pQNz8IpVVg0\r\n<h2 id=\"title1\">Use the product rule to multiply exponential expressions<\/h2>\r\n<b>Exponential notation<\/b> was developed to write repeated multiplication more efficiently. There are times when it is easier or faster to leave the expressions in exponential notation when multiplying or dividing. Let\u2019s look at rules that will allow you to do this.\r\n\r\nFor example, the notation [latex]5^{4}[\/latex]\u00a0can be expanded and written as [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or 625. And don\u2019t forget, the exponent only applies to the number immediately to its left, unless there are parentheses.\r\n\r\nWhat happens if you multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[\/latex]. Expanding each exponent, this can be rewritten as [latex]\\left(2\\cdot2\\cdot2\\right)\\left(2\\cdot2\\cdot2\\cdot2\\right)[\/latex] or [latex]2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2[\/latex]. In exponential form, you would write the product as [latex]2^{7}[\/latex]. Notice that 7 is the sum of the original two exponents, 3 and 4.\r\n\r\nWhat about [latex]{x}^{2}{x}^{6}[\/latex]? This can be written as [latex]\\left(x\\cdot{x}\\right)\\left(x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\right)=x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}[\/latex] or [latex]x^{8}[\/latex]. And, once again, 8 is the sum of the original two exponents. This concept can be generalized in the following way:\r\n<div class=\"textbox shaded\">\r\n<h3>The Product Rule for Exponents<\/h3>\r\nFor any number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>,\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].\r\n\r\nTo multiply exponential terms with the same base, add the exponents.\r\n\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"88\" height=\"78\" \/>Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. \u00a0For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says \"For any number <em>x<\/em>, and any integers <em>a<\/em> and <em>b<\/em>.\"\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex](a^{3})(a^{7})[\/latex]<\/p>\r\n[reveal-answer q=\"356596\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"356596\"]The base of both exponents is <i>a<\/i>, so the product rule applies.\r\n<p style=\"text-align: center;\">[latex]\\left(a^{3}\\right)\\left(a^{7}\\right)[\/latex]<\/p>\r\nAdd the exponents with a common base.\r\n<p style=\"text-align: center;\">[latex]a^{3+7}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(a^{3}\\right)\\left(a^{7}\\right) = a^{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen multiplying more complicated terms, multiply the coefficients and then multiply the variables.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]5a^{4}\\cdot7a^{6}[\/latex]<\/p>\r\n[reveal-answer q=\"215459\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"215459\"]Multiply the coefficients.\r\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{4}\\cdot{a}^{6}[\/latex]<\/p>\r\nThe base of both exponents is <i>a<\/i>, so the product rule applies. Add the exponents.\r\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{4+6}[\/latex]<\/p>\r\nAdd the exponents with a common base.\r\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{10}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]5a^{4}\\cdot7a^{6}=35a^{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659.png\" alt=\"traffic-sign-160659\" width=\"96\" height=\"83\" \/><\/h3>\r\nCaution! Do not try to apply this rule to sums.\r\n\r\nThink about the expression\u00a0[latex]\\left(2+3\\right)^{2}[\/latex]\r\n<p style=\"text-align: center;\">Does [latex]\\left(2+3\\right)^{2}[\/latex] equal [latex]2^{2}+3^{2}[\/latex]?<\/p>\r\nNo, it does not because of the order of operations!\r\n<p style=\"text-align: center;\">[latex]\\left(2+3\\right)^{2}=5^{2}=25[\/latex]<\/p>\r\n<p style=\"text-align: center;\">and<\/p>\r\n<p style=\"text-align: center;\">[latex]2^{2}+3^{2}=4+9=13[\/latex]<\/p>\r\nTherefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next).\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/hA9AT7QsXWo\r\n<h2 id=\"title2\">Use the quotient rule to divide exponential expressions<\/h2>\r\nLet\u2019s look at dividing terms containing exponential expressions. What happens if you 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\nYou can rewrite the expression as: [latex] \\displaystyle \\frac{4\\cdot 4\\cdot 4\\cdot 4\\cdot 4}{4\\cdot 4}[\/latex]. Then you can cancel the common factors of 4 in the numerator and denominator: [latex] \\displaystyle [\/latex]\r\n\r\nFinally, this expression can be rewritten as [latex]4^{3}[\/latex]\u00a0using exponential notation. Notice that the exponent, 3, is the difference between the two exponents in the original expression, 5 and 2.\r\n\r\nSo,\u00a0[latex] \\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[\/latex].\r\n\r\nBe careful that you subtract the exponent in the denominator from the exponent in the numerator.\r\n\r\nSo, to divide two exponential terms with the same base, subtract the exponents.\r\n<div class=\"textbox shaded\">\r\n<h3>The Quotient (Division) Rule for Exponents<\/h3>\r\nFor any non-zero number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>: [latex] \\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate. [latex] \\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}[\/latex]\r\n\r\n[reveal-answer q=\"96156\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"96156\"]These two exponents have the same base, 4. According to the Quotient Rule, you can subtract the power in the denominator from the power in the numerator.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{4}^{9-4}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}=4^{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]\r\n\r\n[reveal-answer q=\"23604\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"23604\"]Separate into numerical and variable factors.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\left( \\frac{12}{2} \\right)\\left( \\frac{{{x}^{4}}}{x} \\right)[\/latex]<\/p>\r\nSince the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle 6\\left( {{x}^{4-1}} \\right)[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]=[latex] \\displaystyle 6{{x}^{3}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"video2\" class=\"no-indent\" style=\"text-align: left;\">In the following video we show another example of how to use the quotient rule to divide exponential expressions<\/p>\r\nhttps:\/\/youtu.be\/Jmf-CPhm3XM\r\n<h2>Raise powers to powers<\/h2>\r\nAnother word for exponent is power. \u00a0You have likely seen or heard an example such as [latex]3^5[\/latex] can be described as 3 raised to the 5th power. In this section we will further expand our capabilities with exponents. We will learn what to do when a term with a\u00a0power\u00a0is raised to another power, and what to do when two numbers or variables are multiplied and both are raised to an exponent. \u00a0We will also learn what to do when numbers or variables that are divided are raised to a power. \u00a0We will begin by raising powers to powers.\r\n\r\nLet\u2019s simplify [latex]\\left(5^{2}\\right)^{4}[\/latex]. In this case, the base is [latex]5^2[\/latex]<sup>\u00a0<\/sup>and the exponent is 4, so you multiply [latex]5^{2}[\/latex]<sup>\u00a0<\/sup>four times: [latex]\\left(5^{2}\\right)^{4}=5^{2}\\cdot5^{2}\\cdot5^{2}\\cdot5^{2}=5^{8}[\/latex]<sup>\u00a0<\/sup>(using the Product Rule\u2014add the exponents).\r\n\r\n[latex]\\left(5^{2}\\right)^{4}[\/latex]<sup>\u00a0<\/sup>is a power of a power. It is the fourth power of 5 to the second power. And we saw above that the answer is [latex]5^{8}[\/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\\cdot4=8[\/latex].\r\n\r\nSo, [latex]\\left(5^{2}\\right)^{4}=5^{2\\cdot4}=5^{8}[\/latex]\u00a0(which equals 390,625, if you do the multiplication).\r\n\r\nLikewise, [latex]\\left(x^{4}\\right)^{3}=x^{4\\cdot3}=x^{12}[\/latex]\r\n\r\nThis leads to another rule for exponents\u2014the <b>Power Rule for Exponents<\/b>. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\\left(2^{3}\\right)^{5}=2^{15}[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>The Power Rule for Exponents<\/h3>\r\nFor any positive number <i>x<\/i> and integers <i>a<\/i> and <i>b<\/i>: [latex]\\left(x^{a}\\right)^{b}=x^{a\\cdot{b}}[\/latex].\r\n\r\nTake a moment to contrast how this is different from the product rule for exponents found on the previous page.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]6\\left(c^{4}\\right)^{2}[\/latex].\r\n\r\n[reveal-answer q=\"841688\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"841688\"]Since you are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses.\r\n<p style=\"text-align: center;\">[latex]6\\left(c^{4}\\right)^{2}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]6\\left(c^{4}\\right)^{2}=6c^{8}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Raise a product to a power<\/h3>\r\nSimplify this expression.\r\n<p style=\"text-align: center;\">[latex]\\left(2a\\right)^{4}=\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)=\\left(2\\cdot2\\cdot2\\cdot2\\right)\\left(a\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}\\right)=\\left(2^{4}\\right)\\left(a^{4}\\right)=16a^{4}[\/latex]<\/p>\r\nNotice that the exponent is applied to each factor of 2<i>a<\/i>. So, we can eliminate the middle steps.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(2a\\right)^{4} = \\left(2^{4}\\right)\\left(a^{4}\\right)\\text{, applying the }4\\text{ to each factor, }2\\text{ and }a\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=16a^{4}\\end{array}[\/latex]<\/p>\r\nThe product of two or more numbers raised to a power is equal to the product of each number raised to the same power.\r\n<div class=\"textbox shaded\">\r\n<h3>A Product Raised to a Power<\/h3>\r\nFor any nonzero numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>, [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].\r\n\r\nHow is this rule different from the power raised to a power rule? How is it different from the product rule for exponents on the previous page?\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify.\u00a0[latex]\\left(2yz\\right)^{6}[\/latex]\r\n\r\n[reveal-answer q=\"368657\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"368657\"]\r\n\r\nApply the exponent to each number in the product.\r\n\r\n[latex]2^{6}y^{6}z^{6}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]\\left(2yz\\right)^{6}=64y^{6}z^{6}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIf the variable has an exponent with it, use the Power Rule: multiply the exponents.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]\\left(\u22127a^{4}b\\right)^{2}[\/latex]\r\n\r\n[reveal-answer q=\"136794\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"136794\"]Apply the exponent 2 to each factor within the parentheses.\r\n\r\n[latex]\\left(\u22127\\right)^{2}\\left(a^{4}\\right)^{2}\\left(b\\right)^{2}[\/latex]\r\n\r\nSquare the coefficient and use the Power Rule to square\u00a0[latex]\\left(a^{4}\\right)^{2}[\/latex].\r\n<p style=\"text-align: center;\">[latex]49a^{4\\cdot2}b^{2}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]49a^{8}b^{2}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(-7a^{4}b\\right)^{2}=49a^{8}b^{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/Hgu9HKDHTUA\r\n<h3>Raise a quotient to a power<\/h3>\r\nNow let\u2019s look at what happens if you raise a quotient to a power. Remember that quotient means divide. Suppose you have [latex] \\displaystyle \\frac{3}{4}[\/latex] and raise it to the 3<sup>rd<\/sup> power.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{\\left( \\frac{3}{4} \\right)}^{3}}=\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)=\\frac{3\\cdot 3\\cdot 3}{4\\cdot 4\\cdot 4}=\\frac{{{3}^{3}}}{{{4}^{3}}}[\/latex]<\/p>\r\nYou can see that raising the quotient to the power of 3 can also be written as the numerator (3) to the power of 3, and the denominator (4) to the power of 3.\r\n\r\nSimilarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to power.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{\\left( \\frac{a}{b} \\right)}^{4}}=\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)=\\frac{a\\cdot a\\cdot a\\cdot a}{b\\cdot b\\cdot b\\cdot b}=\\frac{{{a}^{4}}}{{{b}^{4}}}[\/latex]<\/p>\r\nWhen a quotient is raised to a power, you can apply the power to the numerator and denominator individually, as shown below.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{\\left( \\frac{a}{b} \\right)}^{4}}=\\frac{{{a}^{4}}}{{{b}^{4}}}[\/latex]<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>A Quotient Raised to a Power<\/h3>\r\nFor any number <i>a<\/i>, any non-zero number <i>b<\/i>, and any integer <i>x<\/i>, [latex] \\displaystyle {\\left(\\frac{a}{b}\\right)}^{x}=\\frac{a^{x}}{b^{x}}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}[\/latex]\r\n\r\n[reveal-answer q=\"875425\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"875425\"]Apply the power to each factor individually.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{{{2}^{3}{\\left({x}^{2}\\right)}^{3}{y}^{3}}}{{{x}^{3}}}[\/latex]<\/p>\r\nSeparate into numerical and variable factors.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{2}^{3}}\\cdot \\frac{{{x}^{3\\cdot2}}}{{{x}^{3}}}\\cdot \\frac{{{y}^{3}}}{1}[\/latex]<\/p>\r\nSimplify by taking 2 to the third power and applying the Power and Quotient Rules for exponents\u2014multiply\u00a0and subtract the exponents of matching variables.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle 8\\cdot {{x}^{(6-3)}}\\cdot {{y}^{3}}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle 8{{x}^{3}}{{y}^{3}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}=8{{x}^{3}}{{y}^{3}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you will be shown examples of simplifying quotients that are raised to a power.\r\n\r\nhttps:\/\/youtu.be\/ZbxgDRV35dE\r\n<h2 id=\"title1\">Define and use the zero exponent rule<\/h2>\r\nWhen we defined the quotient rule, we only worked with expressions like the following: [latex]\\frac{{{4}^{9}}}{{{4}^{4}}}[\/latex], where the exponent in the numerator (up) was greater than the one in the denominator (down), so the final exponent after simplifying was always a positive number, and greater than zero. In this section, we will explore what happens when we apply the quotient rule for exponents and get a negative or zero exponent.\r\n<h3>What if the exponent is zero?<\/h3>\r\nTo see how this is defined, let us begin with an example. We will use the idea that dividing any number by itself gives a result of 1.\r\n<p style=\"text-align: center;\">[latex]\\frac{t^{8}}{t^{8}}=\\frac{\\cancel{t^{8}}}{\\cancel{t^{8}}}=1[\/latex]<\/p>\r\nIf we were to simplify the original expression using the quotient rule, we would have\r\n<p style=\"text-align: center;\">[latex]\\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[\/latex]<\/p>\r\nIf we equate the two answers, the result is [latex]{t}^{0}=1[\/latex]. This is true for any nonzero real number, or any variable representing a real number.\r\n<div style=\"text-align: center;\">[latex]{a}^{0}=1[\/latex]<\/div>\r\nThe 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 shaded\">\r\n<h3>Exponents of 0 or 1<\/h3>\r\nAny number or variable raised to a power of 1 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 0 is equal to 1\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\nAs done previously, to evaluate expressions containing exponents of 0 or 1, 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\n1\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<h2 id=\"title2\">Define and use the negative exponent rule<\/h2>\r\nWe proposed another question at the beginning of this section.\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 data-effect=\"italics\">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 1, leaving two <em>h<\/em>s multiplied in the denominator, and a numerator of 1.\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\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\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<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\n16\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the follwoing video you will see examples of simplifying expressions with negative exponents.\r\n\r\nhttps:\/\/youtu.be\/WvFlHjlIITg\r\n<h2>Simplify expressions using a combination of\u00a0exponent rules<\/h2>\r\nOnce the rules of exponents are understood, you can begin simplifying\u00a0more complicated expressions.\u00a0There are many applications and formulas that make use of exponents, and sometimes expressions can get pretty cluttered. Simplifying an expression before evaluating can often make the computation easier, as you will see in the following example which\u00a0makes use of the quotient rule to simplify before substituting 4 for x.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex] \\displaystyle \\frac{24{{x}^{8}}}{2{{x}^{5}}}[\/latex] when [latex]x=4[\/latex].\r\n\r\n[reveal-answer q=\"389478\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"389478\"]Separate into numerical and variable factors.\r\n\r\n[latex] \\displaystyle \\left( \\frac{24}{2} \\right)\\left( \\frac{{{x}^{8}}}{{{x}^{5}}} \\right)[\/latex]\r\n\r\nDivide coefficients, and subtract the exponents of the variables.\r\n\r\n[latex] \\displaystyle 12\\left( {{x}^{8-5}} \\right)[\/latex]\r\n\r\nSimplify.\r\n\r\n[latex] \\displaystyle 12{{x}^{3}}[\/latex]\r\n\r\nSubstitute the value 4 for the variable <i>x<\/i>.\r\n\r\n[latex] \\displaystyle (12)({{4}^{3}})=12\\cdot 64[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{24{{x}^{8}}}{2{{x}^{5}}}[\/latex] = 768\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\nEvaluate [latex] \\displaystyle \\frac{24{{x}^{8}}{{y}^{2}}}{{{(2{{x}^{3}}y)}^{2}}}[\/latex] when [latex]x=4[\/latex] and [latex]y=-2[\/latex].\r\n\r\n[reveal-answer q=\"261413\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"261413\"]In the denominator, notice that a product is being raised to a power.\r\n\r\nUse the rules of exponents to simplify the denominator.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle{\\left(2{x}^{3}y\\right)}^{2}={2}^{2}{x}^{3\\cdot 2}{y}^{2}={2}^{2}{x}^{6}{y}^{2}={4x^{6}y^{2}}[\/latex]<\/p>\r\nHere is the fraction with a simplified denominator:\r\n<p style=\"text-align: center;\">[latex]\\frac{24x^{8}y^{2}}{4x^{6}y^{2}}[\/latex]<\/p>\r\nSeparate into numerical and variable factors to simplify further.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\left( \\frac{24}{4} \\right)\\left( \\frac{{{x}^{8}}}{{{x}^{6}}} \\right)\\left( \\frac{{{y}^{2}}}{{{y}^{2}}}\\right)[\/latex]<\/p>\r\nDivide coefficients, use the Quotient Rule to divide the variables\u2014subtract the exponents.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle 6\\left( {{x}^{8-6}} \\right)\\left( {{y}^{2-2}} \\right)[\/latex]<\/p>\r\nSimplify. Remember that [latex]y^{0}[\/latex]\u00a0is 1.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle 6{{x}^{2}}{{y}^{0}}=6{{x}^{2}}[\/latex]<\/p>\r\nSubstitute the value 4 for the variable <i>x<\/i>.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle (6)({{4}^{2}})=6\\cdot 16[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{24{{x}^{8}}{{y}^{2}}}{{{(2{{x}^{3}}y)}^{2}}}=96[\/latex] when [latex]x=4[\/latex]\u00a0and [latex]y=-2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that you could have worked this problem by substituting 4 for <i>x<\/i> and 2 for <i>y<\/i> in the original expression. You would still get the answer of 96, but the computation would be much more complex. Notice that you didn\u2019t even need to use the value of <i>y <\/i>to evaluate the above expression.\r\n\r\nIn the following video you are shown examples of evaluating an exponential expression for given numbers.\r\n\r\nhttps:\/\/youtu.be\/mD06EyGja2w\r\n\r\nUsually, it is easier to simplify the expression before substituting any values for your variables, but you will get the same answer either way.\u00a0In the next examples, you will see how to simplify expressions using different combinations of the rules for exponents.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]a^{2}\\left(a^{5}\\right)^{3}[\/latex]\r\n\r\n[reveal-answer q=\"526055\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"526055\"]Raise [latex]a^{5}[\/latex]\u00a0to the power of 3 by multiplying the exponents together (the Power Rule).\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{a}^{2}}{{a}^{5\\cdot 3}}[\/latex]<\/p>\r\nSince the exponents share the same base, <i>a<\/i>, they can be combined (the Product Rule).\r\n<p style=\"text-align: center;\">[latex] \\displaystyle {{a}^{2}}{{a}^{15}}\\\\{{a}^{2+15}}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle {{a}^{2}}{{({{a}^{5}})}^{3}}={{a}^{17}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following examples require the use of all the exponent rules we have learned so far. Remember that the product, power, and quotient rules apply when your terms have the same base.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex] \\displaystyle \\frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}[\/latex]\r\n\r\n[reveal-answer q=\"43782\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"43782\"]Use the order of operations.\r\n\r\nParentheses, Exponents, Multiply\/ Divide, Add\/ Subtract\r\n\r\nThere is\u00a0nothing inside parentheses or brackets that we can simplify further, so we will evaluate exponents first.\r\n\r\nUse the Power Rule to simplify\u00a0[latex]\\left(a^{5}\\right)^{3}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left(a^{5}\\right)^{3}=a^{5\\cdot{3}}=a^{15}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">The expression now looks like this:<\/p>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle\\frac{{{a}^{2}}{{a}^{15}}}{8{{a}^{8}}}[\/latex]<\/p>\r\nNow we can multiply, using the Product Rule to simplify the numerator because the bases are the same.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle{{a}^{2}}{{a}^{15}}=a^{17}[\/latex], and the expression looks like this:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\frac{{{{a}^{17}}}}{{8{{a}^{8}}}}[\/latex]<\/p>\r\nNow we can divide using the Quotient Rule.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{{{a}^{17-8}}}{8}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}=\\frac{{{a}^{9}}}{8}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2 id=\"video2\" class=\"no-indent\" style=\"text-align: left;\">Simplify Expressions With Negative Exponents<\/h2>\r\nNow we will add the last layer to our exponent simplifying skills and practice simplifying compound expressions that have negative exponents in them. It is standard convention to write exponents as positive because it is easier for the user to understand the value associated with positive exponents, rather than negative exponents.\r\n\r\nUse the following summary of negative exponents to help you simplify expressions with negative exponents.\r\n<div class=\"textbox shaded\">\r\n<h3>Rules for Negative Exponents<\/h3>\r\nWith\u00a0<em>a<\/em>, <em>b<\/em>, <em>m<\/em>, and <em>n<\/em>\u00a0not equal to zero, and <em>m\u00a0<\/em>and\u00a0<em>n<\/em>\u00a0as integers, the following rules apply:\r\n<p style=\"text-align: center;\">[latex]a^{-m}=\\frac{1}{a^{m}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{1}{a^{-m}}=a^{m}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{a^{-n}}{b^{-m}}=\\frac{b^m}{a^n}[\/latex]<\/p>\r\n\r\n<\/div>\r\nWhen you are simplifying expressions that have many layers of exponents, it is often hard to know where to start. It is common to start in one of two ways:\r\n<ul>\r\n \t<li>Rewrite negative exponents as positive exponents<\/li>\r\n \t<li>Apply the product rule to eliminate any \"outer\" layer exponents such as in the following term: [latex]\\left(5y^3\\right)^2[\/latex]<\/li>\r\n<\/ul>\r\nWe will explore this idea with the following example:\r\n\r\nSimplify. [latex] \\displaystyle {{\\left( 4{{x}^{3}} \\right)}^{5}}\\cdot \\,\\,{{\\left( 2{{x}^{2}} \\right)}^{-4}}[\/latex]\r\n\r\nWrite your answer with positive exponents. The table below shows how to simplify the same expression in two different ways, rewriting negative exponents as positive first, and applying the product rule for exponents first. You will see that there is a column for each method that describes the exponent rule or other steps taken to simplify the expression.\r\n<table>\r\n<thead>\r\n<tr>\r\n<td>Rewrite with positive Exponents First<\/td>\r\n<td>Description of Steps Taken<\/td>\r\n<td>Apply the Product Rule for Exponents First<\/td>\r\n<td>Description of Steps Taken<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex] \\frac{\\left(4x^{3}\\right)^{5}}{\\left(2x^{2}\\right)^{4}}[\/latex]<\/td>\r\n<td>move the term [latex]{{\\left( 2{{x}^{2}} \\right)}^{-4}}[\/latex] to the denominator with a positive exponent<\/td>\r\n<td>[latex] \\left(4^5x^{15}\\right)\\left(2^{-4}x^{-8}\\right)[\/latex]<\/td>\r\n<td>\u00a0Apply the exponent of 5 to each term in expression on the left, and the exponent of -4 to each term in the expression on the right.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]\\frac{\\left(4^5x^{15}\\right)}{\\left(2^4x^{8}\\right)}[\/latex]<\/td>\r\n<td>Use the product rule to apply the outer exponents to the terms inside each set of parentheses.<\/td>\r\n<td>[latex]\\left(4^5\\right)\\left(2^{-4}\\right)\\left(x^{15}\\cdot{x^{-8}}\\right)[\/latex]<\/td>\r\n<td>Regroup the numerical terms\u00a0and the variables to make combining like terms easier<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]\\left(\\frac{4^5}{2^4}\\right)\\left(\\frac{x^{15}}{x^{8}}\\right)[\/latex]<\/td>\r\n<td>Regroup the numerical terms\u00a0and the variables to make combining like terms easier<\/td>\r\n<td>[latex]\\left(4^5\\right)\\left(2^{-4}\\right)\\left(x^{15-8}\\right)[\/latex]<\/td>\r\n<td>\u00a0Use the rule for multiplying terms with exponents to simplify the x terms<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]\\left(\\frac{4^5}{2^4}\\right)\\left(x^{15-8}\\right)[\/latex]<\/td>\r\n<td>Use the quotient rule to simplify the x terms<\/td>\r\n<td>[latex]\\left(\\frac{4^5}{2^4}\\right)\\left(x^{7}\\right)[\/latex]<\/td>\r\n<td>\u00a0Rewrite all the negative exponents with positive exponents<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]\\left(\\frac{1,024}{16}\\right)\\left(x^{7}\\right)[\/latex]<\/td>\r\n<td>Expand the numerical terms<\/td>\r\n<td>[latex]\\left(\\frac{1,024}{16}\\right)\\left(x^{7}\\right)[\/latex]<\/td>\r\n<td>\u00a0Expand the numerical terms<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0\u00a0[latex]64x^{7}[\/latex]<\/td>\r\n<td>Divide the\u00a0numerical terms<\/td>\r\n<td>\u00a0[latex]64x^{7}[\/latex]<\/td>\r\n<td>\u00a0Divide the\u00a0numerical terms<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIf you compare the two columns that describe the steps that were taken to simplify the expression, you will see that they are all nearly the same, except the order is changed slightly. Neither way is better or more correct than the other, it truly is a matter of preference.\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]\\frac{\\left(t^{3}\\right)^2}{\\left(t^2\\right)^{-8}}[\/latex]\r\n\r\nWrite your answer with positive exponents.\r\n[reveal-answer q=\"494485\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"494485\"]\r\n\r\nWe can either rewrite this expression with positive exponents first\u00a0or use the Product Raised to a Power Rule first.\r\n\r\nLet's start by simplifying the numerator and denominator using the Product Raised to a Power Rule.\r\n\r\nNumerator: [latex]\\left(t^{3}\\right)^2=t^{3\\cdot{2}}=t^6[\/latex]\r\n\r\nDenominator: [latex]\\left(t^2\\right)^{-8}=t^{2\\cdot{-8}}=t^{-16}[\/latex]\r\n\r\nNow the expression looks like this:\r\n<p style=\"text-align: center;\">[latex]\\frac{t^6}{t^{-16}}[\/latex]<\/p>\r\nWe can use the quotient rule because we have the same base.\r\n\r\nQuotient Rule:\u00a0[latex]\\frac{t^6}{t^{-16}}=t^{6-\\left(-16\\right)}=t^{6+16}=t^{22}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{\\left(t^{3}\\right)^2}{\\left(t^2\\right)^{-8}}=t^{22}[\/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{\\left(5x\\right)^{-2}y}{x^3y^{-1}}[\/latex]\r\n\r\nWrite your answer with positive exponents.\r\n[reveal-answer q=\"798985\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"798985\"]\r\n\r\nThis time, let's start by rewriting the terms in the expression so they have positive exponents. The terms with negative exponents in the top\u00a0will go to the\u00a0bottom of the fraction, and the terms with negative exponents in the bottom will go to the top.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{\\left(5x\\right)^{-2}y}{x^3y^{-1}}\\\\\\text{ }\\\\=\\frac{\\left({y^{1}}\\right)y}{x^3\\left(5x\\right)^{2}}\\end{array}[\/latex]<\/p>\r\nNote how we left the single y term in the top because it did not have a negative exponent on it, and we left the [latex]x^3[\/latex] term in the bottom because it did not have a negative exponent on it.\r\n\r\nNow\u00a0we can apply the Product Raised to a Power Rule:\r\n<p style=\"text-align: center;\">[latex]\\frac{yy^{1}}{5^{2}x^3x^{2}}[\/latex]<\/p>\r\nUse the product rule to simplify further:\r\n<p style=\"text-align: center;\">[latex]\\frac{yy^{1}}{5^{2}x^3x^{2}}=\\frac{y^2}{25x^{3+2}}=\\frac{y^2}{25x^5}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We can't simplify any further, so our answer is<\/p>\r\n\r\n<h4 style=\"text-align: left;\">Answer<\/h4>\r\n[latex]\\frac{\\left(5x\\right)^{-2}y}{x^3y^{-1}}=\\frac{y^2}{25x^5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next section, you will learn how to write very large and very small numbers using exponents. This practice is widely used in science and engineering.\r\n<h2>Summary<\/h2>\r\n<ul>\r\n \t<li>Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.<\/li>\r\n \t<li>The product rule for exponents:\u00a0For any number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>,\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<\/li>\r\n \t<li>The quotient rule for exponents:\u00a0For any non-zero number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>: [latex] \\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]<\/li>\r\n \t<li>The power rule for exponents:\r\n<ol>\r\n \t<li>For any nonzero numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>, [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].<\/li>\r\n \t<li>For any number <i>a<\/i>, any non-zero number <i>b<\/i>, and any integer <i>x<\/i>, [latex] \\displaystyle {\\left(\\frac{a}{b}\\right)}^{x}=\\frac{a^{x}}{b^{x}}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Product and Quotient Rules\n<ul>\n<li>Use the product rule to multiply exponential expressions<\/li>\n<li>Use the quotient rule to divide exponential expressions<\/li>\n<\/ul>\n<\/li>\n<li>The Power Rule for Exponents\n<ul>\n<li>Use the power rule to simplify expressions involving\u00a0products, quotients, and exponents<\/li>\n<\/ul>\n<\/li>\n<li>Negative and Zero Exponents\n<ul>\n<li>Define and use the zero exponent rule<\/li>\n<li>Define and use the negative exponent rule<\/li>\n<\/ul>\n<\/li>\n<li>Simplify Expressions Using the Exponent Rules\n<ul>\n<li>Simplify expressions using a combination of the exponent rules<\/li>\n<li>Simplify compound exponential expressions with negative exponents<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"attachment_4445\" style=\"width: 292px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4445\" class=\"wp-image-4445\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/31173815\/Screen-Shot-2016-05-31-at-10.38.21-AM-150x150.png\" alt=\"Image of a woman taking a picture with a camera repeated five times in different colors.\" width=\"282\" height=\"282\" \/><\/p>\n<p id=\"caption-attachment-4445\" class=\"wp-caption-text\">Repeated Image<\/p>\n<\/div>\n<h2>Anatomy of exponential\u00a0terms<\/h2>\n<p>We use exponential notation to write repeated multiplication. For example [latex]10\\cdot10\\cdot10[\/latex] can be written more succinctly as [latex]10^{3}[\/latex]. The 10 in [latex]10^{3}[\/latex]<sup>\u00a0<\/sup>is called the <b>base<\/b>. The 3 in [latex]10^{3}[\/latex]<sup>\u00a0<\/sup>is called the <b>exponent<\/b>. The expression [latex]10^{3}[\/latex] is called the exponential expression. Knowing the names for the parts of an exponential expression or term will help you learn how to perform mathematical operations on them.<\/p>\n<p style=\"text-align: center;\">[latex]\\text{base}\\rightarrow10^{3\\leftarrow\\text{exponent}}[\/latex]<\/p>\n<p>[latex]10^{3}[\/latex] is read as \u201c10 to the third power\u201d or \u201c10 cubed.\u201d It means [latex]10\\cdot10\\cdot10[\/latex], or 1,000.<\/p>\n<p>[latex]8^{2}[\/latex]\u00a0is read as \u201c8 to the second power\u201d or \u201c8 squared.\u201d It means [latex]8\\cdot8[\/latex], or 64.<\/p>\n<p>[latex]5^{4}[\/latex]\u00a0is read as \u201c5 to the fourth power.\u201d It means [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or 625.<\/p>\n<p>[latex]b^{5}[\/latex]\u00a0is read as \u201c<i>b<\/i> to the fifth power.\u201d It means [latex]{b}\\cdot{b}\\cdot{b}\\cdot{b}\\cdot{b}[\/latex]. Its value will depend on the value of <i>b<\/i>.<\/p>\n<p>The exponent applies only to the number that it is next to. Therefore, in the expression [latex]xy^{4}[\/latex],\u00a0only the <i>y<\/i> is affected by the 4. [latex]xy^{4}[\/latex]\u00a0means [latex]{x}\\cdot{y}\\cdot{y}\\cdot{y}\\cdot{y}[\/latex]. The <em>x<\/em> in this term is a <strong>coefficient<\/strong> of <em>y<\/em>.<\/p>\n<p>If the exponential expression is negative, such as [latex]\u22123^{4}[\/latex], it means [latex]\u2013\\left(3\\cdot3\\cdot3\\cdot3\\right)[\/latex] or [latex]\u221281[\/latex].<\/p>\n<p>If [latex]\u22123[\/latex] is to be the base, it must be written as [latex]\\left(\u22123\\right)^{4}[\/latex], which means [latex]\u22123\\cdot\u22123\\cdot\u22123\\cdot\u22123[\/latex], or 81.<\/p>\n<p>Likewise,\u00a0[latex]\\left(\u2212x\\right)^{4}=\\left(\u2212x\\right)\\cdot\\left(\u2212x\\right)\\cdot\\left(\u2212x\\right)\\cdot\\left(\u2212x\\right)=x^{4}[\/latex], while [latex]\u2212x^{4}=\u2013\\left(x\\cdot x\\cdot x\\cdot x\\right)[\/latex].<\/p>\n<p>You can see that there is quite a difference, so you have to be very careful! The following examples show how to identify the base and the exponent, as well as how to identify the expanded and exponential format of writing repeated multiplication.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Identify the exponent and the base in the following terms, then simplify:<\/p>\n<ol>\n<li>[latex]7^{2}[\/latex]<\/li>\n<li>[latex]{\\left(\\frac{1}{2}\\right)}^{3}[\/latex]<\/li>\n<li>[latex]2x^{3}[\/latex]<\/li>\n<li>[latex]\\left(-5\\right)^{2}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q211363\">Show Solution<\/span><\/p>\n<p style=\"text-align: left;\">\n<div id=\"q211363\" class=\"hidden-answer\" style=\"display: none\">\n<p>1) [latex]7^{2}[\/latex]<\/p>\n<p>The exponent in this term is 2 and the base is 7. To simplify, expand the term: [latex]7^{2}=7\\cdot{7}=49[\/latex]<\/p>\n<p>2) [latex]{\\left(\\frac{1}{2}\\right)}^{3}[\/latex]<\/p>\n<p>The exponent on this term is 3, and the base is [latex]\\frac{1}{2}[\/latex]. To simplify, expand the multiplication and remember how to multiply fractions: [latex]{\\left(\\frac{1}{2}\\right)}^{3}=\\frac{1}{2}\\cdot{\\frac{1}{2}}\\cdot{\\frac{1}{2}}=\\frac{1}{16}[\/latex]<\/p>\n<p>3) \u00a0[latex]2x^{3}[\/latex]<\/p>\n<p>The exponent on this term is 3, and the base is x, the 2 is not getting the exponent because there are no parentheses that tell us it is. \u00a0This term is in its most simplified form.<\/p>\n<p>4)\u00a0[latex]\\left(-5\\right)^{2}[\/latex]<\/p>\n<p>The exponent on this terms is 2 and the base is [latex]-5[\/latex]. To simplify, expand the multiplication: [latex]\\left(-5\\right)^{2}=-5\\cdot{-5}=25[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p class=\"no-indent\">In the following video you are provided more examples of applying exponents to various bases.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify  Basic Exponential Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ocedY91LHKU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Evaluate expressions<\/h3>\n<p>Evaluating expressions containing exponents is the same as evaluating the linear expressions from earlier in the course. You substitute the value of the variable into the expression and simplify.<\/p>\n<p>You can use the order of operations\u00a0to evaluate the expressions containing exponents. First, evaluate anything in Parentheses or grouping symbols. Next, look for Exponents, followed by Multiplication and Division (reading from left to right), and lastly, Addition and Subtraction (again, reading from left to right).<\/p>\n<p>So, when you evaluate the expression [latex]5x^{3}[\/latex]\u00a0if [latex]x=4[\/latex], first substitute the value 4 for the variable <i>x<\/i>. Then evaluate, using order of operations.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate\u00a0[latex]5x^{3}[\/latex]<i>\u00a0<\/i>if [latex]x=4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q411363\">Show Solution<\/span><\/p>\n<div id=\"q411363\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute 4 for the variable <i>x<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]5\\cdot4^{3}[\/latex]<\/p>\n<p>Evaluate [latex]4^{3}[\/latex]. Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]5\\left(4\\cdot4\\cdot4\\right)=5\\cdot64=320[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]5x^{3}=320[\/latex]\u00a0when [latex]x=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the example below, notice the how adding parentheses can change the outcome when you are simplifying terms with exponents.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]\\left(5x\\right)^{3}[\/latex]\u00a0if [latex]x=4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q362021\">Show Solution<\/span><\/p>\n<div id=\"q362021\" class=\"hidden-answer\" style=\"display: none\">Substitute 4 for the variable <i>x<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(5\\cdot4\\right)3[\/latex]<\/p>\n<p>Multiply inside the parentheses, then apply the exponent\u2014following the rules of PEMDAS.<\/p>\n<p style=\"text-align: center;\">[latex]20^{3}[\/latex]<\/p>\n<p>Evaluate [latex]20^{3}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]20\\cdot20\\cdot20=8,000[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(5x\\right)3=8,000[\/latex] when [latex]x=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The addition of parentheses made quite a difference!\u00a0Parentheses allow you to apply an exponent to variables or numbers that are multiplied, divided, added, or subtracted to each other.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]x^{3}[\/latex] if [latex]x=\u22124[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q86290\">Show Solution<\/span><\/p>\n<div id=\"q86290\" class=\"hidden-answer\" style=\"display: none\">Substitute [latex]\u22124[\/latex] for the variable <i>x<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(\u22124\\right)^{3}[\/latex]<\/p>\n<p>Evaluate. Note how placing parentheses around the [latex]\u22124[\/latex] means the negative sign also gets multiplied.<\/p>\n<p style=\"text-align: center;\">[latex]\u22124\\cdot\u22124\\cdot\u22124[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\u22124\\cdot\u22124\\cdot\u22124=\u221264[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]x^{3}=\u221264[\/latex] when [latex]x=\u22124[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"75\" height=\"66\" \/><\/p>\n<p>Caution! Whether to include a negative sign as part of a base or not often leads to confusion. To clarify\u00a0whether a negative sign is applied before or after the exponent, here is an example.<\/p>\n<p>&nbsp;<\/p>\n<p>What is the difference in the way you would evaluate these two terms?<\/p>\n<ol>\n<li style=\"text-align: left;\">[latex]-{3}^{2}[\/latex]<\/li>\n<li style=\"text-align: left;\">[latex]{\\left(-3\\right)}^{2}[\/latex]<\/li>\n<\/ol>\n<p>To evaluate 1), you would apply the exponent to the three first, then apply the negative sign last, like this:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}-\\left({3}^{2}\\right)\\\\=-\\left(9\\right) = -9\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">To evaluate 2), you would apply the exponent to the 3 and the negative sign:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}{\\left(-3\\right)}^{2}\\\\=\\left(-3\\right)\\cdot\\left(-3\\right)\\\\={ 9}\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">The key to remembering this is to follow the order of operations. The first expression does not include parentheses so you would apply the exponent to the integer 3 first, then apply the negative sign. The second expression includes parentheses, so hopefully you will remember that the negative sign also gets squared.<\/p>\n<\/div>\n<p id=\"video0\" class=\"no-indent\" style=\"text-align: left;\">In the next sections, you will learn how to simplify expressions that contain exponents. Come back to this page if you forget how to apply the order of operations to a term with exponents, or forget which is the base and which is the exponent!<\/p>\n<p class=\"no-indent\" style=\"text-align: left;\">In the following video you are provided with examples of evaluating exponential expressions for a given number.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Evaluate Basic Exponential Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/pQNz8IpVVg0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title1\">Use the product rule to multiply exponential expressions<\/h2>\n<p><b>Exponential notation<\/b> was developed to write repeated multiplication more efficiently. There are times when it is easier or faster to leave the expressions in exponential notation when multiplying or dividing. Let\u2019s look at rules that will allow you to do this.<\/p>\n<p>For example, the notation [latex]5^{4}[\/latex]\u00a0can be expanded and written as [latex]5\\cdot5\\cdot5\\cdot5[\/latex], or 625. And don\u2019t forget, the exponent only applies to the number immediately to its left, unless there are parentheses.<\/p>\n<p>What happens if you multiply two numbers in exponential form with the same base? Consider the expression [latex]{2}^{3}{2}^{4}[\/latex]. Expanding each exponent, this can be rewritten as [latex]\\left(2\\cdot2\\cdot2\\right)\\left(2\\cdot2\\cdot2\\cdot2\\right)[\/latex] or [latex]2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2\\cdot2[\/latex]. In exponential form, you would write the product as [latex]2^{7}[\/latex]. Notice that 7 is the sum of the original two exponents, 3 and 4.<\/p>\n<p>What about [latex]{x}^{2}{x}^{6}[\/latex]? This can be written as [latex]\\left(x\\cdot{x}\\right)\\left(x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\right)=x\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}\\cdot{x}[\/latex] or [latex]x^{8}[\/latex]. And, once again, 8 is the sum of the original two exponents. This concept can be generalized in the following way:<\/p>\n<div class=\"textbox shaded\">\n<h3>The Product Rule for Exponents<\/h3>\n<p>For any number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>,\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<\/p>\n<p>To multiply exponential terms with the same base, add the exponents.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"88\" height=\"78\" \/>Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. \u00a0For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says &#8220;For any number <em>x<\/em>, and any integers <em>a<\/em> and <em>b<\/em>.&#8221;<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex](a^{3})(a^{7})[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q356596\">Show Solution<\/span><\/p>\n<div id=\"q356596\" class=\"hidden-answer\" style=\"display: none\">The base of both exponents is <i>a<\/i>, so the product rule applies.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a^{3}\\right)\\left(a^{7}\\right)[\/latex]<\/p>\n<p>Add the exponents with a common base.<\/p>\n<p style=\"text-align: center;\">[latex]a^{3+7}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(a^{3}\\right)\\left(a^{7}\\right) = a^{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When multiplying more complicated terms, multiply the coefficients and then multiply the variables.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]5a^{4}\\cdot7a^{6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q215459\">Show Solution<\/span><\/p>\n<div id=\"q215459\" class=\"hidden-answer\" style=\"display: none\">Multiply the coefficients.<\/p>\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{4}\\cdot{a}^{6}[\/latex]<\/p>\n<p>The base of both exponents is <i>a<\/i>, so the product rule applies. Add the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{4+6}[\/latex]<\/p>\n<p>Add the exponents with a common base.<\/p>\n<p style=\"text-align: center;\">[latex]35\\cdot{a}^{10}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]5a^{4}\\cdot7a^{6}=35a^{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659.png\" alt=\"traffic-sign-160659\" width=\"96\" height=\"83\" \/><\/h3>\n<p>Caution! Do not try to apply this rule to sums.<\/p>\n<p>Think about the expression\u00a0[latex]\\left(2+3\\right)^{2}[\/latex]<\/p>\n<p style=\"text-align: center;\">Does [latex]\\left(2+3\\right)^{2}[\/latex] equal [latex]2^{2}+3^{2}[\/latex]?<\/p>\n<p>No, it does not because of the order of operations!<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2+3\\right)^{2}=5^{2}=25[\/latex]<\/p>\n<p style=\"text-align: center;\">and<\/p>\n<p style=\"text-align: center;\">[latex]2^{2}+3^{2}=4+9=13[\/latex]<\/p>\n<p>Therefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next).<\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Simplify Exponential Expressions Using the Product Property of Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hA9AT7QsXWo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title2\">Use the quotient rule to divide exponential expressions<\/h2>\n<p>Let\u2019s look at dividing terms containing exponential expressions. What happens if you 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>You can rewrite the expression as: [latex]\\displaystyle \\frac{4\\cdot 4\\cdot 4\\cdot 4\\cdot 4}{4\\cdot 4}[\/latex]. Then you can cancel the common factors of 4 in the numerator and denominator: [latex]\\displaystyle[\/latex]<\/p>\n<p>Finally, this expression can be rewritten as [latex]4^{3}[\/latex]\u00a0using exponential notation. Notice that the exponent, 3, is the difference between the two exponents in the original expression, 5 and 2.<\/p>\n<p>So,\u00a0[latex]\\displaystyle \\frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[\/latex].<\/p>\n<p>Be careful that you subtract the exponent in the denominator from the exponent in the numerator.<\/p>\n<p>So, to divide two exponential terms with the same base, subtract the exponents.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Quotient (Division) Rule for Exponents<\/h3>\n<p>For any non-zero number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>: [latex]\\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate. [latex]\\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q96156\">Show Solution<\/span><\/p>\n<div id=\"q96156\" class=\"hidden-answer\" style=\"display: none\">These two exponents have the same base, 4. According to the Quotient Rule, you can subtract the power in the denominator from the power in the numerator.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{4}^{9-4}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{{{4}^{9}}}{{{4}^{4}}}=4^{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q23604\">Show Solution<\/span><\/p>\n<div id=\"q23604\" class=\"hidden-answer\" style=\"display: none\">Separate into numerical and variable factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\left( \\frac{12}{2} \\right)\\left( \\frac{{{x}^{4}}}{x} \\right)[\/latex]<\/p>\n<p>Since the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle 6\\left( {{x}^{4-1}} \\right)[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{12{{x}^{4}}}{2x}[\/latex]=[latex]\\displaystyle 6{{x}^{3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"video2\" class=\"no-indent\" style=\"text-align: left;\">In the following video we show another example of how to use the quotient rule to divide exponential expressions<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex: Simplify Exponential Expressions Using the Quotient Property of Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Jmf-CPhm3XM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Raise powers to powers<\/h2>\n<p>Another word for exponent is power. \u00a0You have likely seen or heard an example such as [latex]3^5[\/latex] can be described as 3 raised to the 5th power. In this section we will further expand our capabilities with exponents. We will learn what to do when a term with a\u00a0power\u00a0is raised to another power, and what to do when two numbers or variables are multiplied and both are raised to an exponent. \u00a0We will also learn what to do when numbers or variables that are divided are raised to a power. \u00a0We will begin by raising powers to powers.<\/p>\n<p>Let\u2019s simplify [latex]\\left(5^{2}\\right)^{4}[\/latex]. In this case, the base is [latex]5^2[\/latex]<sup>\u00a0<\/sup>and the exponent is 4, so you multiply [latex]5^{2}[\/latex]<sup>\u00a0<\/sup>four times: [latex]\\left(5^{2}\\right)^{4}=5^{2}\\cdot5^{2}\\cdot5^{2}\\cdot5^{2}=5^{8}[\/latex]<sup>\u00a0<\/sup>(using the Product Rule\u2014add the exponents).<\/p>\n<p>[latex]\\left(5^{2}\\right)^{4}[\/latex]<sup>\u00a0<\/sup>is a power of a power. It is the fourth power of 5 to the second power. And we saw above that the answer is [latex]5^{8}[\/latex]. Notice that the new exponent is the same as the product of the original exponents: [latex]2\\cdot4=8[\/latex].<\/p>\n<p>So, [latex]\\left(5^{2}\\right)^{4}=5^{2\\cdot4}=5^{8}[\/latex]\u00a0(which equals 390,625, if you do the multiplication).<\/p>\n<p>Likewise, [latex]\\left(x^{4}\\right)^{3}=x^{4\\cdot3}=x^{12}[\/latex]<\/p>\n<p>This leads to another rule for exponents\u2014the <b>Power Rule for Exponents<\/b>. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, [latex]\\left(2^{3}\\right)^{5}=2^{15}[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>The Power Rule for Exponents<\/h3>\n<p>For any positive number <i>x<\/i> and integers <i>a<\/i> and <i>b<\/i>: [latex]\\left(x^{a}\\right)^{b}=x^{a\\cdot{b}}[\/latex].<\/p>\n<p>Take a moment to contrast how this is different from the product rule for exponents found on the previous page.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]6\\left(c^{4}\\right)^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q841688\">Show Solution<\/span><\/p>\n<div id=\"q841688\" class=\"hidden-answer\" style=\"display: none\">Since you are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses.<\/p>\n<p style=\"text-align: center;\">[latex]6\\left(c^{4}\\right)^{2}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]6\\left(c^{4}\\right)^{2}=6c^{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Raise a product to a power<\/h3>\n<p>Simplify this expression.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2a\\right)^{4}=\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)\\left(2a\\right)=\\left(2\\cdot2\\cdot2\\cdot2\\right)\\left(a\\cdot{a}\\cdot{a}\\cdot{a}\\cdot{a}\\right)=\\left(2^{4}\\right)\\left(a^{4}\\right)=16a^{4}[\/latex]<\/p>\n<p>Notice that the exponent is applied to each factor of 2<i>a<\/i>. So, we can eliminate the middle steps.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(2a\\right)^{4} = \\left(2^{4}\\right)\\left(a^{4}\\right)\\text{, applying the }4\\text{ to each factor, }2\\text{ and }a\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=16a^{4}\\end{array}[\/latex]<\/p>\n<p>The product of two or more numbers raised to a power is equal to the product of each number raised to the same power.<\/p>\n<div class=\"textbox shaded\">\n<h3>A Product Raised to a Power<\/h3>\n<p>For any nonzero numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>, [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].<\/p>\n<p>How is this rule different from the power raised to a power rule? How is it different from the product rule for exponents on the previous page?<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify.\u00a0[latex]\\left(2yz\\right)^{6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q368657\">Show Solution<\/span><\/p>\n<div id=\"q368657\" class=\"hidden-answer\" style=\"display: none\">\n<p>Apply the exponent to each number in the product.<\/p>\n<p>[latex]2^{6}y^{6}z^{6}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(2yz\\right)^{6}=64y^{6}z^{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>If the variable has an exponent with it, use the Power Rule: multiply the exponents.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\left(\u22127a^{4}b\\right)^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q136794\">Show Solution<\/span><\/p>\n<div id=\"q136794\" class=\"hidden-answer\" style=\"display: none\">Apply the exponent 2 to each factor within the parentheses.<\/p>\n<p>[latex]\\left(\u22127\\right)^{2}\\left(a^{4}\\right)^{2}\\left(b\\right)^{2}[\/latex]<\/p>\n<p>Square the coefficient and use the Power Rule to square\u00a0[latex]\\left(a^{4}\\right)^{2}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]49a^{4\\cdot2}b^{2}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]49a^{8}b^{2}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(-7a^{4}b\\right)^{2}=49a^{8}b^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex: Simplify Exponential Expressions Using the Power Property of Exponents\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Hgu9HKDHTUA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Raise a quotient to a power<\/h3>\n<p>Now let\u2019s look at what happens if you raise a quotient to a power. Remember that quotient means divide. Suppose you have [latex]\\displaystyle \\frac{3}{4}[\/latex] and raise it to the 3<sup>rd<\/sup> power.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{\\left( \\frac{3}{4} \\right)}^{3}}=\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)\\left( \\frac{3}{4} \\right)=\\frac{3\\cdot 3\\cdot 3}{4\\cdot 4\\cdot 4}=\\frac{{{3}^{3}}}{{{4}^{3}}}[\/latex]<\/p>\n<p>You can see that raising the quotient to the power of 3 can also be written as the numerator (3) to the power of 3, and the denominator (4) to the power of 3.<\/p>\n<p>Similarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to power.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{\\left( \\frac{a}{b} \\right)}^{4}}=\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)\\left( \\frac{a}{b} \\right)=\\frac{a\\cdot a\\cdot a\\cdot a}{b\\cdot b\\cdot b\\cdot b}=\\frac{{{a}^{4}}}{{{b}^{4}}}[\/latex]<\/p>\n<p>When a quotient is raised to a power, you can apply the power to the numerator and denominator individually, as shown below.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{\\left( \\frac{a}{b} \\right)}^{4}}=\\frac{{{a}^{4}}}{{{b}^{4}}}[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>A Quotient Raised to a Power<\/h3>\n<p>For any number <i>a<\/i>, any non-zero number <i>b<\/i>, and any integer <i>x<\/i>, [latex]\\displaystyle {\\left(\\frac{a}{b}\\right)}^{x}=\\frac{a^{x}}{b^{x}}[\/latex].<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q875425\">Show Solution<\/span><\/p>\n<div id=\"q875425\" class=\"hidden-answer\" style=\"display: none\">Apply the power to each factor individually.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{2}^{3}{\\left({x}^{2}\\right)}^{3}{y}^{3}}}{{{x}^{3}}}[\/latex]<\/p>\n<p>Separate into numerical and variable factors.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{2}^{3}}\\cdot \\frac{{{x}^{3\\cdot2}}}{{{x}^{3}}}\\cdot \\frac{{{y}^{3}}}{1}[\/latex]<\/p>\n<p>Simplify by taking 2 to the third power and applying the Power and Quotient Rules for exponents\u2014multiply\u00a0and subtract the exponents of matching variables.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle 8\\cdot {{x}^{(6-3)}}\\cdot {{y}^{3}}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle 8{{x}^{3}}{{y}^{3}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}=8{{x}^{3}}{{y}^{3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you will be shown examples of simplifying quotients that are raised to a power.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex: Simplify Fractions Raised to Powers (Positive Exponents Only) Version 1\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ZbxgDRV35dE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title1\">Define and use the zero exponent rule<\/h2>\n<p>When we defined the quotient rule, we only worked with expressions like the following: [latex]\\frac{{{4}^{9}}}{{{4}^{4}}}[\/latex], where the exponent in the numerator (up) was greater than the one in the denominator (down), so the final exponent after simplifying was always a positive number, and greater than zero. In this section, we will explore what happens when we apply the quotient rule for exponents and get a negative or zero exponent.<\/p>\n<h3>What if the exponent is zero?<\/h3>\n<p>To see how this is defined, let us begin with an example. We will use the idea that dividing any number by itself gives a result of 1.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{t^{8}}{t^{8}}=\\frac{\\cancel{t^{8}}}{\\cancel{t^{8}}}=1[\/latex]<\/p>\n<p>If we were to simplify the original expression using the quotient rule, we would have<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}[\/latex]<\/p>\n<p>If we equate the two answers, the result is [latex]{t}^{0}=1[\/latex]. This is true for any nonzero real number, or any variable representing a real number.<\/p>\n<div style=\"text-align: center;\">[latex]{a}^{0}=1[\/latex]<\/div>\n<p>The 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 shaded\">\n<h3>Exponents of 0 or 1<\/h3>\n<p>Any number or variable raised to a power of 1 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 0 is equal to 1<\/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>As done previously, to evaluate expressions containing exponents of 0 or 1, 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>1<\/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-7\" 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<h2 id=\"title2\">Define and use the negative exponent rule<\/h2>\n<p>We proposed another question at the beginning of this section.\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 data-effect=\"italics\">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 1, leaving two <em>h<\/em>s multiplied in the denominator, and a numerator of 1.<\/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>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 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<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>16<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the follwoing video you will see examples of simplifying expressions with negative exponents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-8\" 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 using a combination of\u00a0exponent rules<\/h2>\n<p>Once the rules of exponents are understood, you can begin simplifying\u00a0more complicated expressions.\u00a0There are many applications and formulas that make use of exponents, and sometimes expressions can get pretty cluttered. Simplifying an expression before evaluating can often make the computation easier, as you will see in the following example which\u00a0makes use of the quotient rule to simplify before substituting 4 for x.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]\\displaystyle \\frac{24{{x}^{8}}}{2{{x}^{5}}}[\/latex] when [latex]x=4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q389478\">Show Solution<\/span><\/p>\n<div id=\"q389478\" class=\"hidden-answer\" style=\"display: none\">Separate into numerical and variable factors.<\/p>\n<p>[latex]\\displaystyle \\left( \\frac{24}{2} \\right)\\left( \\frac{{{x}^{8}}}{{{x}^{5}}} \\right)[\/latex]<\/p>\n<p>Divide coefficients, and subtract the exponents of the variables.<\/p>\n<p>[latex]\\displaystyle 12\\left( {{x}^{8-5}} \\right)[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p>[latex]\\displaystyle 12{{x}^{3}}[\/latex]<\/p>\n<p>Substitute the value 4 for the variable <i>x<\/i>.<\/p>\n<p>[latex]\\displaystyle (12)({{4}^{3}})=12\\cdot 64[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{24{{x}^{8}}}{2{{x}^{5}}}[\/latex] = 768<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]\\displaystyle \\frac{24{{x}^{8}}{{y}^{2}}}{{{(2{{x}^{3}}y)}^{2}}}[\/latex] when [latex]x=4[\/latex] and [latex]y=-2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q261413\">Show Solution<\/span><\/p>\n<div id=\"q261413\" class=\"hidden-answer\" style=\"display: none\">In the denominator, notice that a product is being raised to a power.<\/p>\n<p>Use the rules of exponents to simplify the denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle{\\left(2{x}^{3}y\\right)}^{2}={2}^{2}{x}^{3\\cdot 2}{y}^{2}={2}^{2}{x}^{6}{y}^{2}={4x^{6}y^{2}}[\/latex]<\/p>\n<p>Here is the fraction with a simplified denominator:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{24x^{8}y^{2}}{4x^{6}y^{2}}[\/latex]<\/p>\n<p>Separate into numerical and variable factors to simplify further.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\left( \\frac{24}{4} \\right)\\left( \\frac{{{x}^{8}}}{{{x}^{6}}} \\right)\\left( \\frac{{{y}^{2}}}{{{y}^{2}}}\\right)[\/latex]<\/p>\n<p>Divide coefficients, use the Quotient Rule to divide the variables\u2014subtract the exponents.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle 6\\left( {{x}^{8-6}} \\right)\\left( {{y}^{2-2}} \\right)[\/latex]<\/p>\n<p>Simplify. Remember that [latex]y^{0}[\/latex]\u00a0is 1.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle 6{{x}^{2}}{{y}^{0}}=6{{x}^{2}}[\/latex]<\/p>\n<p>Substitute the value 4 for the variable <i>x<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle (6)({{4}^{2}})=6\\cdot 16[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{24{{x}^{8}}{{y}^{2}}}{{{(2{{x}^{3}}y)}^{2}}}=96[\/latex] when [latex]x=4[\/latex]\u00a0and [latex]y=-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that you could have worked this problem by substituting 4 for <i>x<\/i> and 2 for <i>y<\/i> in the original expression. You would still get the answer of 96, but the computation would be much more complex. Notice that you didn\u2019t even need to use the value of <i>y <\/i>to evaluate the above expression.<\/p>\n<p>In the following video you are shown examples of evaluating an exponential expression for given numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-9\" title=\"Simplify and Evaluate Compound Exponential Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/mD06EyGja2w?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Usually, it is easier to simplify the expression before substituting any values for your variables, but you will get the same answer either way.\u00a0In the next examples, you will see how to simplify expressions using different combinations of the rules for exponents.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]a^{2}\\left(a^{5}\\right)^{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q526055\">Show Solution<\/span><\/p>\n<div id=\"q526055\" class=\"hidden-answer\" style=\"display: none\">Raise [latex]a^{5}[\/latex]\u00a0to the power of 3 by multiplying the exponents together (the Power Rule).<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{a}^{2}}{{a}^{5\\cdot 3}}[\/latex]<\/p>\n<p>Since the exponents share the same base, <i>a<\/i>, they can be combined (the Product Rule).<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle {{a}^{2}}{{a}^{15}}\\\\{{a}^{2+15}}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle {{a}^{2}}{{({{a}^{5}})}^{3}}={{a}^{17}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following examples require the use of all the exponent rules we have learned so far. Remember that the product, power, and quotient rules apply when your terms have the same base.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\displaystyle \\frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q43782\">Show Solution<\/span><\/p>\n<div id=\"q43782\" class=\"hidden-answer\" style=\"display: none\">Use the order of operations.<\/p>\n<p>Parentheses, Exponents, Multiply\/ Divide, Add\/ Subtract<\/p>\n<p>There is\u00a0nothing inside parentheses or brackets that we can simplify further, so we will evaluate exponents first.<\/p>\n<p>Use the Power Rule to simplify\u00a0[latex]\\left(a^{5}\\right)^{3}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a^{5}\\right)^{3}=a^{5\\cdot{3}}=a^{15}[\/latex]<\/p>\n<p style=\"text-align: left;\">The expression now looks like this:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\frac{{{a}^{2}}{{a}^{15}}}{8{{a}^{8}}}[\/latex]<\/p>\n<p>Now we can multiply, using the Product Rule to simplify the numerator because the bases are the same.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle{{a}^{2}}{{a}^{15}}=a^{17}[\/latex], and the expression looks like this:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\frac{{{{a}^{17}}}}{{8{{a}^{8}}}}[\/latex]<\/p>\n<p>Now we can divide using the Quotient Rule.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{{{a}^{17-8}}}{8}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}=\\frac{{{a}^{9}}}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2 id=\"video2\" class=\"no-indent\" style=\"text-align: left;\">Simplify Expressions With Negative Exponents<\/h2>\n<p>Now we will add the last layer to our exponent simplifying skills and practice simplifying compound expressions that have negative exponents in them. It is standard convention to write exponents as positive because it is easier for the user to understand the value associated with positive exponents, rather than negative exponents.<\/p>\n<p>Use the following summary of negative exponents to help you simplify expressions with negative exponents.<\/p>\n<div class=\"textbox shaded\">\n<h3>Rules for Negative Exponents<\/h3>\n<p>With\u00a0<em>a<\/em>, <em>b<\/em>, <em>m<\/em>, and <em>n<\/em>\u00a0not equal to zero, and <em>m\u00a0<\/em>and\u00a0<em>n<\/em>\u00a0as integers, the following rules apply:<\/p>\n<p style=\"text-align: center;\">[latex]a^{-m}=\\frac{1}{a^{m}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1}{a^{-m}}=a^{m}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{a^{-n}}{b^{-m}}=\\frac{b^m}{a^n}[\/latex]<\/p>\n<\/div>\n<p>When you are simplifying expressions that have many layers of exponents, it is often hard to know where to start. It is common to start in one of two ways:<\/p>\n<ul>\n<li>Rewrite negative exponents as positive exponents<\/li>\n<li>Apply the product rule to eliminate any &#8220;outer&#8221; layer exponents such as in the following term: [latex]\\left(5y^3\\right)^2[\/latex]<\/li>\n<\/ul>\n<p>We will explore this idea with the following example:<\/p>\n<p>Simplify. [latex]\\displaystyle {{\\left( 4{{x}^{3}} \\right)}^{5}}\\cdot \\,\\,{{\\left( 2{{x}^{2}} \\right)}^{-4}}[\/latex]<\/p>\n<p>Write your answer with positive exponents. The table below shows how to simplify the same expression in two different ways, rewriting negative exponents as positive first, and applying the product rule for exponents first. You will see that there is a column for each method that describes the exponent rule or other steps taken to simplify the expression.<\/p>\n<table>\n<thead>\n<tr>\n<td>Rewrite with positive Exponents First<\/td>\n<td>Description of Steps Taken<\/td>\n<td>Apply the Product Rule for Exponents First<\/td>\n<td>Description of Steps Taken<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\frac{\\left(4x^{3}\\right)^{5}}{\\left(2x^{2}\\right)^{4}}[\/latex]<\/td>\n<td>move the term [latex]{{\\left( 2{{x}^{2}} \\right)}^{-4}}[\/latex] to the denominator with a positive exponent<\/td>\n<td>[latex]\\left(4^5x^{15}\\right)\\left(2^{-4}x^{-8}\\right)[\/latex]<\/td>\n<td>\u00a0Apply the exponent of 5 to each term in expression on the left, and the exponent of -4 to each term in the expression on the right.<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]\\frac{\\left(4^5x^{15}\\right)}{\\left(2^4x^{8}\\right)}[\/latex]<\/td>\n<td>Use the product rule to apply the outer exponents to the terms inside each set of parentheses.<\/td>\n<td>[latex]\\left(4^5\\right)\\left(2^{-4}\\right)\\left(x^{15}\\cdot{x^{-8}}\\right)[\/latex]<\/td>\n<td>Regroup the numerical terms\u00a0and the variables to make combining like terms easier<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]\\left(\\frac{4^5}{2^4}\\right)\\left(\\frac{x^{15}}{x^{8}}\\right)[\/latex]<\/td>\n<td>Regroup the numerical terms\u00a0and the variables to make combining like terms easier<\/td>\n<td>[latex]\\left(4^5\\right)\\left(2^{-4}\\right)\\left(x^{15-8}\\right)[\/latex]<\/td>\n<td>\u00a0Use the rule for multiplying terms with exponents to simplify the x terms<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]\\left(\\frac{4^5}{2^4}\\right)\\left(x^{15-8}\\right)[\/latex]<\/td>\n<td>Use the quotient rule to simplify the x terms<\/td>\n<td>[latex]\\left(\\frac{4^5}{2^4}\\right)\\left(x^{7}\\right)[\/latex]<\/td>\n<td>\u00a0Rewrite all the negative exponents with positive exponents<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]\\left(\\frac{1,024}{16}\\right)\\left(x^{7}\\right)[\/latex]<\/td>\n<td>Expand the numerical terms<\/td>\n<td>[latex]\\left(\\frac{1,024}{16}\\right)\\left(x^{7}\\right)[\/latex]<\/td>\n<td>\u00a0Expand the numerical terms<\/td>\n<\/tr>\n<tr>\n<td>\u00a0\u00a0[latex]64x^{7}[\/latex]<\/td>\n<td>Divide the\u00a0numerical terms<\/td>\n<td>\u00a0[latex]64x^{7}[\/latex]<\/td>\n<td>\u00a0Divide the\u00a0numerical terms<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If you compare the two columns that describe the steps that were taken to simplify the expression, you will see that they are all nearly the same, except the order is changed slightly. Neither way is better or more correct than the other, it truly is a matter of preference.<\/p>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify [latex]\\frac{\\left(t^{3}\\right)^2}{\\left(t^2\\right)^{-8}}[\/latex]<\/p>\n<p>Write your answer with positive exponents.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q494485\">Show Solution<\/span><\/p>\n<div id=\"q494485\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can either rewrite this expression with positive exponents first\u00a0or use the Product Raised to a Power Rule first.<\/p>\n<p>Let&#8217;s start by simplifying the numerator and denominator using the Product Raised to a Power Rule.<\/p>\n<p>Numerator: [latex]\\left(t^{3}\\right)^2=t^{3\\cdot{2}}=t^6[\/latex]<\/p>\n<p>Denominator: [latex]\\left(t^2\\right)^{-8}=t^{2\\cdot{-8}}=t^{-16}[\/latex]<\/p>\n<p>Now the expression looks like this:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{t^6}{t^{-16}}[\/latex]<\/p>\n<p>We can use the quotient rule because we have the same base.<\/p>\n<p>Quotient Rule:\u00a0[latex]\\frac{t^6}{t^{-16}}=t^{6-\\left(-16\\right)}=t^{6+16}=t^{22}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{\\left(t^{3}\\right)^2}{\\left(t^2\\right)^{-8}}=t^{22}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify [latex]\\frac{\\left(5x\\right)^{-2}y}{x^3y^{-1}}[\/latex]<\/p>\n<p>Write your answer with positive exponents.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q798985\">Show Solution<\/span><\/p>\n<div id=\"q798985\" class=\"hidden-answer\" style=\"display: none\">\n<p>This time, let&#8217;s start by rewriting the terms in the expression so they have positive exponents. The terms with negative exponents in the top\u00a0will go to the\u00a0bottom of the fraction, and the terms with negative exponents in the bottom will go to the top.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\frac{\\left(5x\\right)^{-2}y}{x^3y^{-1}}\\\\\\text{ }\\\\=\\frac{\\left({y^{1}}\\right)y}{x^3\\left(5x\\right)^{2}}\\end{array}[\/latex]<\/p>\n<p>Note how we left the single y term in the top because it did not have a negative exponent on it, and we left the [latex]x^3[\/latex] term in the bottom because it did not have a negative exponent on it.<\/p>\n<p>Now\u00a0we can apply the Product Raised to a Power Rule:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{yy^{1}}{5^{2}x^3x^{2}}[\/latex]<\/p>\n<p>Use the product rule to simplify further:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{yy^{1}}{5^{2}x^3x^{2}}=\\frac{y^2}{25x^{3+2}}=\\frac{y^2}{25x^5}[\/latex]<\/p>\n<p style=\"text-align: left;\">We can&#8217;t simplify any further, so our answer is<\/p>\n<h4 style=\"text-align: left;\">Answer<\/h4>\n<p>[latex]\\frac{\\left(5x\\right)^{-2}y}{x^3y^{-1}}=\\frac{y^2}{25x^5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next section, you will learn how to write very large and very small numbers using exponents. This practice is widely used in science and engineering.<\/p>\n<h2>Summary<\/h2>\n<ul>\n<li>Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.<\/li>\n<li>The product rule for exponents:\u00a0For any number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>,\u00a0[latex]\\left(x^{a}\\right)\\left(x^{b}\\right) = x^{a+b}[\/latex].<\/li>\n<li>The quotient rule for exponents:\u00a0For any non-zero number <i>x<\/i> and any integers <i>a<\/i> and <i>b<\/i>: [latex]\\displaystyle \\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[\/latex]<\/li>\n<li>The power rule for exponents:\n<ol>\n<li>For any nonzero numbers <i>a<\/i> and <i>b<\/i> and any integer <i>x<\/i>, [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].<\/li>\n<li>For any number <i>a<\/i>, any non-zero number <i>b<\/i>, and any integer <i>x<\/i>, [latex]\\displaystyle {\\left(\\frac{a}{b}\\right)}^{x}=\\frac{a^{x}}{b^{x}}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ul>\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-4294\">\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>Simplify Basic Exponential Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ocedY91LHKU\">https:\/\/youtu.be\/ocedY91LHKU<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: Repeated Image. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Evaluate Basic Exponential Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/pQNz8IpVVg0\">https:\/\/youtu.be\/pQNz8IpVVg0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Evaluate and Simplify Expressions Using the Zero Exponent Rule . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/jKihp_DVDa0\">https:\/\/youtu.be\/jKihp_DVDa0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify and Evaluate Compound Exponential Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/mD06EyGja2w\">https:\/\/youtu.be\/mD06EyGja2w<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify Compound Exponential Expressions 1. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/3cSNZR9pegk\">https:\/\/youtu.be\/3cSNZR9pegk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Simplify Exponential Expressions Using the Power Property of Exponents. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Hgu9HKDHTUA\">https:\/\/youtu.be\/Hgu9HKDHTUA<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Simplify Fractions Raised to Powers (Positive Exponents Only) Version 1. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ZbxgDRV35dE\">https:\/\/youtu.be\/ZbxgDRV35dE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra. <strong>Authored by<\/strong>: Jay Abrams et, al.. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/\">https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Negative Exponents - Basics. 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