{"id":5940,"date":"2016-09-07T21:52:47","date_gmt":"2016-09-07T21:52:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/beginalgebra\/?post_type=chapter&#038;p=5940"},"modified":"2018-01-03T23:56:37","modified_gmt":"2018-01-03T23:56:37","slug":"introduction-to-exponents","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/beginalgebra\/chapter\/introduction-to-exponents\/","title":{"raw":"Terms and Expressions With Exponents","rendered":"Terms and Expressions With Exponents"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the components of a term containing integer exponents<\/li>\r\n \t<li>Evaluate expressions containing integer exponents<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\nA lingua franca is a common language used to make communication possible between people who speak different languages. Math, as a general idea, is sometimes thought of as an example of a common language because formulas and equations don't rely on fluency in a specific language.\r\n\r\nBut even within mathematics a\u00a0common language is needed in order to communicate mathematical ideas clearly and efficiently. <strong>Exponential notation <\/strong>(remember this can also called scientific notation)\u00a0was developed to write repeated multiplication more efficiently. For example, growth occurs in living organisms by the division of cells. One type of cell divides 2 times in an hour. So in 12 hours, the cell will divide [latex]2\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}[\/latex] times. This can be written more efficiently as [latex]2^{12}[\/latex]. Expressing it\u00a0this way is a much more efficient and clear way to express the ways cells divide.\r\n\r\nIn this section we will learn how to simplify and perform mathematical operations such as multiplication and division on terms that have exponents. We will also learn how to use scientific notation to represent very large or very small numbers, and perform mathematical operations on them.\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<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[\/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\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<p class=\"no-indent\" style=\"text-align: left;\"><\/p>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the components of a term containing integer exponents<\/li>\n<li>Evaluate expressions containing integer exponents<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<p>A lingua franca is a common language used to make communication possible between people who speak different languages. Math, as a general idea, is sometimes thought of as an example of a common language because formulas and equations don&#8217;t rely on fluency in a specific language.<\/p>\n<p>But even within mathematics a\u00a0common language is needed in order to communicate mathematical ideas clearly and efficiently. <strong>Exponential notation <\/strong>(remember this can also called scientific notation)\u00a0was developed to write repeated multiplication more efficiently. For example, growth occurs in living organisms by the division of cells. One type of cell divides 2 times in an hour. So in 12 hours, the cell will divide [latex]2\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}\\cdot{2}[\/latex] times. This can be written more efficiently as [latex]2^{12}[\/latex]. Expressing it\u00a0this way is a much more efficient and clear way to express the ways cells divide.<\/p>\n<p>In this section we will learn how to simplify and perform mathematical operations such as multiplication and division on terms that have exponents. We will also learn how to use scientific notation to represent very large or very small numbers, and perform mathematical operations on them.<\/p>\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<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<\/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>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<p class=\"no-indent\" style=\"text-align: left;\">\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-5940\">\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><\/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><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Simplify Basic Exponential Expressions\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/ocedY91LHKU\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot: Repeated Image\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Evaluate Basic Exponential Expressions\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/pQNz8IpVVg0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5940","chapter","type-chapter","status-publish","hentry"],"part":867,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/beginalgebra\/wp-json\/pressbooks\/v2\/chapters\/5940","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/beginalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/beginalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/beginalgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/beginalgebra\/wp-json\/pressbooks\/v2\/chapters\/5940\/revisions"}],"predecessor-version":[{"id":6027,"href":"https:\/\/courses.lumenlearning.com\/beginalgebra\/wp-json\/pressbooks\/v2\/chapters\/5940\/revisions\/6027"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/beginalgebra\/wp-json\/pressbooks\/v2\/parts\/867"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/beginalgebra\/wp-json\/pressbooks\/v2\/chapters\/5940\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/beginalgebra\/wp-json\/wp\/v2\/media?parent=5940"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/beginalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=5940"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/beginalgebra\/wp-json\/wp\/v2\/contributor?post=5940"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/beginalgebra\/wp-json\/wp\/v2\/license?post=5940"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}