{"id":40,"date":"2023-06-21T13:22:26","date_gmt":"2023-06-21T13:22:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/properties-of-real-numbers\/"},"modified":"2023-07-04T04:53:14","modified_gmt":"2023-07-04T04:53:14","slug":"properties-of-real-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/properties-of-real-numbers\/","title":{"raw":"\u25aa   Properties of Real Numbers","rendered":"\u25aa   Properties of Real Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the order of operations to simplify an algebraic expression.<\/li>\r\n \t<li>Use properties of real numbers to simplify algebraic expressions.<\/li>\r\n<\/ul>\r\n<\/div>\r\nWhen we multiply a number by itself, we square it or raise it to a power of 2. For example, [latex]{4}^{2}=4\\cdot 4=16[\/latex]. We can raise any number to any power. In general, the <strong>exponential notation<\/strong> [latex]{a}^{n}[\/latex] means that the number or variable [latex]a[\/latex] is used as a factor [latex]n[\/latex] times.\r\n<div style=\"text-align: center;\">[latex]a^{n}=a\\cdot a\\cdot a\\cdot \\dots \\cdot a[\/latex]<\/div>\r\nIn this notation, [latex]{a}^{n}[\/latex] is read as the <em>n<\/em>th power of [latex]a[\/latex], where [latex]a[\/latex] is called the <strong>base<\/strong> and [latex]n[\/latex] is called the <strong>exponent<\/strong>. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, [latex]24+6\\cdot \\frac{2}{3}-{4}^{2}[\/latex] is a mathematical expression.\r\n\r\nTo evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the <strong>order of operations<\/strong>. This is a sequence of rules for evaluating such expressions.\r\n\r\nRecall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.\r\n\r\nThe next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.\r\n\r\nLet\u2019s take a look at the expression provided.\r\n<div style=\"text-align: center;\">[latex]24+6\\cdot \\dfrac{2}{3}-{4}^{2}[\/latex]<\/div>\r\nThere are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify [latex]{4}^{2}[\/latex] as 16.\r\n<div style=\"text-align: center;\">\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}24+6\\cdot\\frac{2}{3}-4^{2} \\\\ 24+6\\cdot\\frac{2}{3}-16\\end{gathered}[\/latex]<\/div>\r\nNext, perform multiplication or division, left to right.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}24+6\\cdot\\frac{2}{3}-16 \\\\ 24+4-16\\end{gathered}[\/latex]<\/div>\r\nLastly, perform addition or subtraction, left to right.\r\n<div style=\"text-align: center;\">[latex]\\begin{gathered}24+4-16 \\\\ 28-16 \\\\ 12\\end{gathered}[\/latex]<\/div>\r\n<\/div>\r\nTherefore, [latex]24+6\\cdot \\dfrac{2}{3}-{4}^{2}=12[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>How was the fraction above simplified?<\/h3>\r\nRecall, when mulitplying fractions, we multiply the numerators together and place the result over the product of the denominators.\r\n<ul>\r\n \t<li>[latex]\\dfrac{a}{b}\\cdot\\dfrac{c}{d}=\\dfrac{ac}{bd}[\/latex]<\/li>\r\n \t<li>[latex]6\\cdot\\dfrac{2}{3}=\\dfrac{6}{1}\\cdot\\dfrac{2}{3}=\\dfrac{12}{3}=4[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\nFor some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Order of Operations<\/h3>\r\nOperations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym <strong>PEMDAS<\/strong>:\r\n\r\n<strong>P<\/strong>(arentheses)\r\n\r\n<strong>E<\/strong>(xponents)\r\n\r\n<strong>M<\/strong>(ultiplication) and <strong>D<\/strong>(ivision)\r\n\r\n<strong>A<\/strong>(ddition) and <strong>S<\/strong>(ubtraction)\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a mathematical expression, simplify it using the order of operations.<\/h3>\r\n<ol>\r\n \t<li>Simplify any expressions within grouping symbols.<\/li>\r\n \t<li>Simplify any expressions containing exponents or radicals.<\/li>\r\n \t<li>Perform any multiplication and division in order, from left to right.<\/li>\r\n \t<li>Perform any addition and subtraction in order, from left to right.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Order of Operations<\/h3>\r\nUse the order of operations to evaluate each of the following expressions.\r\n<ol>\r\n \t<li>[latex]{\\left(3\\cdot 2\\right)}^{2}-4\\left(6+2\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{{5}^{2}-4}{7}-\\sqrt{11 - 2}[\/latex]<\/li>\r\n \t<li>[latex]6-|5 - 8|+3\\left(4 - 1\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{14 - 3\\cdot 2}{2\\cdot 5-{3}^{2}}[\/latex]<\/li>\r\n \t<li>[latex]7\\left(5\\cdot 3\\right)-2\\left[\\left(6 - 3\\right)-{4}^{2}\\right]+1[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"371324\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"371324\"]\r\n\r\n1.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(3\\cdot 2\\right)^{2} &amp; =\\left(6\\right)^{2}-4\\left(8\\right) &amp;&amp; \\text{Simplify parentheses} \\\\ &amp; =36-4\\left(8\\right) &amp;&amp; \\text{Simplify exponent} \\\\ &amp; =36-32 &amp;&amp; \\text{Simplify multiplication} \\\\ &amp; =4 &amp;&amp; \\text{Simplify subtraction}\\end{align}[\/latex]<\/p>\r\n2.\r\n<p class=\"p1\" style=\"text-align: center;\"><span class=\"s1\">[latex]\\begin{align}\\frac{5^{2}-4}{7}-\\sqrt{11-2} &amp; =\\frac{5^{2}-4}{7}-\\sqrt{9} &amp;&amp; \\text{Simplify grouping systems (radical)} \\\\ &amp; =\\frac{5^{2}-4}{7}-3 &amp;&amp; \\text{Simplify radical} \\\\ &amp; =\\frac{25-4}{7}-3 &amp;&amp; \\text{Simplify exponent} \\\\ &amp; =\\frac{21}{7}-3 &amp;&amp; \\text{Simplify subtraction in numerator} \\\\ &amp; =3-3 &amp;&amp; \\text{Simplify division} \\\\ &amp; =0 &amp;&amp; \\text{Simplify subtraction}\\end{align}[\/latex]<\/span><\/p>\r\nNote that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.\r\n\r\n3.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}6-|5-8|+3\\left(4-1\\right) &amp; =6-|-3|+3\\left(3\\right) &amp;&amp; \\text{Simplify inside grouping system} \\\\ &amp; =6-3+3\\left(3\\right) &amp;&amp; \\text{Simplify absolute value} \\\\ &amp; =6-3+9 &amp;&amp; \\text{Simplify multiplication} \\\\ &amp; =3+9 &amp;&amp; \\text{Simplify subtraction} \\\\ &amp; =12 &amp;&amp; \\text{Simplify addition}\\end{align}[\/latex]<\/p>\r\n4.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{14-3\\cdot2}{2\\cdot5-3^{2}} &amp; =\\frac{14-3\\cdot2}{2\\cdot5-9} &amp;&amp; \\text{Simplify exponent} \\\\ &amp; =\\frac{14-6}{10-9} &amp;&amp; \\text{Simplify products} \\\\ &amp; =\\frac{8}{1} &amp;&amp; \\text{Simplify quotient} \\\\ &amp; =8 &amp;&amp; \\text{Simplify quotient}\\end{align}[\/latex]\r\nIn this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.<\/p>\r\n5.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}7\\left(5\\cdot3\\right)-2[\\left(6-3\\right)-4^{2}]+1 &amp; =7\\left(15\\right)-2[\\left(3\\right)-4^{2}]+1 &amp;&amp; \\text{Simplify inside parentheses} \\\\ &amp; 7\\left(15\\right)-2\\left(3-16\\right)+1 &amp;&amp; \\text{Simplify exponent} \\\\ &amp; =7\\left(15\\right)-2\\left(-13\\right)+1 &amp;&amp; \\text{Subtract} \\\\ &amp; =105+26+1 &amp;&amp; \\text{Multiply} \\\\ &amp; =132 &amp;&amp; \\text{Add}\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse the order of operations to evaluate the following expression:\r\n[latex]\\sqrt{(21-17)}-2(7+9^{2})+\\dfrac{23-5}{3^2}[\/latex]\r\n\r\n[reveal-answer q=\"371333\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"371333\"]\r\n<p style=\"padding-left: 30px;\">[latex]=\\sqrt{(4)}-2(7+81)+\\dfrac{23-5}{3^2}[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]=\\sqrt{(4)}-2(88)+\\dfrac{18}{9}[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]=2-176+2[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]=-172[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=259&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=99379&amp;theme=oea&amp;iframe_resize_id=mom11[\/embed]\r\n\r\n<\/div>\r\nWatch the following video for more examples of using the order of operations to simplify an expression.\r\n<div class=\"textbox examples\">\r\n<h3>Recall: simplifying fractions<\/h3>\r\nTo simplify a fraction, look for common factors in the numerator and the denominator. In the video below, a fraction, [latex]\\dfrac{14}{26}[\/latex] must be simplified. Since\u00a0[latex]\\dfrac{14}{26}=\\dfrac{2\\cdot7}{2\\cdot13}[\/latex], and since [latex]\\dfrac{2}{2} = 1[\/latex], this fraction simplifies to [latex]\\dfrac{7}{13}[\/latex].\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/9suc63qB96o\r\n<h2>Using Properties of Real Numbers<\/h2>\r\nFor some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.\r\n<h3>Commutative Properties<\/h3>\r\nThe <strong>commutative property of addition<\/strong> states that numbers may be added in any order without affecting the sum.\r\n<div style=\"text-align: center;\">[latex]a+b=b+a[\/latex]<\/div>\r\nWe can better see this relationship when using real numbers.\r\n<div style=\"text-align: center;\">[latex]\\left(-2\\right)+7=5\\text{ and }7+\\left(-2\\right)=5[\/latex]<\/div>\r\nSimilarly, the <strong>commutative property of multiplication<\/strong> states that numbers may be multiplied in any order without affecting the product.\r\n<div style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\r\nAgain, consider an example with real numbers.\r\n<div style=\"text-align: center;\">[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44\\text{ and }\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/div>\r\nIt is important to note that neither subtraction nor division is commutative. For example, [latex]17 - 5[\/latex] is not the same as [latex]5 - 17[\/latex]. Similarly, [latex]20\\div 5\\ne 5\\div 20[\/latex].\r\n<h3>Associative Properties<\/h3>\r\nThe <strong>associative property of multiplication<\/strong> tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.\r\n<div style=\"text-align: center;\">[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/div>\r\nConsider this example.\r\n<div style=\"text-align: center;\">[latex]\\left(3\\cdot4\\right)\\cdot5=60\\text{ and }3\\cdot\\left(4\\cdot5\\right)=60[\/latex]<\/div>\r\nThe <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.\r\n<div style=\"text-align: center;\">[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/div>\r\nThis property can be especially helpful when dealing with negative integers. Consider this example.\r\n<div style=\"text-align: center;\">[latex][15+\\left(-9\\right)]+23=29\\text{ and }15+[\\left(-9\\right)+23]=29[\/latex]<\/div>\r\nAre subtraction and division associative? Review these examples.\r\n<div style=\"text-align: center;\">\r\n<div style=\"text-align: center;\">[latex]\\begin{align}8-\\left(3-15\\right) &amp; \\stackrel{?}{=}\\left(8-3\\right)-15 \\\\ 8-\\left(-12\\right) &amp; \\stackrel{?}=5-15 \\\\ 20 &amp; \\neq 20-10 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\n<div style=\"text-align: center;\">[latex]\\begin{align}64\\div\\left(8\\div4\\right)&amp;\\stackrel{?}{=}\\left(64\\div8\\right)\\div4 \\\\ 64\\div2 &amp; \\stackrel{?}{=}8\\div4 \\\\ 32 &amp; \\neq 2 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\n<\/div>\r\nAs we can see, neither subtraction nor division is associative.\r\n<h3>Distributive Property<\/h3>\r\nThe <strong>distributive property<\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.\r\n<div style=\"text-align: center;\">[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/div>\r\nThis property combines both addition and multiplication (and is the only property to do so). Let us consider an example.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223815\/CNX_CAT_Figure_01_01_003.jpg\" alt=\"The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer.\" \/>\r\n\r\nNote that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by \u20137, and adding the products.\r\n\r\nTo be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.\r\n<div style=\"text-align: center;\">[latex]\\begin{align} 6+\\left(3\\cdot 5\\right)&amp; \\stackrel{?}{=} \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ 6+\\left(15\\right)&amp; \\stackrel{?}{=} \\left(9\\right)\\cdot \\left(11\\right) \\\\ 21&amp; \\ne 99 \\end{align}[\/latex]<\/div>\r\nMultiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.\r\n\r\nA special case of the distributive property occurs when a sum of terms is subtracted.\r\n<div style=\"text-align: center;\">[latex]a-b=a+\\left(-b\\right)[\/latex]<\/div>\r\nFor example, consider the difference [latex]12-\\left(5+3\\right)[\/latex]. We can rewrite the difference of the two terms 12 and [latex]\\left(5+3\\right)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\\left(5+3\\right)[\/latex], we add the opposite.\r\n<div style=\"text-align: center;\">[latex]12+\\left(-1\\right)\\cdot \\left(5+3\\right)[\/latex]<\/div>\r\nNow, distribute [latex]-1[\/latex] and simplify the result.\r\n<div style=\"text-align: center;\">\r\n<div style=\"text-align: center;\">[latex]\\begin{align}12+\\left(-1\\right)\\cdot\\left(5+3\\right)&amp;=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3] \\\\&amp;=12+(-5-3) \\\\&amp;=12+\\left(-8\\right) \\\\&amp;=4 \\end{align}[\/latex]<\/div>\r\nThis seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}12-\\left(5+3\\right) &amp;=12+\\left(-5-3\\right) \\\\ &amp;=12+\\left(-8\\right) \\\\ &amp;=4\\end{align}[\/latex]<\/div>\r\n<\/div>\r\n<h3>Identity Properties<\/h3>\r\nThe <strong>identity property of addition<\/strong> states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.\r\n<div style=\"text-align: center;\">\u00a0[latex]a+0=a[\/latex]<\/div>\r\n<div><\/div>\r\nThe <strong>identity property of multiplication<\/strong> states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.\r\n<div style=\"text-align: center;\">[latex]a\\cdot 1=a[\/latex]<\/div>\r\nFor example, we have [latex]\\left(-6\\right)+0=-6[\/latex] and [latex]23\\cdot 1=23[\/latex]. There are no exceptions for these properties; they work for every real number, including 0 and 1.\r\n<h3>Inverse Properties<\/h3>\r\nThe <strong>inverse property of addition<\/strong> states that, for every real number <em>a<\/em>, there is a unique number, called the additive inverse (or opposite), denoted\u2212<em>a<\/em>, that, when added to the original number, results in the additive identity, 0.\r\n<div style=\"text-align: center;\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\r\nFor example, if [latex]a=-8[\/latex], the additive inverse is 8, since [latex]\\left(-8\\right)+8=0[\/latex].\r\n\r\nThe <strong>inverse property of multiplication<\/strong> holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number <em>a<\/em>, there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\\frac{1}{a}[\/latex], that, when multiplied by the original number, results in the multiplicative identity, 1.\r\n<div style=\"text-align: center;\">\r\n<div style=\"text-align: center;\">[latex]a\\cdot \\dfrac{1}{a}=1[\/latex]<\/div>\r\nFor example, if [latex]a=-\\frac{2}{3}[\/latex], the reciprocal, denoted [latex]\\frac{1}{a}[\/latex], is [latex]-\\frac{3}{2}[\/latex]\u00a0because\r\n<div style=\"text-align: center;\">[latex]a\\cdot \\dfrac{1}{a}=\\left(-\\dfrac{2}{3}\\right)\\cdot \\left(-\\dfrac{3}{2}\\right)=1[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Properties of Real Numbers<\/h3>\r\nThe following properties hold for real numbers <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>.\r\n<table summary=\"A table with six rows and three columns. The first entry of the first row is blank while the remaining columns read: Addition and Multiplication. The first entry of the second row reads: Commutative Property. The second column entry reads a plus b equals b plus a. The third column entry reads a times b equals b times a. The first entry of the third row reads Associative Property. The second column entry reads: a plus the quantity b plus c in parenthesis equals the quantity a plus b in parenthesis plus c. The third column entry reads: a times the quantity b times c in parenthesis equals the quantity a times b in parenthesis times c. The first entry of the fourth row reads: Distributive Property. The second and third column are combined on this row and read: a times the quantity b plus c in parenthesis equals a times b plus a times c. The first entry in the fifth row reads: Identity Property. The second column entry reads: There exists a unique real number called the additive identity, 0, such that for any real number a, a + 0 = a. The third column entry reads: There exists a unique real number called the multiplicative inverse, 1, such that for any real number a, a times 1 equals a. The first entry in the sixth row reads: Inverse Property. The second column entry reads: Every real number a has an additive inverse, or opposite, denoted negative a such that, a plus negative a equals zero. The third column entry reads: Every nonzero real\">\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>Addition<\/th>\r\n<th>Multiplication<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><strong>Commutative Property<\/strong><\/td>\r\n<td>[latex]a+b=b+a[\/latex]<\/td>\r\n<td>[latex]a\\cdot b=b\\cdot a[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Associative Property<\/strong><\/td>\r\n<td>[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/td>\r\n<td>[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Distributive Property<\/strong><\/td>\r\n<td>[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Identity Property<\/strong><\/td>\r\n<td>There exists a unique real number called the additive identity, 0, such that, for any real number <em>a<\/em>\r\n<div>[latex]a+0=a[\/latex]<\/div><\/td>\r\n<td>There exists a unique real number called the multiplicative identity, 1, such that, for any real number <em>a<\/em>\r\n<div>[latex]a\\cdot 1=a[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Inverse Property<\/strong><\/td>\r\n<td>Every real number a has an additive inverse, or opposite, denoted <em>\u2013a<\/em>, such that\r\n<div>[latex]a+\\left(-a\\right)=0[\/latex]<\/div><\/td>\r\n<td>Every nonzero real number <em>a<\/em> has a multiplicative inverse, or reciprocal, denoted [latex]\\frac{1}{a}[\/latex], such that\r\n<div>[latex]a\\cdot \\left(\\dfrac{1}{a}\\right)=1[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Using the Inverse property of numbers: simplifying fractions<\/h3>\r\nRecall when simplifying fractions, we look for common factors in the numerator and denominator to \"cancel out.\" What we mean by that is that common factors, [latex]\\dfrac{a}{a}[\/latex], divide to the number 1. This is based on the inverse property of multiplication.\r\n\r\nBecause the inverse property states that\r\n\r\n[latex]a\\cdot \\left(\\dfrac{1}{a}\\right)=1[\/latex],\r\n\r\nand because fraction multiplication gives that\r\n\r\n[latex]\\dfrac{a}{b}\\cdot\\dfrac{c}{d} = \\dfrac {ac}{bd}[\/latex],\r\n\r\nWe have\r\n\r\n[latex]\\dfrac{a}{1}\\cdot \\left(\\dfrac{1}{a}\\right)=\\dfrac{a}{a}=1[\/latex]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Properties of Real Numbers<\/h3>\r\nUse the properties of real numbers to rewrite and simplify each expression. State which properties apply.\r\n<ol>\r\n \t<li>[latex]3\\cdot 6+3\\cdot 4[\/latex]<\/li>\r\n \t<li>[latex]\\left(5+8\\right)+\\left(-8\\right)[\/latex]<\/li>\r\n \t<li>[latex]6-\\left(15+9\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{7}\\cdot \\left(\\frac{2}{3}\\cdot \\dfrac{7}{4}\\right)[\/latex]<\/li>\r\n \t<li>[latex]100\\cdot \\left[0.75+\\left(-2.38\\right)\\right][\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"892710\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"892710\"]\r\n\r\n1.\r\n\r\n[latex]\\begin{align}3\\cdot6+3\\cdot4 &amp;=3\\cdot\\left(6+4\\right) &amp;&amp; \\text{Distributive property} \\\\ &amp;=3\\cdot10 &amp;&amp; \\text{Simplify} \\\\ &amp; =30 &amp;&amp; \\text{Simplify}\\end{align}[\/latex]\r\n\r\n2.\r\n\r\n[latex]\\begin{align}\\left(5+8\\right)+\\left(-8\\right) &amp;=5+\\left[8+\\left(-8\\right)\\right] &amp;&amp;\\text{Associative property of addition} \\\\ &amp;=5+0 &amp;&amp; \\text{Inverse property of addition} \\\\ &amp;=5 &amp;&amp;\\text{Identity property of addition}\\end{align}[\/latex]\r\n\r\n3.\r\n\r\n[latex]\\begin{align}6-\\left(15+9\\right) &amp; =6+(-15-9) &amp;&amp; \\text{Distributive property} \\\\ &amp; =6+\\left(-24\\right) &amp;&amp; \\text{Simplify} \\\\ &amp; =-18 &amp;&amp; \\text{Simplify}\\end{align}[\/latex]\r\n\r\n4.\r\n\r\n[latex]\\begin{align}\\frac{4}{7}\\cdot\\left(\\frac{2}{3}\\cdot\\frac{7}{4}\\right) &amp; =\\frac{4}{7} \\cdot\\left(\\frac{7}{4}\\cdot\\frac{2}{3}\\right) &amp;&amp; \\text{Commutative property of multiplication} \\\\ &amp; =\\left(\\frac{4}{7}\\cdot\\frac{7}{4}\\right)\\cdot\\frac{2}{3} &amp;&amp; \\text{Associative property of multiplication} \\\\ &amp; =1\\cdot\\frac{2}{3} &amp;&amp; \\text{Inverse property of multiplication} \\\\ &amp; =\\frac{2}{3} &amp;&amp; \\text{Identity property of multiplication}\\end{align}[\/latex]\r\n\r\n5.\r\n\r\n[latex]\\begin{align}100\\cdot[0.75+\\left(-2.38\\right)] &amp; =100\\cdot0.75+100\\cdot\\left(-2.38\\right) &amp;&amp; \\text{Distributive property} \\\\ &amp; =75+\\left(-238\\right) &amp;&amp; \\text{Simplify} \\\\ &amp; =-163 &amp;&amp; \\text{Simplify}\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse the properties of real numbers to rewrite and simplify each expression. State which properties apply.\r\n<ol>\r\n \t<li>[latex]\\left(-\\dfrac{23}{5}\\right)\\cdot \\left[11\\cdot \\left(-\\dfrac{5}{23}\\right)\\right][\/latex]<\/li>\r\n \t<li>[latex]5\\cdot \\left(6.2+0.4\\right)[\/latex]<\/li>\r\n \t<li>[latex]18-\\left(7 - 15\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{17}{18}+\\cdot \\left[\\dfrac{4}{9}+\\left(-\\dfrac{17}{18}\\right)\\right][\/latex]<\/li>\r\n \t<li>[latex]6\\cdot \\left(-3\\right)+6\\cdot 3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"881536\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"881536\"]\r\n<ol>\r\n \t<li>11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;<\/li>\r\n \t<li>33, distributive property;<\/li>\r\n \t<li>26, distributive property;<\/li>\r\n \t<li>[latex]\\dfrac{4}{9}[\/latex], commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;<\/li>\r\n \t<li>0, distributive property, inverse property of addition, identity property of addition<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92360&amp;theme=oea&amp;iframe_resize_id=mom115[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92361&amp;theme=oea&amp;iframe_resize_id=mom120[\/embed]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the order of operations to simplify an algebraic expression.<\/li>\n<li>Use properties of real numbers to simplify algebraic expressions.<\/li>\n<\/ul>\n<\/div>\n<p>When we multiply a number by itself, we square it or raise it to a power of 2. For example, [latex]{4}^{2}=4\\cdot 4=16[\/latex]. We can raise any number to any power. In general, the <strong>exponential notation<\/strong> [latex]{a}^{n}[\/latex] means that the number or variable [latex]a[\/latex] is used as a factor [latex]n[\/latex] times.<\/p>\n<div style=\"text-align: center;\">[latex]a^{n}=a\\cdot a\\cdot a\\cdot \\dots \\cdot a[\/latex]<\/div>\n<p>In this notation, [latex]{a}^{n}[\/latex] is read as the <em>n<\/em>th power of [latex]a[\/latex], where [latex]a[\/latex] is called the <strong>base<\/strong> and [latex]n[\/latex] is called the <strong>exponent<\/strong>. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, [latex]24+6\\cdot \\frac{2}{3}-{4}^{2}[\/latex] is a mathematical expression.<\/p>\n<p>To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the <strong>order of operations<\/strong>. This is a sequence of rules for evaluating such expressions.<\/p>\n<p>Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.<\/p>\n<p>The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.<\/p>\n<p>Let\u2019s take a look at the expression provided.<\/p>\n<div style=\"text-align: center;\">[latex]24+6\\cdot \\dfrac{2}{3}-{4}^{2}[\/latex]<\/div>\n<p>There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify [latex]{4}^{2}[\/latex] as 16.<\/p>\n<div style=\"text-align: center;\">\n<div style=\"text-align: center;\">[latex]\\begin{gathered}24+6\\cdot\\frac{2}{3}-4^{2} \\\\ 24+6\\cdot\\frac{2}{3}-16\\end{gathered}[\/latex]<\/div>\n<p>Next, perform multiplication or division, left to right.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}24+6\\cdot\\frac{2}{3}-16 \\\\ 24+4-16\\end{gathered}[\/latex]<\/div>\n<p>Lastly, perform addition or subtraction, left to right.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{gathered}24+4-16 \\\\ 28-16 \\\\ 12\\end{gathered}[\/latex]<\/div>\n<\/div>\n<p>Therefore, [latex]24+6\\cdot \\dfrac{2}{3}-{4}^{2}=12[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>How was the fraction above simplified?<\/h3>\n<p>Recall, when mulitplying fractions, we multiply the numerators together and place the result over the product of the denominators.<\/p>\n<ul>\n<li>[latex]\\dfrac{a}{b}\\cdot\\dfrac{c}{d}=\\dfrac{ac}{bd}[\/latex]<\/li>\n<li>[latex]6\\cdot\\dfrac{2}{3}=\\dfrac{6}{1}\\cdot\\dfrac{2}{3}=\\dfrac{12}{3}=4[\/latex]<\/li>\n<\/ul>\n<\/div>\n<p>For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Order of Operations<\/h3>\n<p>Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym <strong>PEMDAS<\/strong>:<\/p>\n<p><strong>P<\/strong>(arentheses)<\/p>\n<p><strong>E<\/strong>(xponents)<\/p>\n<p><strong>M<\/strong>(ultiplication) and <strong>D<\/strong>(ivision)<\/p>\n<p><strong>A<\/strong>(ddition) and <strong>S<\/strong>(ubtraction)<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a mathematical expression, simplify it using the order of operations.<\/h3>\n<ol>\n<li>Simplify any expressions within grouping symbols.<\/li>\n<li>Simplify any expressions containing exponents or radicals.<\/li>\n<li>Perform any multiplication and division in order, from left to right.<\/li>\n<li>Perform any addition and subtraction in order, from left to right.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Order of Operations<\/h3>\n<p>Use the order of operations to evaluate each of the following expressions.<\/p>\n<ol>\n<li>[latex]{\\left(3\\cdot 2\\right)}^{2}-4\\left(6+2\\right)[\/latex]<\/li>\n<li>[latex]\\dfrac{{5}^{2}-4}{7}-\\sqrt{11 - 2}[\/latex]<\/li>\n<li>[latex]6-|5 - 8|+3\\left(4 - 1\\right)[\/latex]<\/li>\n<li>[latex]\\dfrac{14 - 3\\cdot 2}{2\\cdot 5-{3}^{2}}[\/latex]<\/li>\n<li>[latex]7\\left(5\\cdot 3\\right)-2\\left[\\left(6 - 3\\right)-{4}^{2}\\right]+1[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q371324\">Show Solution<\/span><\/p>\n<div id=\"q371324\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(3\\cdot 2\\right)^{2} & =\\left(6\\right)^{2}-4\\left(8\\right) && \\text{Simplify parentheses} \\\\ & =36-4\\left(8\\right) && \\text{Simplify exponent} \\\\ & =36-32 && \\text{Simplify multiplication} \\\\ & =4 && \\text{Simplify subtraction}\\end{align}[\/latex]<\/p>\n<p>2.<\/p>\n<p class=\"p1\" style=\"text-align: center;\"><span class=\"s1\">[latex]\\begin{align}\\frac{5^{2}-4}{7}-\\sqrt{11-2} & =\\frac{5^{2}-4}{7}-\\sqrt{9} && \\text{Simplify grouping systems (radical)} \\\\ & =\\frac{5^{2}-4}{7}-3 && \\text{Simplify radical} \\\\ & =\\frac{25-4}{7}-3 && \\text{Simplify exponent} \\\\ & =\\frac{21}{7}-3 && \\text{Simplify subtraction in numerator} \\\\ & =3-3 && \\text{Simplify division} \\\\ & =0 && \\text{Simplify subtraction}\\end{align}[\/latex]<\/span><\/p>\n<p>Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.<\/p>\n<p>3.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}6-|5-8|+3\\left(4-1\\right) & =6-|-3|+3\\left(3\\right) && \\text{Simplify inside grouping system} \\\\ & =6-3+3\\left(3\\right) && \\text{Simplify absolute value} \\\\ & =6-3+9 && \\text{Simplify multiplication} \\\\ & =3+9 && \\text{Simplify subtraction} \\\\ & =12 && \\text{Simplify addition}\\end{align}[\/latex]<\/p>\n<p>4.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\frac{14-3\\cdot2}{2\\cdot5-3^{2}} & =\\frac{14-3\\cdot2}{2\\cdot5-9} && \\text{Simplify exponent} \\\\ & =\\frac{14-6}{10-9} && \\text{Simplify products} \\\\ & =\\frac{8}{1} && \\text{Simplify quotient} \\\\ & =8 && \\text{Simplify quotient}\\end{align}[\/latex]<br \/>\nIn this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.<\/p>\n<p>5.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}7\\left(5\\cdot3\\right)-2[\\left(6-3\\right)-4^{2}]+1 & =7\\left(15\\right)-2[\\left(3\\right)-4^{2}]+1 && \\text{Simplify inside parentheses} \\\\ & 7\\left(15\\right)-2\\left(3-16\\right)+1 && \\text{Simplify exponent} \\\\ & =7\\left(15\\right)-2\\left(-13\\right)+1 && \\text{Subtract} \\\\ & =105+26+1 && \\text{Multiply} \\\\ & =132 && \\text{Add}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use the order of operations to evaluate the following expression:<br \/>\n[latex]\\sqrt{(21-17)}-2(7+9^{2})+\\dfrac{23-5}{3^2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q371333\">Show Solution<\/span><\/p>\n<div id=\"q371333\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"padding-left: 30px;\">[latex]=\\sqrt{(4)}-2(7+81)+\\dfrac{23-5}{3^2}[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]=\\sqrt{(4)}-2(88)+\\dfrac{18}{9}[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]=2-176+2[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]=-172[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm259\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=259&#38;theme=oea&#38;iframe_resize_id=ohm259&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm99379\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=99379&#38;theme=oea&#38;iframe_resize_id=ohm99379&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video for more examples of using the order of operations to simplify an expression.<\/p>\n<div class=\"textbox examples\">\n<h3>Recall: simplifying fractions<\/h3>\n<p>To simplify a fraction, look for common factors in the numerator and the denominator. In the video below, a fraction, [latex]\\dfrac{14}{26}[\/latex] must be simplified. Since\u00a0[latex]\\dfrac{14}{26}=\\dfrac{2\\cdot7}{2\\cdot13}[\/latex], and since [latex]\\dfrac{2}{2} = 1[\/latex], this fraction simplifies to [latex]\\dfrac{7}{13}[\/latex].<\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Order of Operations with a Fraction Containing a Square Root\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9suc63qB96o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Using Properties of Real Numbers<\/h2>\n<p>For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.<\/p>\n<h3>Commutative Properties<\/h3>\n<p>The <strong>commutative property of addition<\/strong> states that numbers may be added in any order without affecting the sum.<\/p>\n<div style=\"text-align: center;\">[latex]a+b=b+a[\/latex]<\/div>\n<p>We can better see this relationship when using real numbers.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(-2\\right)+7=5\\text{ and }7+\\left(-2\\right)=5[\/latex]<\/div>\n<p>Similarly, the <strong>commutative property of multiplication<\/strong> states that numbers may be multiplied in any order without affecting the product.<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\n<p>Again, consider an example with real numbers.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44\\text{ and }\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/div>\n<p>It is important to note that neither subtraction nor division is commutative. For example, [latex]17 - 5[\/latex] is not the same as [latex]5 - 17[\/latex]. Similarly, [latex]20\\div 5\\ne 5\\div 20[\/latex].<\/p>\n<h3>Associative Properties<\/h3>\n<p>The <strong>associative property of multiplication<\/strong> tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.<\/p>\n<div style=\"text-align: center;\">[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/div>\n<p>Consider this example.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(3\\cdot4\\right)\\cdot5=60\\text{ and }3\\cdot\\left(4\\cdot5\\right)=60[\/latex]<\/div>\n<p>The <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.<\/p>\n<div style=\"text-align: center;\">[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/div>\n<p>This property can be especially helpful when dealing with negative integers. Consider this example.<\/p>\n<div style=\"text-align: center;\">[latex][15+\\left(-9\\right)]+23=29\\text{ and }15+[\\left(-9\\right)+23]=29[\/latex]<\/div>\n<p>Are subtraction and division associative? Review these examples.<\/p>\n<div style=\"text-align: center;\">\n<div style=\"text-align: center;\">[latex]\\begin{align}8-\\left(3-15\\right) & \\stackrel{?}{=}\\left(8-3\\right)-15 \\\\ 8-\\left(-12\\right) & \\stackrel{?}=5-15 \\\\ 20 & \\neq 20-10 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex]\\begin{align}64\\div\\left(8\\div4\\right)&\\stackrel{?}{=}\\left(64\\div8\\right)\\div4 \\\\ 64\\div2 & \\stackrel{?}{=}8\\div4 \\\\ 32 & \\neq 2 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<\/div>\n<p>As we can see, neither subtraction nor division is associative.<\/p>\n<h3>Distributive Property<\/h3>\n<p>The <strong>distributive property<\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/div>\n<p>This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223815\/CNX_CAT_Figure_01_01_003.jpg\" alt=\"The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer.\" \/><\/p>\n<p>Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by \u20137, and adding the products.<\/p>\n<p>To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align} 6+\\left(3\\cdot 5\\right)& \\stackrel{?}{=} \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ 6+\\left(15\\right)& \\stackrel{?}{=} \\left(9\\right)\\cdot \\left(11\\right) \\\\ 21& \\ne 99 \\end{align}[\/latex]<\/div>\n<p>Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.<\/p>\n<p>A special case of the distributive property occurs when a sum of terms is subtracted.<\/p>\n<div style=\"text-align: center;\">[latex]a-b=a+\\left(-b\\right)[\/latex]<\/div>\n<p>For example, consider the difference [latex]12-\\left(5+3\\right)[\/latex]. We can rewrite the difference of the two terms 12 and [latex]\\left(5+3\\right)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\\left(5+3\\right)[\/latex], we add the opposite.<\/p>\n<div style=\"text-align: center;\">[latex]12+\\left(-1\\right)\\cdot \\left(5+3\\right)[\/latex]<\/div>\n<p>Now, distribute [latex]-1[\/latex] and simplify the result.<\/p>\n<div style=\"text-align: center;\">\n<div style=\"text-align: center;\">[latex]\\begin{align}12+\\left(-1\\right)\\cdot\\left(5+3\\right)&=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3] \\\\&=12+(-5-3) \\\\&=12+\\left(-8\\right) \\\\&=4 \\end{align}[\/latex]<\/div>\n<p>This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}12-\\left(5+3\\right) &=12+\\left(-5-3\\right) \\\\ &=12+\\left(-8\\right) \\\\ &=4\\end{align}[\/latex]<\/div>\n<\/div>\n<h3>Identity Properties<\/h3>\n<p>The <strong>identity property of addition<\/strong> states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.<\/p>\n<div style=\"text-align: center;\">\u00a0[latex]a+0=a[\/latex]<\/div>\n<div><\/div>\n<p>The <strong>identity property of multiplication<\/strong> states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot 1=a[\/latex]<\/div>\n<p>For example, we have [latex]\\left(-6\\right)+0=-6[\/latex] and [latex]23\\cdot 1=23[\/latex]. There are no exceptions for these properties; they work for every real number, including 0 and 1.<\/p>\n<h3>Inverse Properties<\/h3>\n<p>The <strong>inverse property of addition<\/strong> states that, for every real number <em>a<\/em>, there is a unique number, called the additive inverse (or opposite), denoted\u2212<em>a<\/em>, that, when added to the original number, results in the additive identity, 0.<\/p>\n<div style=\"text-align: center;\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\n<p>For example, if [latex]a=-8[\/latex], the additive inverse is 8, since [latex]\\left(-8\\right)+8=0[\/latex].<\/p>\n<p>The <strong>inverse property of multiplication<\/strong> holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number <em>a<\/em>, there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\\frac{1}{a}[\/latex], that, when multiplied by the original number, results in the multiplicative identity, 1.<\/p>\n<div style=\"text-align: center;\">\n<div style=\"text-align: center;\">[latex]a\\cdot \\dfrac{1}{a}=1[\/latex]<\/div>\n<p>For example, if [latex]a=-\\frac{2}{3}[\/latex], the reciprocal, denoted [latex]\\frac{1}{a}[\/latex], is [latex]-\\frac{3}{2}[\/latex]\u00a0because<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot \\dfrac{1}{a}=\\left(-\\dfrac{2}{3}\\right)\\cdot \\left(-\\dfrac{3}{2}\\right)=1[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Properties of Real Numbers<\/h3>\n<p>The following properties hold for real numbers <em>a<\/em>, <em>b<\/em>, and <em>c<\/em>.<\/p>\n<table summary=\"A table with six rows and three columns. The first entry of the first row is blank while the remaining columns read: Addition and Multiplication. The first entry of the second row reads: Commutative Property. The second column entry reads a plus b equals b plus a. The third column entry reads a times b equals b times a. The first entry of the third row reads Associative Property. The second column entry reads: a plus the quantity b plus c in parenthesis equals the quantity a plus b in parenthesis plus c. The third column entry reads: a times the quantity b times c in parenthesis equals the quantity a times b in parenthesis times c. The first entry of the fourth row reads: Distributive Property. The second and third column are combined on this row and read: a times the quantity b plus c in parenthesis equals a times b plus a times c. The first entry in the fifth row reads: Identity Property. The second column entry reads: There exists a unique real number called the additive identity, 0, such that for any real number a, a + 0 = a. The third column entry reads: There exists a unique real number called the multiplicative inverse, 1, such that for any real number a, a times 1 equals a. The first entry in the sixth row reads: Inverse Property. The second column entry reads: Every real number a has an additive inverse, or opposite, denoted negative a such that, a plus negative a equals zero. The third column entry reads: Every nonzero real\">\n<thead>\n<tr>\n<th><\/th>\n<th>Addition<\/th>\n<th>Multiplication<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong>Commutative Property<\/strong><\/td>\n<td>[latex]a+b=b+a[\/latex]<\/td>\n<td>[latex]a\\cdot b=b\\cdot a[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Associative Property<\/strong><\/td>\n<td>[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/td>\n<td>[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Distributive Property<\/strong><\/td>\n<td>[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Identity Property<\/strong><\/td>\n<td>There exists a unique real number called the additive identity, 0, such that, for any real number <em>a<\/em><\/p>\n<div>[latex]a+0=a[\/latex]<\/div>\n<\/td>\n<td>There exists a unique real number called the multiplicative identity, 1, such that, for any real number <em>a<\/em><\/p>\n<div>[latex]a\\cdot 1=a[\/latex]<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td><strong>Inverse Property<\/strong><\/td>\n<td>Every real number a has an additive inverse, or opposite, denoted <em>\u2013a<\/em>, such that<\/p>\n<div>[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\n<\/td>\n<td>Every nonzero real number <em>a<\/em> has a multiplicative inverse, or reciprocal, denoted [latex]\\frac{1}{a}[\/latex], such that<\/p>\n<div>[latex]a\\cdot \\left(\\dfrac{1}{a}\\right)=1[\/latex]<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Using the Inverse property of numbers: simplifying fractions<\/h3>\n<p>Recall when simplifying fractions, we look for common factors in the numerator and denominator to &#8220;cancel out.&#8221; What we mean by that is that common factors, [latex]\\dfrac{a}{a}[\/latex], divide to the number 1. This is based on the inverse property of multiplication.<\/p>\n<p>Because the inverse property states that<\/p>\n<p>[latex]a\\cdot \\left(\\dfrac{1}{a}\\right)=1[\/latex],<\/p>\n<p>and because fraction multiplication gives that<\/p>\n<p>[latex]\\dfrac{a}{b}\\cdot\\dfrac{c}{d} = \\dfrac {ac}{bd}[\/latex],<\/p>\n<p>We have<\/p>\n<p>[latex]\\dfrac{a}{1}\\cdot \\left(\\dfrac{1}{a}\\right)=\\dfrac{a}{a}=1[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using Properties of Real Numbers<\/h3>\n<p>Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.<\/p>\n<ol>\n<li>[latex]3\\cdot 6+3\\cdot 4[\/latex]<\/li>\n<li>[latex]\\left(5+8\\right)+\\left(-8\\right)[\/latex]<\/li>\n<li>[latex]6-\\left(15+9\\right)[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{7}\\cdot \\left(\\frac{2}{3}\\cdot \\dfrac{7}{4}\\right)[\/latex]<\/li>\n<li>[latex]100\\cdot \\left[0.75+\\left(-2.38\\right)\\right][\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q892710\">Show Solution<\/span><\/p>\n<div id=\"q892710\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<\/p>\n<p>[latex]\\begin{align}3\\cdot6+3\\cdot4 &=3\\cdot\\left(6+4\\right) && \\text{Distributive property} \\\\ &=3\\cdot10 && \\text{Simplify} \\\\ & =30 && \\text{Simplify}\\end{align}[\/latex]<\/p>\n<p>2.<\/p>\n<p>[latex]\\begin{align}\\left(5+8\\right)+\\left(-8\\right) &=5+\\left[8+\\left(-8\\right)\\right] &&\\text{Associative property of addition} \\\\ &=5+0 && \\text{Inverse property of addition} \\\\ &=5 &&\\text{Identity property of addition}\\end{align}[\/latex]<\/p>\n<p>3.<\/p>\n<p>[latex]\\begin{align}6-\\left(15+9\\right) & =6+(-15-9) && \\text{Distributive property} \\\\ & =6+\\left(-24\\right) && \\text{Simplify} \\\\ & =-18 && \\text{Simplify}\\end{align}[\/latex]<\/p>\n<p>4.<\/p>\n<p>[latex]\\begin{align}\\frac{4}{7}\\cdot\\left(\\frac{2}{3}\\cdot\\frac{7}{4}\\right) & =\\frac{4}{7} \\cdot\\left(\\frac{7}{4}\\cdot\\frac{2}{3}\\right) && \\text{Commutative property of multiplication} \\\\ & =\\left(\\frac{4}{7}\\cdot\\frac{7}{4}\\right)\\cdot\\frac{2}{3} && \\text{Associative property of multiplication} \\\\ & =1\\cdot\\frac{2}{3} && \\text{Inverse property of multiplication} \\\\ & =\\frac{2}{3} && \\text{Identity property of multiplication}\\end{align}[\/latex]<\/p>\n<p>5.<\/p>\n<p>[latex]\\begin{align}100\\cdot[0.75+\\left(-2.38\\right)] & =100\\cdot0.75+100\\cdot\\left(-2.38\\right) && \\text{Distributive property} \\\\ & =75+\\left(-238\\right) && \\text{Simplify} \\\\ & =-163 && \\text{Simplify}\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.<\/p>\n<ol>\n<li>[latex]\\left(-\\dfrac{23}{5}\\right)\\cdot \\left[11\\cdot \\left(-\\dfrac{5}{23}\\right)\\right][\/latex]<\/li>\n<li>[latex]5\\cdot \\left(6.2+0.4\\right)[\/latex]<\/li>\n<li>[latex]18-\\left(7 - 15\\right)[\/latex]<\/li>\n<li>[latex]\\dfrac{17}{18}+\\cdot \\left[\\dfrac{4}{9}+\\left(-\\dfrac{17}{18}\\right)\\right][\/latex]<\/li>\n<li>[latex]6\\cdot \\left(-3\\right)+6\\cdot 3[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q881536\">Show Solution<\/span><\/p>\n<div id=\"q881536\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;<\/li>\n<li>33, distributive property;<\/li>\n<li>26, distributive property;<\/li>\n<li>[latex]\\dfrac{4}{9}[\/latex], commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;<\/li>\n<li>0, distributive property, inverse property of addition, identity property of addition<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm92360\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92360&#38;theme=oea&#38;iframe_resize_id=ohm92360&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm92361\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92361&#38;theme=oea&#38;iframe_resize_id=ohm92361&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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-40\">\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>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>Evaluate a Mathematical Expression With the Desmos Calculator. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/uM13cNyQGPM\">https:\/\/youtu.be\/uM13cNyQGPM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 259. <strong>Authored by<\/strong>: Sousa, James. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 99379. <strong>Authored by<\/strong>: Davis, Desiree. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 92360. <strong>Authored by<\/strong>: Jenck, Michael for Lumen Learning. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 92361. <strong>Authored by<\/strong>: Jenck, Michael for Lumen Learning. <strong>License<\/strong>: <em>Other<\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Simplifying Expressions With Square Roots. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9suc63qB96o\">https:\/\/youtu.be\/9suc63qB96o<\/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, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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":395986,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 259\",\"author\":\"Sousa, James\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"other\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 99379\",\"author\":\"Davis, Desiree\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"other\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 92360\",\"author\":\"Jenck, Michael for Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"other\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 92361\",\"author\":\"Jenck, Michael for Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"other\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"original\",\"description\":\"Evaluate a Mathematical Expression With the Desmos Calculator\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/uM13cNyQGPM\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Simplifying Expressions With Square Roots\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/9suc63qB96o\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"7b693694-02d0-4b11-88f9-bf3d0c8366c9","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-40","chapter","type-chapter","status-publish","hentry"],"part":31,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/40","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/40\/revisions"}],"predecessor-version":[{"id":833,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/40\/revisions\/833"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/31"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/40\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=40"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=40"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=40"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=40"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}