{"id":9470,"date":"2017-05-02T22:56:25","date_gmt":"2017-05-02T22:56:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9470"},"modified":"2020-09-11T00:36:03","modified_gmt":"2020-09-11T00:36:03","slug":"rewriting-expressions-using-the-commutative-and-associative-properties","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/rewriting-expressions-using-the-commutative-and-associative-properties\/","title":{"raw":"5.2.a - Using the Commutative and Associative Properties","rendered":"5.2.a &#8211; Using the Commutative and Associative Properties"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the associative and commutative properties of addition and multiplication<\/li>\r\n \t<li>Use the associative and commutative properties of addition and multiplication to rewrite\u00a0algebraic expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\nFor some activities we perform, the order of certain processes\u00a0does not matter, but the order of others do. 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 addition and multiplication.\r\n<h2>The Commutative Properties<\/h2>\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\r\nThink about adding two numbers, such as [latex]5[\/latex] and [latex]3[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{cccc}\\hfill 5+3\\hfill &amp; &amp; &amp; \\hfill 3+5\\hfill \\\\ \\hfill 8\\hfill &amp; &amp; &amp; \\hfill 8\\hfill \\end{array}[\/latex]<\/p>\r\nThe results are the same. [latex]5+3=3+5[\/latex]\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nShow that numbers may be added in any order without affecting the sum. [latex]\\left(-2\\right)+7=5[\/latex]\r\n[reveal-answer q=\"279824\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"279824\"]\r\n\r\n[latex]7+\\left(-2\\right)=5[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice, the order in which we add does not matter. The same is true when multiplying [latex]5[\/latex] and [latex]3[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{cccc}\\hfill 5\\cdot 3\\hfill &amp; &amp; &amp; \\hfill 3\\cdot 5\\hfill \\\\ \\hfill 15\\hfill &amp; &amp; &amp; \\hfill 15\\hfill \\end{array}[\/latex]<\/p>\r\nAgain, the results are the same! [latex]5\\cdot 3=3\\cdot 5[\/latex]. The order in which we multiply does not matter.\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 class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nShow that numbers may be multiplied\u00a0in any order without affecting the product.[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44[\/latex]\r\n[reveal-answer q=\"112050\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"112050\"]\r\n\r\n[latex]\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nThese examples illustrate the commutative properties of addition and multiplication.\r\n<div class=\"textbox shaded\">\r\n<h3>Commutative Properties<\/h3>\r\n<strong>Commutative Property of Addition<\/strong>: if [latex]a[\/latex] and [latex]b[\/latex] are real numbers, then\r\n<p style=\"padding-left: 30px\">[latex]a+b=b+a[\/latex]<\/p>\r\n<p style=\"text-align: left\"><strong>Commutative Property of Multiplication<\/strong>: if [latex]a[\/latex] and [latex]b[\/latex] are real numbers, then<\/p>\r\n<p style=\"padding-left: 30px\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/p>\r\n\r\n<\/div>\r\nThe commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse the commutative properties to rewrite the following expressions:\r\n1. [latex]-1+3=[\/latex]\r\n2. [latex]4\\cdot 9=[\/latex]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466089427\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-1+3=[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the commutative property of addition to change the order.<\/td>\r\n<td>[latex]-1+3=3+\\left(-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466783247\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4\\cdot 9=[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the commutative property of multiplication to change the order.<\/td>\r\n<td>[latex]4\\cdot 9=9\\cdot 4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145966[\/ohm_question]\r\n\r\n[ohm_question]145968[\/ohm_question]\r\n\r\n<\/div>\r\nWhat about subtraction? Does order matter when we subtract numbers? Does [latex]7 - 3[\/latex] give the same result as [latex]3 - 7?[\/latex]\r\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}\\hfill 7 - 3\\hfill &amp; &amp; \\hfill 3 - 7\\hfill \\\\ \\hfill 4\\hfill &amp; &amp; \\hfill -4\\hfill \\\\ &amp; \\hfill 4\\ne -4\\hfill &amp; \\end{array}[\/latex]\r\nThe results are not the same. [latex]7 - 3\\ne 3 - 7[\/latex]<\/p>\r\nSince changing the order of the subtraction did not give the same result, we can say that subtraction is not commutative.\r\n\r\nLet\u2019s see what happens when we divide two numbers. Is division commutative?\r\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}\\hfill 12\\div 4\\hfill &amp; &amp; \\hfill 4\\div 12\\hfill \\\\ \\hfill \\frac{12}{4}\\hfill &amp; &amp; \\hfill \\frac{4}{12}\\hfill \\\\ \\hfill 3\\hfill &amp; &amp; \\hfill \\frac{1}{3}\\hfill \\\\ &amp; \\hfill 3\\ne \\frac{1}{3}\\hfill &amp; \\end{array}[\/latex]<\/p>\r\nThe results are not the same. So [latex]12\\div 4\\ne 4\\div 12[\/latex]\r\n\r\nSince changing the order of the division did not give the same result, division is not commutative.\r\n\r\nAddition and multiplication are commutative. Subtraction and division are not commutative.\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\" wp-image-980 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183526\/traffic-sign-160659-300x265.png\" alt=\"traffic-sign-160659\" width=\"61\" height=\"55\" \/>\r\n<h2>Caution! 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].<\/h2>\r\n<\/div>\r\n<h2>The Associative Properties<\/h2>\r\nSuppose you were asked to simplify this expression.\r\n<p style=\"text-align: center\">[latex]7+8+2[\/latex]<\/p>\r\nHow would you do it and what would your answer be?\r\n\r\nSome people would think [latex]7+8\\text{ is }15[\/latex] and then [latex]15+2\\text{ is }17[\/latex]. Others might start with [latex]8+2\\text{ makes }10[\/latex] and then [latex]7+10\\text{ makes }17[\/latex].\r\n\r\nBoth ways give the same result, as shown below. (Remember that parentheses are grouping symbols that indicate which operations should be done first.)\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222322\/CNX_BMath_Figure_07_02_001.png\" alt=\"The image shows an equation. The left side of the equation shows the quantity 7 plus 8 in parentheses plus 2. The right side of the equation show 7 plus the quantity 8 plus 2. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 17 below.\" \/>\r\nWhen adding three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Addition.\r\n<div style=\"text-align: left\">The <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.<\/div>\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 class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nShow that regrouping addition does not affect the sum. [latex][15+\\left(-9\\right)]+23=29[\/latex]\r\n[reveal-answer q=\"898684\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"898684\"]\r\n\r\n[latex]15+[\\left(-9\\right)+23]=29[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe same principle holds true for multiplication as well. Suppose we want to find the value of the following expression:\r\n<p style=\"text-align: center\">[latex]5\\cdot \\frac{1}{3}\\cdot 3[\/latex]<\/p>\r\nChanging the grouping of the numbers gives the same result.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222323\/CNX_BMath_Figure_07_02_002.png\" alt=\"The image shows an equation. The left side of the equation shows the quantity 5 times 1 third in parentheses times 3. The right side of the equation show 5 times the quantity 1 third times 3. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 5 below.\" \/>\r\nWhen multiplying three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Multiplication.\r\n\r\nIf we multiply three numbers, changing the grouping does not affect the product.\r\n\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 class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nShow that you can regroup numbers that are multiplied together and not affect the product.[latex]\\left(3\\cdot4\\right)\\cdot5=60[\/latex]\r\n[reveal-answer q=\"786302\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"786302\"]\r\n\r\n[latex]3\\cdot\\left(4\\cdot5\\right)=60[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou probably know this, but the terminology may be new to you. These examples illustrate the <em>Associative Properties<\/em>.\r\n<div class=\"textbox shaded\">\r\n<h3>Associative Properties<\/h3>\r\n<strong>Associative Property of Addition<\/strong>: if [latex]a,b[\/latex], and [latex]c[\/latex] are real numbers, then\r\n<p style=\"padding-left: 30px\">[latex]\\left(a+b\\right)+c=a+\\left(b+c\\right)[\/latex]<\/p>\r\n<strong>Associative Property of Multiplication<\/strong>: if [latex]a,b[\/latex], and [latex]c[\/latex] are real numbers, then\r\n<p style=\"padding-left: 30px\">[latex]\\left(a\\cdot b\\right)\\cdot c=a\\cdot \\left(b\\cdot c\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div style=\"text-align: left\"><\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nUse the associative properties to rewrite the following:\r\n\r\n1. [latex]\\left(3+0.6\\right)+0.4=[\/latex]\r\n2. [latex]\\left(-4\\cdot \\frac{2}{5}\\right)\\cdot 15=[\/latex]\r\n[reveal-answer q=\"529914\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"529914\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467319983\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(3+0.6\\right)+0.4=[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Change the grouping.<\/td>\r\n<td>[latex]\\left(3+0.6\\right)+0.4=3+\\left(0.6+0.4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that [latex]0.6+0.4[\/latex] is [latex]1[\/latex], so the addition will be easier if we group as shown on the right.\r\n<table id=\"eip-id1168467353185\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(-4\\cdot \\frac{2}{5}\\right)\\cdot 15=[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Change the grouping.<\/td>\r\n<td>[latex]\\left(-4\\cdot \\frac{2}{5}\\right)\\cdot 15=-4\\cdot \\left(\\frac{2}{5}\\cdot 15\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that [latex]\\frac{2}{5}\\cdot 15[\/latex] is [latex]6[\/latex]. The multiplication will be easier if we group as shown on the right.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145970[\/ohm_question]\r\n\r\n[ohm_question]145971[\/ohm_question]\r\n\r\n<\/div>\r\n<div style=\"text-align: center\"><\/div>\r\nAre subtraction and division associative? Review these examples.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the associative property to explore whether subtraction and division are associative.\r\n\r\n1) [latex]8-\\left(3-15\\right)\\stackrel{?}{=}\\left(8-3\\right)-15[\/latex]\r\n\r\n2) [latex]64\\div\\left(8\\div4\\right)\\stackrel{?}{=}\\left(64\\div8\\right)\\div4[\/latex]\r\n\r\n[reveal-answer q=\"515666\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"515666\"]\r\n\r\n1) [latex]8-\\left(3-15\\right)\\stackrel{?}{=}\\left(8-3\\right)-15[\/latex]\r\n\r\n[latex]\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,8-\\left(-12\\right)=5-15[\/latex]\r\n\r\n[latex]\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,20\\neq-10[\/latex]\r\n\r\n2) [latex]64\\div\\left(8\\div4\\right)\\stackrel{?}{=}\\left(64\\div8\\right)\\div4[\/latex]\r\n\r\n[latex]\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,64\\div2\\stackrel{?}{=}8\\div4[\/latex]\r\n\r\n[latex]\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,32\\neq 2[\/latex]\r\n\r\nAs we can see, neither subtraction nor division is associative.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the associative and commutative properties of addition and multiplication<\/li>\n<li>Use the associative and commutative properties of addition and multiplication to rewrite\u00a0algebraic expressions<\/li>\n<\/ul>\n<\/div>\n<p>For some activities we perform, the order of certain processes\u00a0does not matter, but the order of others do. 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 addition and multiplication.<\/p>\n<h2>The Commutative Properties<\/h2>\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<p>Think about adding two numbers, such as [latex]5[\/latex] and [latex]3[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{cccc}\\hfill 5+3\\hfill & & & \\hfill 3+5\\hfill \\\\ \\hfill 8\\hfill & & & \\hfill 8\\hfill \\end{array}[\/latex]<\/p>\n<p>The results are the same. [latex]5+3=3+5[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Show that numbers may be added in any order without affecting the sum. [latex]\\left(-2\\right)+7=5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q279824\">Show Solution<\/span><\/p>\n<div id=\"q279824\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]7+\\left(-2\\right)=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice, the order in which we add does not matter. The same is true when multiplying [latex]5[\/latex] and [latex]3[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{cccc}\\hfill 5\\cdot 3\\hfill & & & \\hfill 3\\cdot 5\\hfill \\\\ \\hfill 15\\hfill & & & \\hfill 15\\hfill \\end{array}[\/latex]<\/p>\n<p>Again, the results are the same! [latex]5\\cdot 3=3\\cdot 5[\/latex]. The order in which we multiply does not matter.<br \/>\nSimilarly, 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 class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Show that numbers may be multiplied\u00a0in any order without affecting the product.[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q112050\">Show Solution<\/span><\/p>\n<div id=\"q112050\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>These examples illustrate the commutative properties of addition and multiplication.<\/p>\n<div class=\"textbox shaded\">\n<h3>Commutative Properties<\/h3>\n<p><strong>Commutative Property of Addition<\/strong>: if [latex]a[\/latex] and [latex]b[\/latex] are real numbers, then<\/p>\n<p style=\"padding-left: 30px\">[latex]a+b=b+a[\/latex]<\/p>\n<p style=\"text-align: left\"><strong>Commutative Property of Multiplication<\/strong>: if [latex]a[\/latex] and [latex]b[\/latex] are real numbers, then<\/p>\n<p style=\"padding-left: 30px\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/p>\n<\/div>\n<p>The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use the commutative properties to rewrite the following expressions:<br \/>\n1. [latex]-1+3=[\/latex]<br \/>\n2. [latex]4\\cdot 9=[\/latex]<\/p>\n<p>Solution:<\/p>\n<table id=\"eip-id1168466089427\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-1+3=[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the commutative property of addition to change the order.<\/td>\n<td>[latex]-1+3=3+\\left(-1\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466783247\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4\\cdot 9=[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the commutative property of multiplication to change the order.<\/td>\n<td>[latex]4\\cdot 9=9\\cdot 4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145966\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145966&theme=oea&iframe_resize_id=ohm145966&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145968\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145968&theme=oea&iframe_resize_id=ohm145968&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>What about subtraction? Does order matter when we subtract numbers? Does [latex]7 - 3[\/latex] give the same result as [latex]3 - 7?[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}\\hfill 7 - 3\\hfill & & \\hfill 3 - 7\\hfill \\\\ \\hfill 4\\hfill & & \\hfill -4\\hfill \\\\ & \\hfill 4\\ne -4\\hfill & \\end{array}[\/latex]<br \/>\nThe results are not the same. [latex]7 - 3\\ne 3 - 7[\/latex]<\/p>\n<p>Since changing the order of the subtraction did not give the same result, we can say that subtraction is not commutative.<\/p>\n<p>Let\u2019s see what happens when we divide two numbers. Is division commutative?<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}\\hfill 12\\div 4\\hfill & & \\hfill 4\\div 12\\hfill \\\\ \\hfill \\frac{12}{4}\\hfill & & \\hfill \\frac{4}{12}\\hfill \\\\ \\hfill 3\\hfill & & \\hfill \\frac{1}{3}\\hfill \\\\ & \\hfill 3\\ne \\frac{1}{3}\\hfill & \\end{array}[\/latex]<\/p>\n<p>The results are not the same. So [latex]12\\div 4\\ne 4\\div 12[\/latex]<\/p>\n<p>Since changing the order of the division did not give the same result, division is not commutative.<\/p>\n<p>Addition and multiplication are commutative. Subtraction and division are not commutative.<\/p>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-980 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183526\/traffic-sign-160659-300x265.png\" alt=\"traffic-sign-160659\" width=\"61\" height=\"55\" \/><\/p>\n<h2>Caution! 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].<\/h2>\n<\/div>\n<h2>The Associative Properties<\/h2>\n<p>Suppose you were asked to simplify this expression.<\/p>\n<p style=\"text-align: center\">[latex]7+8+2[\/latex]<\/p>\n<p>How would you do it and what would your answer be?<\/p>\n<p>Some people would think [latex]7+8\\text{ is }15[\/latex] and then [latex]15+2\\text{ is }17[\/latex]. Others might start with [latex]8+2\\text{ makes }10[\/latex] and then [latex]7+10\\text{ makes }17[\/latex].<\/p>\n<p>Both ways give the same result, as shown below. (Remember that parentheses are grouping symbols that indicate which operations should be done first.)<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222322\/CNX_BMath_Figure_07_02_001.png\" alt=\"The image shows an equation. The left side of the equation shows the quantity 7 plus 8 in parentheses plus 2. The right side of the equation show 7 plus the quantity 8 plus 2. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 17 below.\" \/><br \/>\nWhen adding three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Addition.<\/p>\n<div style=\"text-align: left\">The <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.<\/div>\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 class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Show that regrouping addition does not affect the sum. [latex][15+\\left(-9\\right)]+23=29[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q898684\">Show Solution<\/span><\/p>\n<div id=\"q898684\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]15+[\\left(-9\\right)+23]=29[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The same principle holds true for multiplication as well. Suppose we want to find the value of the following expression:<\/p>\n<p style=\"text-align: center\">[latex]5\\cdot \\frac{1}{3}\\cdot 3[\/latex]<\/p>\n<p>Changing the grouping of the numbers gives the same result.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222323\/CNX_BMath_Figure_07_02_002.png\" alt=\"The image shows an equation. The left side of the equation shows the quantity 5 times 1 third in parentheses times 3. The right side of the equation show 5 times the quantity 1 third times 3. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 5 below.\" \/><br \/>\nWhen multiplying three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Multiplication.<\/p>\n<p>If we multiply three numbers, changing the grouping does not affect the product.<\/p>\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 class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Show that you can regroup numbers that are multiplied together and not affect the product.[latex]\\left(3\\cdot4\\right)\\cdot5=60[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q786302\">Show Solution<\/span><\/p>\n<div id=\"q786302\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]3\\cdot\\left(4\\cdot5\\right)=60[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You probably know this, but the terminology may be new to you. These examples illustrate the <em>Associative Properties<\/em>.<\/p>\n<div class=\"textbox shaded\">\n<h3>Associative Properties<\/h3>\n<p><strong>Associative Property of Addition<\/strong>: if [latex]a,b[\/latex], and [latex]c[\/latex] are real numbers, then<\/p>\n<p style=\"padding-left: 30px\">[latex]\\left(a+b\\right)+c=a+\\left(b+c\\right)[\/latex]<\/p>\n<p><strong>Associative Property of Multiplication<\/strong>: if [latex]a,b[\/latex], and [latex]c[\/latex] are real numbers, then<\/p>\n<p style=\"padding-left: 30px\">[latex]\\left(a\\cdot b\\right)\\cdot c=a\\cdot \\left(b\\cdot c\\right)[\/latex]<\/p>\n<\/div>\n<div style=\"text-align: left\"><\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Use the associative properties to rewrite the following:<\/p>\n<p>1. [latex]\\left(3+0.6\\right)+0.4=[\/latex]<br \/>\n2. [latex]\\left(-4\\cdot \\frac{2}{5}\\right)\\cdot 15=[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q529914\">Show Solution<\/span><\/p>\n<div id=\"q529914\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467319983\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\left(3+0.6\\right)+0.4=[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Change the grouping.<\/td>\n<td>[latex]\\left(3+0.6\\right)+0.4=3+\\left(0.6+0.4\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that [latex]0.6+0.4[\/latex] is [latex]1[\/latex], so the addition will be easier if we group as shown on the right.<\/p>\n<table id=\"eip-id1168467353185\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\left(-4\\cdot \\frac{2}{5}\\right)\\cdot 15=[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Change the grouping.<\/td>\n<td>[latex]\\left(-4\\cdot \\frac{2}{5}\\right)\\cdot 15=-4\\cdot \\left(\\frac{2}{5}\\cdot 15\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that [latex]\\frac{2}{5}\\cdot 15[\/latex] is [latex]6[\/latex]. The multiplication will be easier if we group as shown on the right.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145970\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145970&theme=oea&iframe_resize_id=ohm145970&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm145971\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145971&theme=oea&iframe_resize_id=ohm145971&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div style=\"text-align: center\"><\/div>\n<p>Are subtraction and division associative? Review these examples.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the associative property to explore whether subtraction and division are associative.<\/p>\n<p>1) [latex]8-\\left(3-15\\right)\\stackrel{?}{=}\\left(8-3\\right)-15[\/latex]<\/p>\n<p>2) [latex]64\\div\\left(8\\div4\\right)\\stackrel{?}{=}\\left(64\\div8\\right)\\div4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q515666\">Show Solution<\/span><\/p>\n<div id=\"q515666\" class=\"hidden-answer\" style=\"display: none\">\n<p>1) [latex]8-\\left(3-15\\right)\\stackrel{?}{=}\\left(8-3\\right)-15[\/latex]<\/p>\n<p>[latex]\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,8-\\left(-12\\right)=5-15[\/latex]<\/p>\n<p>[latex]\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,20\\neq-10[\/latex]<\/p>\n<p>2) [latex]64\\div\\left(8\\div4\\right)\\stackrel{?}{=}\\left(64\\div8\\right)\\div4[\/latex]<\/p>\n<p>[latex]\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,64\\div2\\stackrel{?}{=}8\\div4[\/latex]<\/p>\n<p>[latex]\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,32\\neq 2[\/latex]<\/p>\n<p>As we can see, neither subtraction nor division is associative.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-9470\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 145973, 145970, 145971, 145966, 145968. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Use the Commutative and Associate Properties of Real Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/UOcMDyJA7Yw\">https:\/\/youtu.be\/UOcMDyJA7Yw<\/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>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Use the Commutative and Associate Properties of Real Numbers\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/UOcMDyJA7Yw\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 145973, 145970, 145971, 145966, 145968\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"012fe512ffbe42f997698f14cbb039bf, 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