{"id":1702,"date":"2016-06-24T23:00:04","date_gmt":"2016-06-24T23:00:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=1702"},"modified":"2019-07-24T20:53:43","modified_gmt":"2019-07-24T20:53:43","slug":"read-use-properties-of-real-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-use-properties-of-real-numbers\/","title":{"raw":"Use Properties of Real Numbers","rendered":"Use Properties of Real Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define and use the commutative property of addition and multiplication<\/li>\r\n \t<li>Define and use the associative property of addition and multiplication<\/li>\r\n \t<li>Define and use the distributive property<\/li>\r\n \t<li>Define and use the identity property of addition and multiplication<\/li>\r\n \t<li>Define and use the inverse property of addition and multiplication<\/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<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 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\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\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\r\nCaution! 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].\r\n\r\n<\/div>\r\n<h3>Associative Properties - Grouping<\/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 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\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\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<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<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the distributive property to show that [latex]4\\cdot[12+(-7)]=20[\/latex]\r\n[reveal-answer q=\"907389\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"907389\"]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200212\/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\n<span style=\"font-size: 1rem; text-align: initial;\">Note that [latex]4[\/latex] is outside the grouping symbols, so we distribute the \u00a0[latex]4[\/latex] by multiplying it by [latex]12[\/latex], multiplying it by [latex]\u20137[\/latex], and adding the products.<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo be more precise when describing this property, we say that multiplication distributes over addition.\r\n\r\nThe reverse is not true as we can see in this example.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 6+\\left(3\\cdot 5\\right)&amp; \\stackrel{?}{=}&amp; \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ \\hfill 6+\\left(15\\right)&amp; \\stackrel{?}{=}&amp; \\left(9\\right)\\cdot \\left(11\\right)\\hfill \\\\ \\hfill 21&amp; \\ne &amp; \\text{ }99\\hfill \\end{array}[\/latex]<\/p>\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\n<div style=\"text-align: center;\">\r\n\r\nFor example, consider the difference [latex]12-\\left(5+3\\right)[\/latex]. We can rewrite the difference of the two terms [latex]12[\/latex] 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>[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>[latex]\\begin{array}{l}12-\\left(5+3\\right)=12+\\left(-1\\right)\\cdot\\left(5+3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3]\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+\\left(-8\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nRewrite the last example by changing the sign of each term and adding the results.\r\n[reveal-answer q=\"719333\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"719333\"]\r\n\r\n[latex]\\begin{array}{l}12-\\left(5+3\\right)=12+\\left(-5-3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+\\left(-8\\right) \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex][\/hidden-answer]\r\n\r\n<\/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.\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;\">[latex]a+0=a[\/latex]<\/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\n<div style=\"text-align: center;\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p style=\"text-align: left;\">Show that the identity property of addition and multiplication are true for [latex]-6 \\text{ and }23[\/latex].<\/p>\r\n<p style=\"text-align: left;\">[reveal-answer q=\"587790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"587790\"]<\/p>\r\n<p style=\"text-align: left;\">[latex]\\left(-6\\right)+0=-6[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex]23+0=23[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex]-6\\cdot1=-6[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[latex]23\\cdot 1=23[\/latex]<\/p>\r\n<p style=\"text-align: left;\">There are no exceptions for these properties; they work for every real number, including [latex]0[\/latex] and [latex]1[\/latex].<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<span style=\"color: #6c64ad; font-size: 1em; font-weight: 600;\">Inverse Properties<\/span>\r\n\r\n<\/div>\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\u00a0<em>a<\/em>, that, when added to the original number, results in the additive identity, [latex]0[\/latex].\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 [latex]8[\/latex], since [latex]\\left(-8\\right)+8=0[\/latex].\r\n\r\nThe <strong>inverse property of multiplication<\/strong> holds for all real numbers except [latex]0[\/latex] because the reciprocal of [latex]0[\/latex] 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]\\dfrac{1}{a}[\/latex], that, when multiplied by the original number, results in the multiplicative identity, [latex]1[\/latex].\r\n<div style=\"text-align: center;\">[latex]a\\cdot\\dfrac{1}{a}\\normalsize =1[\/latex]<\/div>\r\n<div style=\"text-align: center;\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p style=\"text-align: left;\">1) Define the additive inverse of\u00a0[latex]a=-8[\/latex], and use it to illustrate the inverse property of addition.<\/p>\r\n<p style=\"text-align: left;\">2) Write the reciprocal of\u00a0[latex]a=-\\dfrac{2}{3}[\/latex], and use it to illustrate the inverse property of multiplication.<\/p>\r\n<p style=\"text-align: left;\">[reveal-answer q=\"468875\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"468875\"]<\/p>\r\n<p style=\"text-align: left;\">1) The additive inverse is [latex]8[\/latex], and\u00a0[latex]\\left(-8\\right)+8=0[\/latex]<\/p>\r\n<p style=\"text-align: left;\">2) The reciprocal is [latex]-\\dfrac{3}{2}[\/latex]\u00a0and\u00a0[latex]\\left(-\\dfrac{2}{3}\\normalsize\\right)\\cdot \\left(-\\dfrac{3}{2}\\normalsize\\right)=1[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3 style=\"text-align: left;\">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 style=\"width: 70%;\" 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<tbody>\r\n<tr>\r\n<th style=\"text-align: center;\"><\/th>\r\n<th style=\"text-align: center;\"><strong>Addition<\/strong><\/th>\r\n<th style=\"text-align: center;\"><strong>Multiplication<\/strong><\/th>\r\n<\/tr>\r\n<\/tbody>\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<td><\/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 style=\"text-align: center;\">[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 style=\"text-align: center;\">[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 [latex]\u2013a[\/latex], such that\r\n<div style=\"text-align: center;\">[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]\\dfrac{1}{a}[\/latex], such that\r\n<div style=\"text-align: center;\">[latex]a\\cdot \\left(\\dfrac{1}{a}\\normalsize\\right)=1[\/latex]<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the properties of real numbers to rewrite and simplify each expression. State which properties apply.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]3\\left(6+4\\right)[\/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}\\normalsize\\cdot \\left(\\dfrac{2}{3}\\normalsize\\cdot\\dfrac{7}{4}\\normalsize\\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=\"823624\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"823624\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\3\\cdot\\left(6+4\\right)=3\\cdot6+3\\cdot4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Distributive property} \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=18+12\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=30\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\\\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{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\6-\\left(15+9\\right) \\hfill&amp; =6+[\\left(-15\\right)+\\left(-9\\right)] \\hfill&amp; \\text{Distributive property} \\\\ \\hfill&amp; =6+\\left(-24\\right) \\hfill&amp; \\text{Simplify} \\\\ \\hfill&amp; =-18 \\hfill&amp; \\text{Simplify}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\\\\\\\\\\\frac{4}{7}\\cdot\\left(\\frac{2}{3}\\cdot\\frac{7}{4}\\right) \\hfill&amp; =\\frac{4}{7} \\cdot\\left(\\frac{7}{4}\\cdot\\frac{2}{3}\\right) \\hfill&amp; \\text{Commutative property of multiplication} \\\\ \\hfill&amp; =\\left(\\frac{4}{7}\\cdot\\frac{7}{4}\\right)\\cdot\\frac{2}{3}\\hfill&amp; \\text{Associative property of multiplication} \\\\ \\hfill&amp; =1\\cdot\\frac{2}{3} \\hfill&amp; \\text{Inverse property of multiplication} \\\\ \\hfill&amp; =\\frac{2}{3} \\hfill&amp; \\text{Identity property of multiplication}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{l}\\\\\\\\100\\cdot[0.75+\\left(-2.38\\right)] \\hfill&amp; =100\\cdot0.75+100\\cdot\\left(-2.38\\right)\\hfill&amp; \\text{Distributive property} \\\\ \\hfill&amp; =75+\\left(-238\\right) \\hfill&amp; \\text{Simplify} \\\\ \\hfill&amp; =-163 \\hfill&amp; \\text{Simplify}\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/8SFm8Os_4C8","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define and use the commutative property of addition and multiplication<\/li>\n<li>Define and use the associative property of addition and multiplication<\/li>\n<li>Define and use the distributive property<\/li>\n<li>Define and use the identity property of addition and multiplication<\/li>\n<li>Define and use the inverse property of addition and multiplication<\/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<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 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>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 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<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<p>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].<\/p>\n<\/div>\n<h3>Associative Properties &#8211; Grouping<\/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 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<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>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<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<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the distributive property to show that [latex]4\\cdot[12+(-7)]=20[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q907389\">Show Solution<\/span><\/p>\n<div id=\"q907389\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200212\/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><span style=\"font-size: 1rem; text-align: initial;\">Note that [latex]4[\/latex] is outside the grouping symbols, so we distribute the \u00a0[latex]4[\/latex] by multiplying it by [latex]12[\/latex], multiplying it by [latex]\u20137[\/latex], and adding the products.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To be more precise when describing this property, we say that multiplication distributes over addition.<\/p>\n<p>The reverse is not true as we can see in this example.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill 6+\\left(3\\cdot 5\\right)& \\stackrel{?}{=}& \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ \\hfill 6+\\left(15\\right)& \\stackrel{?}{=}& \\left(9\\right)\\cdot \\left(11\\right)\\hfill \\\\ \\hfill 21& \\ne & \\text{ }99\\hfill \\end{array}[\/latex]<\/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<div style=\"text-align: center;\">\n<p>For example, consider the difference [latex]12-\\left(5+3\\right)[\/latex]. We can rewrite the difference of the two terms [latex]12[\/latex] 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>[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>[latex]\\begin{array}{l}12-\\left(5+3\\right)=12+\\left(-1\\right)\\cdot\\left(5+3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3]\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+\\left(-8\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Rewrite the last example by changing the sign of each term and adding the results.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q719333\">Show Solution<\/span><\/p>\n<div id=\"q719333\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{array}{l}12-\\left(5+3\\right)=12+\\left(-5-3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=12+\\left(-8\\right) \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/p><\/div>\n<\/div>\n<\/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.<\/p>\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;\">[latex]a+0=a[\/latex]<\/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<div style=\"text-align: center;\">\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p style=\"text-align: left;\">Show that the identity property of addition and multiplication are true for [latex]-6 \\text{ and }23[\/latex].<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q587790\">Show Solution<\/span><\/p>\n<div id=\"q587790\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex]\\left(-6\\right)+0=-6[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]23+0=23[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]-6\\cdot1=-6[\/latex]<\/p>\n<p style=\"text-align: left;\">[latex]23\\cdot 1=23[\/latex]<\/p>\n<p style=\"text-align: left;\">There are no exceptions for these properties; they work for every real number, including [latex]0[\/latex] and [latex]1[\/latex].<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<p><span style=\"color: #6c64ad; font-size: 1em; font-weight: 600;\">Inverse Properties<\/span><\/p>\n<\/div>\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\u00a0<em>a<\/em>, that, when added to the original number, results in the additive identity, [latex]0[\/latex].<\/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 [latex]8[\/latex], since [latex]\\left(-8\\right)+8=0[\/latex].<\/p>\n<p>The <strong>inverse property of multiplication<\/strong> holds for all real numbers except [latex]0[\/latex] because the reciprocal of [latex]0[\/latex] 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]\\dfrac{1}{a}[\/latex], that, when multiplied by the original number, results in the multiplicative identity, [latex]1[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot\\dfrac{1}{a}\\normalsize =1[\/latex]<\/div>\n<div style=\"text-align: center;\">\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p style=\"text-align: left;\">1) Define the additive inverse of\u00a0[latex]a=-8[\/latex], and use it to illustrate the inverse property of addition.<\/p>\n<p style=\"text-align: left;\">2) Write the reciprocal of\u00a0[latex]a=-\\dfrac{2}{3}[\/latex], and use it to illustrate the inverse property of multiplication.<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q468875\">Show Solution<\/span><\/p>\n<div id=\"q468875\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">1) The additive inverse is [latex]8[\/latex], and\u00a0[latex]\\left(-8\\right)+8=0[\/latex]<\/p>\n<p style=\"text-align: left;\">2) The reciprocal is [latex]-\\dfrac{3}{2}[\/latex]\u00a0and\u00a0[latex]\\left(-\\dfrac{2}{3}\\normalsize\\right)\\cdot \\left(-\\dfrac{3}{2}\\normalsize\\right)=1[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3 style=\"text-align: left;\">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 style=\"width: 70%;\" 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<tbody>\n<tr>\n<th style=\"text-align: center;\"><\/th>\n<th style=\"text-align: center;\"><strong>Addition<\/strong><\/th>\n<th style=\"text-align: center;\"><strong>Multiplication<\/strong><\/th>\n<\/tr>\n<\/tbody>\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<td><\/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 style=\"text-align: center;\">[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 style=\"text-align: center;\">[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 [latex]\u2013a[\/latex], such that<\/p>\n<div style=\"text-align: center;\">[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]\\dfrac{1}{a}[\/latex], such that<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot \\left(\\dfrac{1}{a}\\normalsize\\right)=1[\/latex]<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]3\\left(6+4\\right)[\/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}\\normalsize\\cdot \\left(\\dfrac{2}{3}\\normalsize\\cdot\\dfrac{7}{4}\\normalsize\\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=\"q823624\">Show Solution<\/span><\/p>\n<div id=\"q823624\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\begin{array}{l}\\\\\\\\3\\cdot\\left(6+4\\right)=3\\cdot6+3\\cdot4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Distributive property} \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=18+12\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify} \\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=30\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{Simplify}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{l}\\\\\\\\\\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{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{l}\\\\\\\\6-\\left(15+9\\right) \\hfill& =6+[\\left(-15\\right)+\\left(-9\\right)] \\hfill& \\text{Distributive property} \\\\ \\hfill& =6+\\left(-24\\right) \\hfill& \\text{Simplify} \\\\ \\hfill& =-18 \\hfill& \\text{Simplify}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{l}\\\\\\\\\\\\\\\\\\frac{4}{7}\\cdot\\left(\\frac{2}{3}\\cdot\\frac{7}{4}\\right) \\hfill& =\\frac{4}{7} \\cdot\\left(\\frac{7}{4}\\cdot\\frac{2}{3}\\right) \\hfill& \\text{Commutative property of multiplication} \\\\ \\hfill& =\\left(\\frac{4}{7}\\cdot\\frac{7}{4}\\right)\\cdot\\frac{2}{3}\\hfill& \\text{Associative property of multiplication} \\\\ \\hfill& =1\\cdot\\frac{2}{3} \\hfill& \\text{Inverse property of multiplication} \\\\ \\hfill& =\\frac{2}{3} \\hfill& \\text{Identity property of multiplication}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{l}\\\\\\\\100\\cdot[0.75+\\left(-2.38\\right)] \\hfill& =100\\cdot0.75+100\\cdot\\left(-2.38\\right)\\hfill& \\text{Distributive property} \\\\ \\hfill& =75+\\left(-238\\right) \\hfill& \\text{Simplify} \\\\ \\hfill& =-163 \\hfill& \\text{Simplify}\\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Properties of Real Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8SFm8Os_4C8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-1702\">\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>Properties of Real Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/8SFm8Os_4C8\">https:\/\/youtu.be\/8SFm8Os_4C8<\/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: Using Properties of Real Numbers. <strong>Located at<\/strong>: <a 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