{"id":1108,"date":"2021-10-13T16:31:26","date_gmt":"2021-10-13T16:31:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1108"},"modified":"2026-03-27T17:18:19","modified_gmt":"2026-03-27T17:18:19","slug":"1-2-2-arithmetic-operations-on-integers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/1-2-2-arithmetic-operations-on-integers\/","title":{"raw":"1.2.2: Multiplication and Division of Integers","rendered":"1.2.2: Multiplication and Division of Integers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h1>Learning Outcomes<\/h1>\r\n<ul>\r\n \t<li>Simplify expressions using multiplication of integers<\/li>\r\n \t<li>Simplify expressions using division of integers<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>KEY words<\/h1>\r\n<ul>\r\n \t<li><strong>Product<\/strong>: The result of multiplying two or more numbers<\/li>\r\n \t<li><strong>Quotient<\/strong>: The result of dividing two numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Multiplication of Integers<\/h2>\r\nSince multiplication is mathematical shorthand for repeated addition, our money model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.\r\n\r\nSince multiplication is repeated addition, [latex]a\\cdot b[\/latex] means adding [latex]a[\/latex] to itself [latex]b[\/latex] times. This means that if we multiply our $5 by 3, we add $5 to itself 3 times, which gives us $15. However, if we multiply a $5 debt by 3, we end up with a $15 debt. So, [latex]3\\cdot\\left(-5\\right)=-15[\/latex].\r\n\r\nWhen we multiply two integers, the product is positive<span style=\"font-size: 1em;\">, when the signs are the same,<\/span><span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">and when the signs are different, the product is negative. <em><strong>Product<\/strong><\/em> is the name given to the result of multiplying two or more numbers together.<\/span>\r\n<div class=\"textbox shaded\">\r\n<h3>MULTIPLICATION of Signed Numbers<\/h3>\r\nThe sign of the product of two numbers depends on their signs.\r\n<table style=\"border-collapse: collapse; width: 49.5173%; height: 159px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%;\">Same Signs<\/th>\r\n<th style=\"width: 50%;\">Quotient<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">(+)(+)<\/td>\r\n<td style=\"width: 50%;\">(+)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">(-)(-)<\/td>\r\n<td style=\"width: 50%;\">(+)<\/td>\r\n<\/tr>\r\n<tr>\r\n<th style=\"width: 50%;\">Different Signs<\/th>\r\n<th style=\"width: 50%;\">Quotient<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">(+)(-)<\/td>\r\n<td style=\"width: 50%;\">(-)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">(-)(+)<\/td>\r\n<td style=\"width: 50%;\">(-)<\/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\nMultiply each of the following:\r\n<ol>\r\n \t<li>[latex]-9\\cdot 3[\/latex]<\/li>\r\n \t<li>[latex]-2\\left(-5\\right)[\/latex]<\/li>\r\n \t<li>[latex]4\\left(-8\\right)[\/latex]<\/li>\r\n \t<li>[latex]7\\cdot 6[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1168469461529\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-9\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are different, so the product is negative.<\/td>\r\n<td>[latex]-27[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467249422\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>2.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-2\\left(-5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are the same, so the product is positive.<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466068278\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>3.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4\\left(-8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are different, so the product is negative.<\/td>\r\n<td>[latex]-32[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466077286\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>4.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]7\\cdot 6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are the same, so the product is positive.<\/td>\r\n<td>[latex]42[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145306&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\nWatch the following video for more examples of how to multiply integers.\r\n\r\nhttps:\/\/youtu.be\/QY-Za42DItw\r\n<h3>Properties of Multiplication<\/h3>\r\nWe know that [latex]2\\cdot\\,3=3\\cdot\\,2[\/latex]. But is\u00a0[latex]-2\\cdot\\,3=3\\cdot\\left(-2\\right)[\/latex]? \u00a0Well\u00a0[latex]-2\\cdot\\,3=-6[\/latex] and \u00a0[latex]3\\cdot\\left(-2\\right)=-6[\/latex]. So,\u00a0[latex]-2\\cdot\\,3=3\\cdot\\left(-2\\right)[\/latex]. In fact, this is true for all integer values and is called the\u00a0<strong><em>commutative property of multiplication<\/em><\/strong><strong>.<\/strong>\u00a0T<span style=\"font-size: 1em;\">he order<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0that we multiply integers\u00a0doesn't matter. <\/span>\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">It is also true that multiplying any integer by [latex]1[\/latex] has no effect on the integer. For example, [latex]-5\\cdot\\,1=-5[\/latex]. Because [latex]1[\/latex] does not change the identity of any integer it is multiplied onto. \u00a0[latex]1[\/latex] is called the\u00a0<\/span><em style=\"font-size: 1rem; text-align: initial;\"><strong>multiplicative identity<\/strong><\/em><span style=\"font-size: 1rem; text-align: initial;\">.<\/span>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3>COMMUTATIVE PROPERTY OF multiplication<\/h3>\r\nFor any integers [latex]a[\/latex] and [latex]b[\/latex], [latex]a\\cdot\\,b=b\\cdot\\,a[\/latex].\r\n\r\nThe order in which we multiply integers does not matter.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>THE MULTIPLICATIVE IDENTITY<\/h3>\r\n[latex]1\\cdot\\,a=a[\/latex], for any integer [latex]a[\/latex]<span style=\"font-size: 1rem; text-align: initial;\">.<\/span>\r\n\r\nMultiplying an integer by [latex]1[\/latex] does not change the value of the integer.\r\n\r\n[latex]1[\/latex] is the multiplicative identity.\r\n\r\n<\/div>\r\nWhat about multiplying by [latex]-1[\/latex]?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill -1\\cdot 4\\hfill &amp; &amp; \\hfill -1\\left(-3\\right)\\hfill \\\\ \\hfill -4\\hfill &amp; &amp; \\hfill 3\\hfill \\\\ \\hfill -4\\text{ is the opposite of }\\mathbf{\\text{4}}\\hfill &amp; &amp; \\hfill \\mathbf{\\text{3}}\\text{ is the opposite of }-3\\hfill \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Each time we multiply an integer by [latex]-1[\/latex], we get its opposite.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplication by [latex]-1[\/latex]<\/h3>\r\nMultiplying an integer by [latex]-1[\/latex] gives the opposite of the integer.\r\n\r\n[latex]-1\\cdot\\,a=-a[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply each of the following:\r\n<ol>\r\n \t<li>[latex]-1\\cdot 7[\/latex]<\/li>\r\n \t<li>[latex]-1\\left(-11\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"459967\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"459967\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468284281\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are different, so the product will be negative.<\/td>\r\n<td>[latex]-1\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that [latex]\u22127[\/latex] is the opposite of [latex]7[\/latex].<\/td>\r\n<td>[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469855058\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>2.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are the same, so the product will be positive.<\/td>\r\n<td>[latex]-1\\left(-11\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that [latex]11[\/latex] is the opposite of [latex]\u221211[\/latex].<\/td>\r\n<td>[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nZero is another number that has the same result for all integers. Any integer multiplied by [latex]0[\/latex] results in a product of [latex]0[\/latex]. In other words, [latex]a\\cdot\\,0=0[\/latex] for any integer [latex]a[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFind the product:\r\n<ul>\r\n \t<li>[latex]-3(0)=0[\/latex]<\/li>\r\n \t<li>[latex]-4(-7)(0)=0[\/latex]<\/li>\r\n \t<li>[latex]5(-4)(-3)(7)(-2)(8)(0)=0[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplication by [latex]0[\/latex]<\/h3>\r\nMultiplying an integer by [latex]0[\/latex] gives [latex]0[\/latex].\r\n\r\n[latex]0\\cdot\\,a=0[\/latex]\r\n\r\n<\/div>\r\n<h3>Multiplying more than two integers<\/h3>\r\nWhen we multiply more than two integers, we multiply them two at a time.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMultiply [latex]-2\\cdot 4\\cdot \\left(-5\\right)[\/latex]\r\n<h4>Solution<\/h4>\r\nMultiplying from left to right: [latex]-2\\cdot 4\\cdot \\left(-5\\right)=-8\\cdot \\left(-5\\right)=40[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nMultiply [latex]2\\cdot\\left(-10\\right)\\cdot\\left(-7\\right)\\cdot\\,5[\/latex]\r\n\r\n[reveal-answer q=\"H2878\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"H2878\"]\r\n\r\n[latex]2\\cdot\\left(-10\\right)\\cdot\\left(-7\\right)\\cdot 5=700[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAnother way to solve the problem [latex]2\\cdot\\left(-10\\right)\\cdot\\left(-7\\right)\\cdot 5[\/latex] is to notice that [latex]2\\cdot 5=10[\/latex] is part of problem. To use this requires reorganizing the problem using the commutative property to:\u00a0[latex]2\\cdot 5\\cdot\\left(-10\\right)\\cdot\\left(-7\\right)[\/latex]. Then\u00a0[latex]2\\cdot 5=10[\/latex] and\u00a0[latex]\\left(-10\\right)\\cdot\\left(-7\\right)=70[\/latex]. Multiplying these products gives:\u00a0[latex]10\\cdot\\left(70\\right)=700[\/latex]. The same answer as before. Regrouping the numbers is an example of the\u00a0<strong><i>associative property of multiplication<\/i><\/strong>.\r\n<div class=\"textbox shaded\">\r\n<h3>THE ASSOCIATIVE PROPERTY OF multiplication<\/h3>\r\n[latex]\\left(a\\cdot\\,b\\right)\\cdot\\,c=a\\cdot\\,\\left(b\\cdot\\,c\\right)[\/latex] for any integers [latex]a, b, c.[\/latex]\r\n\r\nRegrouping the integers results in the same product.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nMultiply: [latex]-1\\cdot 8\\cdot\\left(-12\\right)\\cdot 5[\/latex]\r\n\r\n[reveal-answer q=\"H5850\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"H5850\"]\r\n\r\n[latex]\\:\\:-1\\cdot 8\\cdot\\left(-12\\right)\\cdot5[\/latex]\r\n\r\n= [latex]\\left(8\\cdot 5 \\right)\\cdot\\left(-1\\right)\\cdot\\left(-12\\right)[\/latex]\r\n\r\n=\u00a0[latex]\\left(40\\right)\\cdot\\left(12\\right)[\/latex]\r\n\r\n= [latex]480[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Division of Integers<\/h2>\r\nDivision is the inverse operation of multiplication. So, [latex]15\\div 3=5[\/latex] because [latex]5\\cdot 3=15[\/latex]. In words, this expression says that [latex]15[\/latex] can be divided into [latex]3[\/latex] groups of [latex]5[\/latex] because adding five to itself three times gives [latex]15[\/latex]. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccc}5\\cdot 3=15\\text{ so }15\\div 3=5\\hfill &amp; &amp; &amp; &amp; -5\\left(3\\right)=-15\\text{ so }-15\\div 3=-5\\hfill \\\\ \\left(-5\\right)\\left(-3\\right)=15\\text{ so }15\\div \\left(-3\\right)=-5\\hfill &amp; &amp; &amp; &amp; 5\\left(-3\\right)=-15\\text{ so }-15\\div -3=5\\hfill \\end{array}[\/latex]<\/p>\r\nDivision of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.\r\n<div class=\"textbox shaded\">\r\n<h3>Division of Signed Numbers<\/h3>\r\nThe sign of the quotient of two numbers depends on their signs.\r\n<table style=\"border-collapse: collapse; width: 47.785235%; height: 72px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 50%; height: 12px;\">Same Signs<\/th>\r\n<th style=\"width: 50%; height: 12px;\">Quotient<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">(+)(+)<\/td>\r\n<td style=\"width: 50%; height: 12px;\">(+)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">(-)(-)<\/td>\r\n<td style=\"width: 50%; height: 12px;\">(+)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 50%; height: 12px;\">Different Signs<\/th>\r\n<th style=\"width: 50%; height: 12px;\">Quotient<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">(+)(-)<\/td>\r\n<td style=\"width: 50%; height: 12px;\">(-)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 50%; height: 12px;\">(-)(+)<\/td>\r\n<td style=\"width: 50%; height: 12px;\">(-)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nRemember, you can always check the answer to a division problem by multiplying.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide each of the following and check the answer:\r\n<ol>\r\n \t<li>[latex]-27\\div 3[\/latex]<\/li>\r\n \t<li>[latex]-100\\div \\left(-4\\right)[\/latex]<\/li>\r\n<\/ol>\r\nSolution\r\n<table id=\"eip-id1168467157728\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 15.7812px;\">\r\n<th style=\"height: 15.7812px;\">1.<\/th>\r\n<td style=\"height: 15.7812px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]-27\\div 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">The signs are different, so the quotient is negative.<\/td>\r\n<td style=\"height: 15px;\">[latex]-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check<\/td>\r\n<td>[latex](-9)(3)=-27[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467157714\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>2.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-100\\div \\left(-4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are the same, so the quotient is positive.<\/td>\r\n<td>[latex]25[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check<\/td>\r\n<td>[layex]25(-4)=-100[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145323[\/ohm_question]\r\n\r\n<\/div>\r\nJust as we saw with multiplication, when we divide an integer by [latex]1[\/latex], the result is the same number. What happens when we divide an integer by [latex]-1?[\/latex] Let\u2019s divide a positive number and then a negative number by [latex]-1[\/latex] to see what we get.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cccc}8\\div \\left(-1\\right)\\hfill &amp; &amp; &amp; -9\\div \\left(-1\\right)\\hfill \\\\ -8\\hfill &amp; &amp; &amp; 9\\hfill \\\\ \\hfill \\text{-8 is the opposite of 8}\\hfill &amp; &amp; &amp; \\hfill \\text{9 is the opposite of -9}\\hfill \\end{array}[\/latex]<\/p>\r\nWhen we divide a number by [latex]-1[\/latex] we get its opposite.\r\n<div class=\"textbox shaded\">\r\n<h3>Division by [latex]-1[\/latex]<\/h3>\r\n[latex]a\\div \\left(-1\\right)=-a[\/latex], for any integer [latex]a[\/latex]\r\n\r\nDividing a number by [latex]-1[\/latex] gives its opposite.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide each of the following:\r\n<ol>\r\n \t<li>[latex]16\\div \\left(-1\\right)[\/latex]<\/li>\r\n \t<li>[latex]-20\\div \\left(-1\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"343427\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"343427\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466004748\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]16\\div \\left(-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The dividend, [latex]16[\/latex], is being divided by [latex]\u20131.[\/latex]<\/td>\r\n<td>[latex]-16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Dividing a number by [latex]\u20131[\/latex] gives its opposite.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Notice that the signs were different, so the result was negative.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469488934\" class=\"unstyled\" style=\"width: 85%; height: 84px;\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"height: 12px;\">2.<\/th>\r\n<td style=\"height: 12px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px;\"><\/td>\r\n<td style=\"height: 24px;\">[latex]-20\\div \\left(-1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px;\">The dividend, [latex]\u201320[\/latex], is being divided by [latex]\u20131.[\/latex]<\/td>\r\n<td style=\"height: 24px;\">[latex]20[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px;\">Dividing a number by [latex]\u20131[\/latex] gives its opposite.<\/td>\r\n<td style=\"height: 12px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px;\">Notice that the signs were the same, so the result was positive.<\/td>\r\n<td style=\"height: 12px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145326[\/ohm_question]\r\n\r\n<\/div>\r\nDividing any number [latex]\\text{(except 0)}[\/latex] by itself, produces a quotient of [latex]1[\/latex]. Also, any number divided by [latex]1[\/latex] produces a quotient of the number. These two ideas are stated in the Division Properties of One.\r\n<div class=\"textbox shaded\">\r\n<h3>Division Properties of One<\/h3>\r\n<table id=\"eip-735\" summary=\"a\">\r\n<tbody>\r\n<tr>\r\n<th>Property<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px;\">Any number (except 0) divided by itself is one.<\/td>\r\n<td style=\"height: 14px;\">[latex]a\\div a=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14.4585px;\">\r\n<td style=\"height: 14.4585px;\">Any number divided by one is the same number.<\/td>\r\n<td style=\"height: 14.4585px;\">[latex]a\\div 1=a[\/latex]<\/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\nDivide. Then check by multiplying:\r\n<ol id=\"eip-id1168288568257\" class=\"circled\">\r\n \t<li>[latex]11\\div 11[\/latex]<\/li>\r\n \t<li>[latex]\\frac{19}{1}[\/latex]<\/li>\r\n \t<li>[latex]1\\overline{)7}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"519474\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"519474\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168288480300\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]11\\div 11[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A number divided by itself is [latex]1[\/latex].<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]1\\cdot 11[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]11\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<table id=\"eip-id1168288536381\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\">\r\n<tbody>\r\n<tr>\r\n<th>2.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\frac{19}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A number divided by [latex]1[\/latex] equals itself.<\/td>\r\n<td>[latex]19[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]19\\cdot 1[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]19\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<table id=\"eip-id1168289599719\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\">\r\n<tbody>\r\n<tr>\r\n<th>3.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]1\\overline{)7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A number divided by [latex]1[\/latex] equals itself.<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]7\\cdot 1[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]7\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144635[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nSuppose we have [latex]\\text{\\$0}[\/latex], and want to divide it among [latex]3[\/latex] people. How much would each person get? Each person would get [latex]\\text{\\$0}[\/latex]. Zero divided by any number is [latex]0[\/latex].\r\n\r\nNow suppose that we want to divide [latex]\\text{\\$10}[\/latex] by [latex]0[\/latex]. That means we would want to find a number that we multiply by [latex]0[\/latex], to get [latex]10[\/latex]. This cannot happen, because [latex]0[\/latex] times any number is [latex]0[\/latex]. Division by zero is said to be <strong><em>undefined<\/em><\/strong>.\r\n\r\nThese two ideas make up the Division Properties of Zero.\r\n<div class=\"textbox shaded\">\r\n<h3>Division Properties of Zero<\/h3>\r\n<table id=\"eip-158\" summary=\"a\">\r\n<tbody>\r\n<tr>\r\n<th>Property<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Zero divided by any number is [latex]0[\/latex].<\/td>\r\n<td>[latex]0\\div a=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Dividing a number by zero is undefined.<\/td>\r\n<td>[latex]a\\div 0 = [\/latex] undefined<\/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\nDivide. Check by multiplying:\r\n<ol>\r\n \t<li>[latex]0\\div 3[\/latex]<\/li>\r\n \t<li>[latex]\\frac{10}{0}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"208505\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"208505\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168288542869\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \">\r\n<tbody>\r\n<tr>\r\n<th>1.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]0\\div 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Zero divided by any number is zero.<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]0\\cdot 3[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<table id=\"eip-id1168288419363\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \">\r\n<tbody>\r\n<tr>\r\n<th>2.<\/th>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]10\/0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Division by zero is undefined.<\/td>\r\n<td>undefined<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144478[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFind the quotient:\r\n<ul>\r\n \t<li>[latex]0\\div (-24)[\/latex]<\/li>\r\n \t<li>[latex]45\\div 0[\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"520669\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"520669\"]\r\n<ul>\r\n \t<li>0<\/li>\r\n \t<li>undefined<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video for more examples of how to divide integers with the same and different signs.\r\n\r\nhttps:\/\/youtu.be\/z5ZFiyLi5Y0","rendered":"<div class=\"textbox learning-objectives\">\n<h1>Learning Outcomes<\/h1>\n<ul>\n<li>Simplify expressions using multiplication of integers<\/li>\n<li>Simplify expressions using division of integers<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>KEY words<\/h1>\n<ul>\n<li><strong>Product<\/strong>: The result of multiplying two or more numbers<\/li>\n<li><strong>Quotient<\/strong>: The result of dividing two numbers<\/li>\n<\/ul>\n<\/div>\n<h2>Multiplication of Integers<\/h2>\n<p>Since multiplication is mathematical shorthand for repeated addition, our money model can easily be applied to show multiplication of integers. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.<\/p>\n<p>Since multiplication is repeated addition, [latex]a\\cdot b[\/latex] means adding [latex]a[\/latex] to itself [latex]b[\/latex] times. This means that if we multiply our $5 by 3, we add $5 to itself 3 times, which gives us $15. However, if we multiply a $5 debt by 3, we end up with a $15 debt. So, [latex]3\\cdot\\left(-5\\right)=-15[\/latex].<\/p>\n<p>When we multiply two integers, the product is positive<span style=\"font-size: 1em;\">, when the signs are the same,<\/span><span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">and when the signs are different, the product is negative. <em><strong>Product<\/strong><\/em> is the name given to the result of multiplying two or more numbers together.<\/span><\/p>\n<div class=\"textbox shaded\">\n<h3>MULTIPLICATION of Signed Numbers<\/h3>\n<p>The sign of the product of two numbers depends on their signs.<\/p>\n<table style=\"border-collapse: collapse; width: 49.5173%; height: 159px;\">\n<tbody>\n<tr>\n<th style=\"width: 50%;\">Same Signs<\/th>\n<th style=\"width: 50%;\">Quotient<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">(+)(+)<\/td>\n<td style=\"width: 50%;\">(+)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">(-)(-)<\/td>\n<td style=\"width: 50%;\">(+)<\/td>\n<\/tr>\n<tr>\n<th style=\"width: 50%;\">Different Signs<\/th>\n<th style=\"width: 50%;\">Quotient<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">(+)(-)<\/td>\n<td style=\"width: 50%;\">(-)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">(-)(+)<\/td>\n<td style=\"width: 50%;\">(-)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply each of the following:<\/p>\n<ol>\n<li>[latex]-9\\cdot 3[\/latex]<\/li>\n<li>[latex]-2\\left(-5\\right)[\/latex]<\/li>\n<li>[latex]4\\left(-8\\right)[\/latex]<\/li>\n<li>[latex]7\\cdot 6[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1168469461529\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<th>1.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-9\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The signs are different, so the product is negative.<\/td>\n<td>[latex]-27[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467249422\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<th>2.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-2\\left(-5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The signs are the same, so the product is positive.<\/td>\n<td>[latex]10[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466068278\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<th>3.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4\\left(-8\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The signs are different, so the product is negative.<\/td>\n<td>[latex]-32[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466077286\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<th>4.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]7\\cdot 6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The signs are the same, so the product is positive.<\/td>\n<td>[latex]42[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145306&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video for more examples of how to multiply integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Multiplying Integers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QY-Za42DItw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Properties of Multiplication<\/h3>\n<p>We know that [latex]2\\cdot\\,3=3\\cdot\\,2[\/latex]. But is\u00a0[latex]-2\\cdot\\,3=3\\cdot\\left(-2\\right)[\/latex]? \u00a0Well\u00a0[latex]-2\\cdot\\,3=-6[\/latex] and \u00a0[latex]3\\cdot\\left(-2\\right)=-6[\/latex]. So,\u00a0[latex]-2\\cdot\\,3=3\\cdot\\left(-2\\right)[\/latex]. In fact, this is true for all integer values and is called the\u00a0<strong><em>commutative property of multiplication<\/em><\/strong><strong>.<\/strong>\u00a0T<span style=\"font-size: 1em;\">he order<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0that we multiply integers\u00a0doesn&#8217;t matter. <\/span><\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">It is also true that multiplying any integer by [latex]1[\/latex] has no effect on the integer. For example, [latex]-5\\cdot\\,1=-5[\/latex]. Because [latex]1[\/latex] does not change the identity of any integer it is multiplied onto. \u00a0[latex]1[\/latex] is called the\u00a0<\/span><em style=\"font-size: 1rem; text-align: initial;\"><strong>multiplicative identity<\/strong><\/em><span style=\"font-size: 1rem; text-align: initial;\">.<\/span><\/p>\n<div>\n<div class=\"textbox shaded\">\n<h3>COMMUTATIVE PROPERTY OF multiplication<\/h3>\n<p>For any integers [latex]a[\/latex] and [latex]b[\/latex], [latex]a\\cdot\\,b=b\\cdot\\,a[\/latex].<\/p>\n<p>The order in which we multiply integers does not matter.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>THE MULTIPLICATIVE IDENTITY<\/h3>\n<p>[latex]1\\cdot\\,a=a[\/latex], for any integer [latex]a[\/latex]<span style=\"font-size: 1rem; text-align: initial;\">.<\/span><\/p>\n<p>Multiplying an integer by [latex]1[\/latex] does not change the value of the integer.<\/p>\n<p>[latex]1[\/latex] is the multiplicative identity.<\/p>\n<\/div>\n<p>What about multiplying by [latex]-1[\/latex]?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill -1\\cdot 4\\hfill & & \\hfill -1\\left(-3\\right)\\hfill \\\\ \\hfill -4\\hfill & & \\hfill 3\\hfill \\\\ \\hfill -4\\text{ is the opposite of }\\mathbf{\\text{4}}\\hfill & & \\hfill \\mathbf{\\text{3}}\\text{ is the opposite of }-3\\hfill \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Each time we multiply an integer by [latex]-1[\/latex], we get its opposite.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplication by [latex]-1[\/latex]<\/h3>\n<p>Multiplying an integer by [latex]-1[\/latex] gives the opposite of the integer.<\/p>\n<p>[latex]-1\\cdot\\,a=-a[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply each of the following:<\/p>\n<ol>\n<li>[latex]-1\\cdot 7[\/latex]<\/li>\n<li>[latex]-1\\left(-11\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q459967\">Show Solution<\/span><\/p>\n<div id=\"q459967\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468284281\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<th>1.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td>The signs are different, so the product will be negative.<\/td>\n<td>[latex]-1\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Notice that [latex]\u22127[\/latex] is the opposite of [latex]7[\/latex].<\/td>\n<td>[latex]-7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469855058\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<th>2.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td>The signs are the same, so the product will be positive.<\/td>\n<td>[latex]-1\\left(-11\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Notice that [latex]11[\/latex] is the opposite of [latex]\u221211[\/latex].<\/td>\n<td>[latex]11[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Zero is another number that has the same result for all integers. Any integer multiplied by [latex]0[\/latex] results in a product of [latex]0[\/latex]. In other words, [latex]a\\cdot\\,0=0[\/latex] for any integer [latex]a[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Find the product:<\/p>\n<ul>\n<li>[latex]-3(0)=0[\/latex]<\/li>\n<li>[latex]-4(-7)(0)=0[\/latex]<\/li>\n<li>[latex]5(-4)(-3)(7)(-2)(8)(0)=0[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Multiplication by [latex]0[\/latex]<\/h3>\n<p>Multiplying an integer by [latex]0[\/latex] gives [latex]0[\/latex].<\/p>\n<p>[latex]0\\cdot\\,a=0[\/latex]<\/p>\n<\/div>\n<h3>Multiplying more than two integers<\/h3>\n<p>When we multiply more than two integers, we multiply them two at a time.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Multiply [latex]-2\\cdot 4\\cdot \\left(-5\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Multiplying from left to right: [latex]-2\\cdot 4\\cdot \\left(-5\\right)=-8\\cdot \\left(-5\\right)=40[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Multiply [latex]2\\cdot\\left(-10\\right)\\cdot\\left(-7\\right)\\cdot\\,5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qH2878\">Show Answer<\/span><\/p>\n<div id=\"qH2878\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]2\\cdot\\left(-10\\right)\\cdot\\left(-7\\right)\\cdot 5=700[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Another way to solve the problem [latex]2\\cdot\\left(-10\\right)\\cdot\\left(-7\\right)\\cdot 5[\/latex] is to notice that [latex]2\\cdot 5=10[\/latex] is part of problem. To use this requires reorganizing the problem using the commutative property to:\u00a0[latex]2\\cdot 5\\cdot\\left(-10\\right)\\cdot\\left(-7\\right)[\/latex]. Then\u00a0[latex]2\\cdot 5=10[\/latex] and\u00a0[latex]\\left(-10\\right)\\cdot\\left(-7\\right)=70[\/latex]. Multiplying these products gives:\u00a0[latex]10\\cdot\\left(70\\right)=700[\/latex]. The same answer as before. Regrouping the numbers is an example of the\u00a0<strong><i>associative property of multiplication<\/i><\/strong>.<\/p>\n<div class=\"textbox shaded\">\n<h3>THE ASSOCIATIVE PROPERTY OF multiplication<\/h3>\n<p>[latex]\\left(a\\cdot\\,b\\right)\\cdot\\,c=a\\cdot\\,\\left(b\\cdot\\,c\\right)[\/latex] for any integers [latex]a, b, c.[\/latex]<\/p>\n<p>Regrouping the integers results in the same product.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Multiply: [latex]-1\\cdot 8\\cdot\\left(-12\\right)\\cdot 5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qH5850\">Show Answer<\/span><\/p>\n<div id=\"qH5850\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\:\\:-1\\cdot 8\\cdot\\left(-12\\right)\\cdot5[\/latex]<\/p>\n<p>= [latex]\\left(8\\cdot 5 \\right)\\cdot\\left(-1\\right)\\cdot\\left(-12\\right)[\/latex]<\/p>\n<p>=\u00a0[latex]\\left(40\\right)\\cdot\\left(12\\right)[\/latex]<\/p>\n<p>= [latex]480[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Division of Integers<\/h2>\n<p>Division is the inverse operation of multiplication. So, [latex]15\\div 3=5[\/latex] because [latex]5\\cdot 3=15[\/latex]. In words, this expression says that [latex]15[\/latex] can be divided into [latex]3[\/latex] groups of [latex]5[\/latex] because adding five to itself three times gives [latex]15[\/latex]. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccccc}5\\cdot 3=15\\text{ so }15\\div 3=5\\hfill & & & & -5\\left(3\\right)=-15\\text{ so }-15\\div 3=-5\\hfill \\\\ \\left(-5\\right)\\left(-3\\right)=15\\text{ so }15\\div \\left(-3\\right)=-5\\hfill & & & & 5\\left(-3\\right)=-15\\text{ so }-15\\div -3=5\\hfill \\end{array}[\/latex]<\/p>\n<p>Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division of Signed Numbers<\/h3>\n<p>The sign of the quotient of two numbers depends on their signs.<\/p>\n<table style=\"border-collapse: collapse; width: 47.785235%; height: 72px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 50%; height: 12px;\">Same Signs<\/th>\n<th style=\"width: 50%; height: 12px;\">Quotient<\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">(+)(+)<\/td>\n<td style=\"width: 50%; height: 12px;\">(+)<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">(-)(-)<\/td>\n<td style=\"width: 50%; height: 12px;\">(+)<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<th style=\"width: 50%; height: 12px;\">Different Signs<\/th>\n<th style=\"width: 50%; height: 12px;\">Quotient<\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">(+)(-)<\/td>\n<td style=\"width: 50%; height: 12px;\">(-)<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 50%; height: 12px;\">(-)(+)<\/td>\n<td style=\"width: 50%; height: 12px;\">(-)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Remember, you can always check the answer to a division problem by multiplying.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide each of the following and check the answer:<\/p>\n<ol>\n<li>[latex]-27\\div 3[\/latex]<\/li>\n<li>[latex]-100\\div \\left(-4\\right)[\/latex]<\/li>\n<\/ol>\n<p>Solution<\/p>\n<table id=\"eip-id1168467157728\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr style=\"height: 15.7812px;\">\n<th style=\"height: 15.7812px;\">1.<\/th>\n<td style=\"height: 15.7812px;\"><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]-27\\div 3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">The signs are different, so the quotient is negative.<\/td>\n<td style=\"height: 15px;\">[latex]-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check<\/td>\n<td>[latex](-9)(3)=-27[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467157714\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<th>2.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-100\\div \\left(-4\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The signs are the same, so the quotient is positive.<\/td>\n<td>[latex]25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check<\/td>\n<td>[layex]25(-4)=-100[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145323\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145323&theme=oea&iframe_resize_id=ohm145323&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Just as we saw with multiplication, when we divide an integer by [latex]1[\/latex], the result is the same number. What happens when we divide an integer by [latex]-1?[\/latex] Let\u2019s divide a positive number and then a negative number by [latex]-1[\/latex] to see what we get.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cccc}8\\div \\left(-1\\right)\\hfill & & & -9\\div \\left(-1\\right)\\hfill \\\\ -8\\hfill & & & 9\\hfill \\\\ \\hfill \\text{-8 is the opposite of 8}\\hfill & & & \\hfill \\text{9 is the opposite of -9}\\hfill \\end{array}[\/latex]<\/p>\n<p>When we divide a number by [latex]-1[\/latex] we get its opposite.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division by [latex]-1[\/latex]<\/h3>\n<p>[latex]a\\div \\left(-1\\right)=-a[\/latex], for any integer [latex]a[\/latex]<\/p>\n<p>Dividing a number by [latex]-1[\/latex] gives its opposite.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide each of the following:<\/p>\n<ol>\n<li>[latex]16\\div \\left(-1\\right)[\/latex]<\/li>\n<li>[latex]-20\\div \\left(-1\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q343427\">Show Solution<\/span><\/p>\n<div id=\"q343427\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466004748\" class=\"unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<th>1.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]16\\div \\left(-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The dividend, [latex]16[\/latex], is being divided by [latex]\u20131.[\/latex]<\/td>\n<td>[latex]-16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Dividing a number by [latex]\u20131[\/latex] gives its opposite.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Notice that the signs were different, so the result was negative.<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469488934\" class=\"unstyled\" style=\"width: 85%; height: 84px;\" summary=\".\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"height: 12px;\">2.<\/th>\n<td style=\"height: 12px;\"><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px;\"><\/td>\n<td style=\"height: 24px;\">[latex]-20\\div \\left(-1\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px;\">The dividend, [latex]\u201320[\/latex], is being divided by [latex]\u20131.[\/latex]<\/td>\n<td style=\"height: 24px;\">[latex]20[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px;\">Dividing a number by [latex]\u20131[\/latex] gives its opposite.<\/td>\n<td style=\"height: 12px;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px;\">Notice that the signs were the same, so the result was positive.<\/td>\n<td style=\"height: 12px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145326\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145326&theme=oea&iframe_resize_id=ohm145326&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Dividing any number [latex]\\text{(except 0)}[\/latex] by itself, produces a quotient of [latex]1[\/latex]. Also, any number divided by [latex]1[\/latex] produces a quotient of the number. These two ideas are stated in the Division Properties of One.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division Properties of One<\/h3>\n<table id=\"eip-735\" summary=\"a\">\n<tbody>\n<tr>\n<th>Property<\/th>\n<td><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px;\">Any number (except 0) divided by itself is one.<\/td>\n<td style=\"height: 14px;\">[latex]a\\div a=1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14.4585px;\">\n<td style=\"height: 14.4585px;\">Any number divided by one is the same number.<\/td>\n<td style=\"height: 14.4585px;\">[latex]a\\div 1=a[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide. Then check by multiplying:<\/p>\n<ol id=\"eip-id1168288568257\" class=\"circled\">\n<li>[latex]11\\div 11[\/latex]<\/li>\n<li>[latex]\\frac{19}{1}[\/latex]<\/li>\n<li>[latex]1\\overline{)7}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q519474\">Show Solution<\/span><\/p>\n<div id=\"q519474\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168288480300\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\">\n<tbody>\n<tr>\n<th>1.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]11\\div 11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>A number divided by itself is [latex]1[\/latex].<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]1\\cdot 11[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]11\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table id=\"eip-id1168288536381\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\">\n<tbody>\n<tr>\n<th>2.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{19}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>A number divided by [latex]1[\/latex] equals itself.<\/td>\n<td>[latex]19[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]19\\cdot 1[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]19\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table id=\"eip-id1168289599719\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\">\n<tbody>\n<tr>\n<th>3.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]1\\overline{)7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>A number divided by [latex]1[\/latex] equals itself.<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]7\\cdot 1[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]7\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144635\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144635&theme=oea&iframe_resize_id=ohm144635&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Suppose we have [latex]\\text{\\$0}[\/latex], and want to divide it among [latex]3[\/latex] people. How much would each person get? Each person would get [latex]\\text{\\$0}[\/latex]. Zero divided by any number is [latex]0[\/latex].<\/p>\n<p>Now suppose that we want to divide [latex]\\text{\\$10}[\/latex] by [latex]0[\/latex]. That means we would want to find a number that we multiply by [latex]0[\/latex], to get [latex]10[\/latex]. This cannot happen, because [latex]0[\/latex] times any number is [latex]0[\/latex]. Division by zero is said to be <strong><em>undefined<\/em><\/strong>.<\/p>\n<p>These two ideas make up the Division Properties of Zero.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division Properties of Zero<\/h3>\n<table id=\"eip-158\" summary=\"a\">\n<tbody>\n<tr>\n<th>Property<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Zero divided by any number is [latex]0[\/latex].<\/td>\n<td>[latex]0\\div a=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Dividing a number by zero is undefined.<\/td>\n<td>[latex]a\\div 0 =[\/latex] undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide. Check by multiplying:<\/p>\n<ol>\n<li>[latex]0\\div 3[\/latex]<\/li>\n<li>[latex]\\frac{10}{0}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q208505\">Show Solution<\/span><\/p>\n<div id=\"q208505\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168288542869\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\">\n<tbody>\n<tr>\n<th>1.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]0\\div 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Zero divided by any number is zero.<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]0\\cdot 3[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]0\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table id=\"eip-id1168288419363\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\">\n<tbody>\n<tr>\n<th>2.<\/th>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]10\/0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Division by zero is undefined.<\/td>\n<td>undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144478\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144478&theme=oea&iframe_resize_id=ohm144478&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Find the quotient:<\/p>\n<ul>\n<li>[latex]0\\div (-24)[\/latex]<\/li>\n<li>[latex]45\\div 0[\/latex]<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q520669\">Show Answer<\/span><\/p>\n<div id=\"q520669\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>0<\/li>\n<li>undefined<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video for more examples of how to divide integers with the same and different signs.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Dividing Integers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/z5ZFiyLi5Y0?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-1108\">\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>Multiplying by Zero; Dividing by Zero; Multiplicative Properties. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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>Ex: Dividing Integers. <strong>Authored by<\/strong>: James Sousa. <strong>Provided by<\/strong>: Mathispower4u.com. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/z5ZFiyLi5Y0\">https:\/\/youtu.be\/z5ZFiyLi5Y0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID: 145323, 145326. <strong>Authored by<\/strong>: Alyson Day. <strong>Project<\/strong>: IMathAS Community License CC-BY + GPL. <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":422605,"menu_order":4,"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\":\"Ex: Dividing Integers\",\"author\":\"James Sousa\",\"organization\":\"Mathispower4u.com\",\"url\":\"https:\/\/youtu.be\/z5ZFiyLi5Y0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Multiplying by Zero; Dividing by Zero; Multiplicative Properties\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID: 145323, 145326\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"IMathAS Community License CC-BY + GPL\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1108","chapter","type-chapter","status-publish","hentry"],"part":587,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1108","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/422605"}],"version-history":[{"count":29,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1108\/revisions"}],"predecessor-version":[{"id":3210,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1108\/revisions\/3210"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/587"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/1108\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=1108"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1108"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=1108"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=1108"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}