{"id":3580,"date":"2019-12-31T17:58:14","date_gmt":"2019-12-31T17:58:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3580"},"modified":"2021-02-05T23:49:11","modified_gmt":"2021-02-05T23:49:11","slug":"multiplying-integers","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/multiplying-integers\/","title":{"raw":"Multiplying Integers","rendered":"Multiplying Integers"},"content":{"raw":"<div class=\"textbox learning-objectives\"><h3>Learning Outcomes<\/h3><ul><li>Multiply integers with different signs<\/li><li>Multiply integers with the same sign<\/li><li>Multiply an integer by -1<\/li><\/ul><\/div>Since multiplication is mathematical shorthand for repeated addition, our counter 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.\n\nWe remember that [latex]a\\cdot b[\/latex] means add [latex]a,b[\/latex] times. Here, we are using the model shown in the graphic below&nbsp;just to help us discover the pattern.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220519\/CNX_BMath_Figure_03_04_001.png\" alt=\"This image has two columns. The first column has 5 times 3. Underneath, it states add 5, 3 times. Under this there are 3 rows of 5 blue circles labeled 15 positives and 5 times 3 equals 15. The second column has negative 5 times 3. Underneath it states add negative 5, 3 times. Under this there are 3 rows of 5 red circles labeled 15 negatives and negative 5 times 3 equals 15.\">\nNow consider what it means to multiply [latex]5[\/latex] by [latex]-3[\/latex]. It means subtract [latex]5,3[\/latex] times. Looking at subtraction as <em>taking away<\/em>, it means to take away [latex]5,3[\/latex] times. But there is nothing to take away, so we start by adding neutral pairs as shown in the graphic below.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220522\/CNX_BMath_Figure_03_04_004-1.png\" alt=\"This figure has 2 columns. The first column has 5 times negative 3. Underneath it states take away 5, 3 times. Under this there are 3 rows of 5 red circles. A downward arrow points to six rows of alternating colored circles in rows of fives. The first row includes 5 red circles, followed by five blue circles, then 5 red, five blue, five red, and five blue. All of the rows of blue circles are circled. The non-circled rows are labeled 15 negatives. Under the label is 5 times negative 3 equals negative 15. The second column has negative 5 times negative 3. Underneath it states take away negative 5, 3 times. Then there are 6 rows of 5 circles alternating in color. The first row is 5 blue circles followed by 5 red circles. All of the red rows are circled. The non-circles rows are labeled 15 positives. Under the label is negative 5 times negative 3 equals 15.\">\nIn both cases, we started with [latex]\\mathbf{\\text{15}}[\/latex] neutral pairs. In the case on the left, we took away [latex]\\mathbf{\\text{5}},\\mathbf{\\text{3}}[\/latex] times and the result was [latex]-\\mathbf{\\text{15}}[\/latex]. To multiply [latex]\\left(-5\\right)\\left(-3\\right)[\/latex], we took away [latex]-\\mathbf{\\text{5}},\\mathbf{\\text{3}}[\/latex] times and the result was [latex]\\mathbf{\\text{15}}[\/latex]. So we found that\n\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}5\\cdot 3=15\\hfill &amp; &amp; -5\\left(3\\right)=-15\\hfill \\\\ 5\\left(-3\\right)=-15\\hfill &amp; &amp; \\left(-5\\right)\\left(-3\\right)=15\\hfill \\end{array}[\/latex]\n\nNotice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.\n\n<div class=\"textbox shaded\"><h3>Multiplication of Signed Numbers<\/h3>The sign of the product of two numbers depends on their signs.\n\n<table id=\"fs-id1385641\" class=\"unnumbered\" style=\"width: 85%\" summary=\"This is a table with two columns. The first column states same signs. The second column states product. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under product, it states negative.\"><thead><tr valign=\"top\"><th>Same signs<\/th><th>Product<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td>\u2022Two positives\u2022Two negatives\n\n<\/td><td>PositivePositive\n\n<\/td><\/tr><\/tbody><\/table><table id=\"fs-id2295230\" class=\"unnumbered\" style=\"width: 85%\" summary=\"This is a table with two columns. The first column states same signs. The second column states product. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under product, it states negative.\"><thead><tr valign=\"top\"><th>Different signs<\/th><th>Product<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td>\u2022Positive \u2022 negative\u2022Negative \u2022 positive\n\n<\/td><td>NegativeNegative\n\n<\/td><\/tr><\/tbody><\/table><\/div>&nbsp;\n\n<div class=\"textbox exercises\"><h3>example<\/h3>Multiply each of the following:\n\n<ol><li>[latex]-9\\cdot 3[\/latex]<\/li><li>[latex]-2\\left(-5\\right)[\/latex]<\/li><li>[latex]4\\left(-8\\right)[\/latex]<\/li><li>[latex]7\\cdot 6[\/latex]<\/li><\/ol>Solution:\n\n<table id=\"eip-id1168469461529\" class=\"unstyled\" style=\"width: 85%\" summary=\".\"><tbody><tr><td>1.<\/td><td><\/td><\/tr><tr><td><\/td><td>[latex]-9\\cdot 3[\/latex]<\/td><\/tr><tr><td>Multiply, noting that the signs are different and so the product is negative.<\/td><td>[latex]-27[\/latex]<\/td><\/tr><\/tbody><\/table><table id=\"eip-id1168467249422\" class=\"unstyled\" style=\"width: 85%\" summary=\".\"><tbody><tr><td>2.<\/td><td><\/td><\/tr><tr><td><\/td><td>[latex]-2\\left(-5\\right)[\/latex]<\/td><\/tr><tr><td>Multiply, noting that the signs are the same and so the product is positive.<\/td><td>[latex]10[\/latex]<\/td><\/tr><\/tbody><\/table><table id=\"eip-id1168466068278\" class=\"unstyled\" style=\"width: 85%\" summary=\".\"><tbody><tr><td>3.<\/td><td><\/td><\/tr><tr><td><\/td><td>[latex]4\\left(-8\\right)[\/latex]<\/td><\/tr><tr><td>Multiply, noting that the signs are different and so the product is negative.<\/td><td>[latex]-32[\/latex]<\/td><\/tr><\/tbody><\/table><table id=\"eip-id1168466077286\" class=\"unstyled\" style=\"width: 85%\" summary=\".\"><tbody><tr><td>4.<\/td><td><\/td><\/tr><tr><td><\/td><td>[latex]7\\cdot 6[\/latex]<\/td><\/tr><tr><td>The signs are the same, so the product is positive.<\/td><td>[latex]42[\/latex]<\/td><\/tr><\/tbody><\/table><\/div>&nbsp;\n\n<div class=\"textbox key-takeaways\"><h3>try it<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145306&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\n\n\n\n<\/div>Watch the following video for more examples of how to multiply integers with different signs, and the same sign.\n\nhttps:\/\/youtu.be\/QY-Za42DItw\n\nWhen we multiply a number by [latex]1[\/latex], the result is the same number. What happens when we multiply a number by [latex]-1?[\/latex] Let\u2019s multiply a positive number and then a negative number by [latex]-1[\/latex] to see what we get.\n\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]\n\n<p style=\"text-align: left\">Each time we multiply a number by [latex]-1[\/latex], we get its opposite.\n\n<div class=\"textbox shaded\"><h3>Multiplication by [latex]-1[\/latex]<\/h3>Multiplying a number by [latex]-1[\/latex] gives its opposite.\n\n[latex]-1a=-a[\/latex]\n\n<\/div>&nbsp;\n\n<div class=\"textbox exercises\"><h3>example<\/h3>Multiply each of the following:\n\n<ol><li>[latex]-1\\cdot 7[\/latex]<\/li><li>[latex]-1\\left(-11\\right)[\/latex]<\/li><\/ol>[reveal-answer q=\"459967\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"459967\"]\n\nSolution:\n\n<table id=\"eip-id1168468284281\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\"><tbody><tr><td>1.<\/td><td><\/td><\/tr><tr><td>The signs are different, so the product will be negative.<\/td><td>[latex]-1\\cdot 7[\/latex]<\/td><\/tr><tr><td>Notice that [latex]\u22127[\/latex] is the opposite of [latex]7[\/latex].<\/td><td>[latex]-7[\/latex]<\/td><\/tr><\/tbody><\/table><table id=\"eip-id1168469855058\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\"><tbody><tr><td>2.<\/td><td><\/td><\/tr><tr><td>The signs are the same, so the product will be positive.<\/td><td>[latex]-1\\left(-11\\right)[\/latex]<\/td><\/tr><tr><td>Notice that [latex]11[\/latex] is the opposite of [latex]\u221211[\/latex].<\/td><td>[latex]11[\/latex]<\/td><\/tr><\/tbody><\/table>[\/hidden-answer]\n\n<\/div>&nbsp;\n\n<div class=\"textbox key-takeaways\"><h3>try it<\/h3>[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145319&amp;theme=oea&amp;iframe_resize_id=mom1[\/embed]\n\n\n\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply integers with different signs<\/li>\n<li>Multiply integers with the same sign<\/li>\n<li>Multiply an integer by -1<\/li>\n<\/ul>\n<\/div>\n<p>Since multiplication is mathematical shorthand for repeated addition, our counter 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>We remember that [latex]a\\cdot b[\/latex] means add [latex]a,b[\/latex] times. Here, we are using the model shown in the graphic below&nbsp;just to help us discover the pattern.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220519\/CNX_BMath_Figure_03_04_001.png\" alt=\"This image has two columns. The first column has 5 times 3. Underneath, it states add 5, 3 times. Under this there are 3 rows of 5 blue circles labeled 15 positives and 5 times 3 equals 15. The second column has negative 5 times 3. Underneath it states add negative 5, 3 times. Under this there are 3 rows of 5 red circles labeled 15 negatives and negative 5 times 3 equals 15.\" \/><br \/>\nNow consider what it means to multiply [latex]5[\/latex] by [latex]-3[\/latex]. It means subtract [latex]5,3[\/latex] times. Looking at subtraction as <em>taking away<\/em>, it means to take away [latex]5,3[\/latex] times. But there is nothing to take away, so we start by adding neutral pairs as shown in the graphic below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220522\/CNX_BMath_Figure_03_04_004-1.png\" alt=\"This figure has 2 columns. The first column has 5 times negative 3. Underneath it states take away 5, 3 times. Under this there are 3 rows of 5 red circles. A downward arrow points to six rows of alternating colored circles in rows of fives. The first row includes 5 red circles, followed by five blue circles, then 5 red, five blue, five red, and five blue. All of the rows of blue circles are circled. The non-circled rows are labeled 15 negatives. Under the label is 5 times negative 3 equals negative 15. The second column has negative 5 times negative 3. Underneath it states take away negative 5, 3 times. Then there are 6 rows of 5 circles alternating in color. The first row is 5 blue circles followed by 5 red circles. All of the red rows are circled. The non-circles rows are labeled 15 positives. Under the label is negative 5 times negative 3 equals 15.\" \/><br \/>\nIn both cases, we started with [latex]\\mathbf{\\text{15}}[\/latex] neutral pairs. In the case on the left, we took away [latex]\\mathbf{\\text{5}},\\mathbf{\\text{3}}[\/latex] times and the result was [latex]-\\mathbf{\\text{15}}[\/latex]. To multiply [latex]\\left(-5\\right)\\left(-3\\right)[\/latex], we took away [latex]-\\mathbf{\\text{5}},\\mathbf{\\text{3}}[\/latex] times and the result was [latex]\\mathbf{\\text{15}}[\/latex]. So we found that<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}5\\cdot 3=15\\hfill & & -5\\left(3\\right)=-15\\hfill \\\\ 5\\left(-3\\right)=-15\\hfill & & \\left(-5\\right)\\left(-3\\right)=15\\hfill \\end{array}[\/latex]<\/p>\n<p>Notice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.<\/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 id=\"fs-id1385641\" class=\"unnumbered\" style=\"width: 85%\" summary=\"This is a table with two columns. The first column states same signs. The second column states product. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under product, it states negative.\">\n<thead>\n<tr valign=\"top\">\n<th>Same signs<\/th>\n<th>Product<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>\u2022Two positives\u2022Two negatives<\/p>\n<\/td>\n<td>PositivePositive<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"fs-id2295230\" class=\"unnumbered\" style=\"width: 85%\" summary=\"This is a table with two columns. The first column states same signs. The second column states product. The first row under same signs states two positives and under that two negatives. In the right column under product, it states positive. Under this row is another column head, this time stating different signs. In the row under this column head, it states positive and negative, and under that, negative and positive. In the right column under product, it states negative.\">\n<thead>\n<tr valign=\"top\">\n<th>Different signs<\/th>\n<th>Product<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>\u2022Positive \u2022 negative\u2022Negative \u2022 positive<\/p>\n<\/td>\n<td>NegativeNegative<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\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<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-9\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply, noting that the signs are different and 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<td>2.<\/td>\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>Multiply, noting that the signs are the same and 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<td>3.<\/td>\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>Multiply, noting that the signs are different and 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<td>4.<\/td>\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<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145306\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145306&#38;theme=oea&#38;iframe_resize_id=ohm145306&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video for more examples of how to multiply integers with different signs, and the same sign.<\/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<p>When we multiply a number by [latex]1[\/latex], the result is the same number. What happens when we multiply a number by [latex]-1?[\/latex] Let\u2019s multiply 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}{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 a number 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 a number by [latex]-1[\/latex] gives its opposite.<\/p>\n<p>[latex]-1a=-a[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\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<td>1.<\/td>\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<td>2.<\/td>\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>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm145319\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145319&#38;theme=oea&#38;iframe_resize_id=ohm145319&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3580\">\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>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: Multiplying Integers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/QY-Za42DItw\">https:\/\/youtu.be\/QY-Za42DItw<\/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: 145306, 145319. <strong>Authored by<\/strong>: Alyson Day. <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>: IMathAS Community License CC-BY + GPL<\/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":25777,"menu_order":13,"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: Multiplying Integers\",\"author\":\"James 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