{"id":189,"date":"2023-06-21T13:22:42","date_gmt":"2023-06-21T13:22:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/operations-on-complex-numbers\/"},"modified":"2023-07-04T03:45:26","modified_gmt":"2023-07-04T03:45:26","slug":"operations-on-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/operations-on-complex-numbers\/","title":{"raw":"\u25aa   Add, Subtract, and Multiply Complex Numbers","rendered":"\u25aa   Add, Subtract, and Multiply Complex Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Add and subtract complex numbers<\/li>\r\n \t<li>Multiply complex numbers<\/li>\r\n<\/ul>\r\n<\/div>\r\nJust as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.\r\n<div class=\"textbox examples\">\r\n<h3>recall doing operations on algebraic expressions<\/h3>\r\nPerforming arithmetic on complex numbers is very similar to adding, subtracting, and multiplying algebraic variable expressions. Recall that doing so involves combining <strong>like terms<\/strong>, carefully subtracting, and using the distributive property.\r\n\r\nComplex numbers of the form [latex]a+bi[\/latex] each contain a real part [latex]a[\/latex] and an imaginary part [latex]bi[\/latex]. Real parts are <strong>like terms<\/strong> with real parts. Likewise, imaginary parts are like with other imaginary parts.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Addition and Subtraction of Complex Numbers<\/h3>\r\nAdding complex numbers:\r\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)+\\left(c+di\\right)=\\left(a+c\\right)+\\left(b+d\\right)i[\/latex]<\/p>\r\nSubtracting complex numbers:\r\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)-\\left(c+di\\right)=\\left(a-c\\right)+\\left(b-d\\right)i[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given two complex numbers, find the sum or difference.<\/h3>\r\n<ol>\r\n \t<li>Identify the real and imaginary parts of each number.<\/li>\r\n \t<li>Add or subtract the real parts.<\/li>\r\n \t<li>Add or subtract the imaginary parts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Adding Complex Numbers<\/h3>\r\nAdd [latex]3 - 4i[\/latex] and [latex]2+5i[\/latex].\r\n\r\n[reveal-answer q=\"5937\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"5937\"]\r\n\r\nWe add the real parts and add the imaginary parts.\r\n<p style=\"text-align: center;\">[latex]\\left(3 - 4i\\right)+\\left(2+5i\\right)=\\left(3+2\\right)+\\left(-4+5\\right)i=5+i[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSubtract [latex]2+5i[\/latex] from [latex]3 - 4i[\/latex].\r\n\r\n[reveal-answer q=\"732700\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"732700\"]\r\n\r\n[latex]\\left(3 - 4i\\right)-\\left(2+5i\\right)=1 - 9i[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question]61710[\/ohm_question]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/SGhTjioGqqA\r\n<h2>Multiplying Complex Numbers<\/h2>\r\nMultiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.\r\n<h2>Multiplying a Complex Number by a Real Number<\/h2>\r\nLet\u2019s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}3(6+2i)&amp;=(3\\cdot6)+(3\\cdot2i)&amp;&amp;\\text{Distribute.}\\\\&amp;=18+6i&amp;&amp;\\text{Simplify.}\\end{align}[\/latex]<\/p>\r\n\r\n<div class=\"textbox\">\r\n<h3>How To: Given a complex number and a real number, multiply to find the product.<\/h3>\r\n<ol>\r\n \t<li>Use the distributive property.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Multiplying a Complex Number by a Real Number<\/h3>\r\nFind the product [latex]4\\left(2+5i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"928099\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"928099\"]\r\n<p style=\"text-align: center;\">[latex]4\\left(2+5i\\right)=\\left(4\\cdot 2\\right)+\\left(4\\cdot 5i\\right)=8+20i[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the product [latex]-4\\left(2+6i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"568092\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"568092\"]\r\n\r\n[latex]-8 - 24i[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question]40462[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Multiplying Complex Numbers Together<\/h2>\r\nNow, let\u2019s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get\r\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci+bd{i}^{2}[\/latex]<\/p>\r\nBecause [latex]{i}^{2}=-1[\/latex], we have\r\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci-bd[\/latex]<\/p>\r\nTo simplify, we combine the real parts, and we combine the imaginary parts.\r\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=\\left(ac-bd\\right)+\\left(ad+bc\\right)i[\/latex]<\/p>\r\n\r\n<div class=\"textbox\">\r\n<h3>How To: Given two complex numbers, multiply to find the product.<\/h3>\r\n<ol>\r\n \t<li>Use the distributive property or the FOIL method.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Multiplying a Complex Number by a Complex Number<\/h3>\r\nMultiply [latex]\\left(4+3i\\right)\\left(2 - 5i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"388605\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"388605\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(4+3i\\right)\\left(2 - 5i\\right)&amp;=4\\cdot 2 + 4\\cdot \\left(-5i\\right)+3i\\cdot2+3i\\cdot \\left(-5i\\right)\\\\ &amp;=8-20i+6i-15i^2\\\\&amp;=8+15-20i+6i\\\\ &amp;=23 - 14i\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nMultiply [latex]\\left(3 - 4i\\right)\\left(2+3i\\right)[\/latex].\r\n\r\n[reveal-answer q=\"576399\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"576399\"]\r\n\r\n[latex]18+i[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question]3903[\/ohm_question]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/O9xQaIi0NX0","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Add and subtract complex numbers<\/li>\n<li>Multiply complex numbers<\/li>\n<\/ul>\n<\/div>\n<p>Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.<\/p>\n<div class=\"textbox examples\">\n<h3>recall doing operations on algebraic expressions<\/h3>\n<p>Performing arithmetic on complex numbers is very similar to adding, subtracting, and multiplying algebraic variable expressions. Recall that doing so involves combining <strong>like terms<\/strong>, carefully subtracting, and using the distributive property.<\/p>\n<p>Complex numbers of the form [latex]a+bi[\/latex] each contain a real part [latex]a[\/latex] and an imaginary part [latex]bi[\/latex]. Real parts are <strong>like terms<\/strong> with real parts. Likewise, imaginary parts are like with other imaginary parts.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Addition and Subtraction of Complex Numbers<\/h3>\n<p>Adding complex numbers:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)+\\left(c+di\\right)=\\left(a+c\\right)+\\left(b+d\\right)i[\/latex]<\/p>\n<p>Subtracting complex numbers:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)-\\left(c+di\\right)=\\left(a-c\\right)+\\left(b-d\\right)i[\/latex]<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given two complex numbers, find the sum or difference.<\/h3>\n<ol>\n<li>Identify the real and imaginary parts of each number.<\/li>\n<li>Add or subtract the real parts.<\/li>\n<li>Add or subtract the imaginary parts.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Adding Complex Numbers<\/h3>\n<p>Add [latex]3 - 4i[\/latex] and [latex]2+5i[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q5937\">Show Solution<\/span><\/p>\n<div id=\"q5937\" class=\"hidden-answer\" style=\"display: none\">\n<p>We add the real parts and add the imaginary parts.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(3 - 4i\\right)+\\left(2+5i\\right)=\\left(3+2\\right)+\\left(-4+5\\right)i=5+i[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Subtract [latex]2+5i[\/latex] from [latex]3 - 4i[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q732700\">Show Solution<\/span><\/p>\n<div id=\"q732700\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(3 - 4i\\right)-\\left(2+5i\\right)=1 - 9i[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm61710\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61710&theme=oea&iframe_resize_id=ohm61710&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Adding and Subtracting Complex Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/SGhTjioGqqA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Multiplying Complex Numbers<\/h2>\n<p>Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.<\/p>\n<h2>Multiplying a Complex Number by a Real Number<\/h2>\n<p>Let\u2019s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}3(6+2i)&=(3\\cdot6)+(3\\cdot2i)&&\\text{Distribute.}\\\\&=18+6i&&\\text{Simplify.}\\end{align}[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a complex number and a real number, multiply to find the product.<\/h3>\n<ol>\n<li>Use the distributive property.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying a Complex Number by a Real Number<\/h3>\n<p>Find the product [latex]4\\left(2+5i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q928099\">Show Solution<\/span><\/p>\n<div id=\"q928099\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]4\\left(2+5i\\right)=\\left(4\\cdot 2\\right)+\\left(4\\cdot 5i\\right)=8+20i[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the product [latex]-4\\left(2+6i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q568092\">Show Solution<\/span><\/p>\n<div id=\"q568092\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-8 - 24i[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm40462\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=40462&theme=oea&iframe_resize_id=ohm40462&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Multiplying Complex Numbers Together<\/h2>\n<p>Now, let\u2019s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci+bd{i}^{2}[\/latex]<\/p>\n<p>Because [latex]{i}^{2}=-1[\/latex], we have<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci-bd[\/latex]<\/p>\n<p>To simplify, we combine the real parts, and we combine the imaginary parts.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=\\left(ac-bd\\right)+\\left(ad+bc\\right)i[\/latex]<\/p>\n<div class=\"textbox\">\n<h3>How To: Given two complex numbers, multiply to find the product.<\/h3>\n<ol>\n<li>Use the distributive property or the FOIL method.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Multiplying a Complex Number by a Complex Number<\/h3>\n<p>Multiply [latex]\\left(4+3i\\right)\\left(2 - 5i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q388605\">Show Solution<\/span><\/p>\n<div id=\"q388605\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\left(4+3i\\right)\\left(2 - 5i\\right)&=4\\cdot 2 + 4\\cdot \\left(-5i\\right)+3i\\cdot2+3i\\cdot \\left(-5i\\right)\\\\ &=8-20i+6i-15i^2\\\\&=8+15-20i+6i\\\\ &=23 - 14i\\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Multiply [latex]\\left(3 - 4i\\right)\\left(2+3i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q576399\">Show Solution<\/span><\/p>\n<div id=\"q576399\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]18+i[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm3903\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3903&theme=oea&iframe_resize_id=ohm3903&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 2:  Multiply Complex Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/O9xQaIi0NX0?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-189\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 120193. <strong>Provided by<\/strong>: LumenLearning. <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><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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Adding and Subtracting Complex Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/SGhTjioGqqA\">https:\/\/youtu.be\/SGhTjioGqqA<\/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 61710. <strong>Authored by<\/strong>: Day, Alyson. <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><li>Question ID 40462. <strong>Authored by<\/strong>: Jenck,Michael. <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><li>Question ID 3903. <strong>Authored by<\/strong>: Lippman,David. <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><li>Question ID 61715. <strong>Authored by<\/strong>: Day, Alyson. <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><li>Ex 2: Multiply Complex Numbers. <strong>Authored by<\/strong>: Sousa, James  (Mathispower4u). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/O9xQaIi0NX0\">https:\/\/youtu.be\/O9xQaIi0NX0<\/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>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Ex: Dividing Complex Numbers . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/XBJjbJAwM1c\">https:\/\/youtu.be\/XBJjbJAwM1c<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":26,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Adding and Subtracting Complex Numbers\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/SGhTjioGqqA\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 61710\",\"author\":\"Day, 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