{"id":16447,"date":"2019-10-03T15:48:54","date_gmt":"2019-10-03T15:48:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/16-4-2-operations-with-complex-numbers\/"},"modified":"2024-05-02T15:48:19","modified_gmt":"2024-05-02T15:48:19","slug":"16-4-2-operations-with-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/16-4-2-operations-with-complex-numbers\/","title":{"raw":"Multiplying and Dividing Complex Numbers","rendered":"Multiplying and Dividing Complex Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Multiply complex numbers<\/li>\r\n \t<li>Find conjugates of complex numbers<\/li>\r\n \t<li>Divide complex numbers<\/li>\r\n \t<li>Simplify\u00a0powers of [latex]i[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section id=\"fs-id1165137417169\">\r\n<p id=\"fs-id1165137832911\">Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.<\/p>\r\n\r\n<h3>\u00a0Multiplying a Complex Number by a Real Number<\/h3>\r\n<img class=\"wp-image-2535 size-full aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2016\/06\/23153423\/CNX_Precalc_Figure_03_01_0062.jpg\" alt=\"Showing how distribution works for complex numbers. For 3(6+2i), 3 is multiplied to both the real and imaginary parts. So we have (3)(6)+(3)(2i) = 18 + 6i. \" width=\"487\" height=\"87\" \/>\r\n\r\nLet us begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for [latex]3(6+2i)[\/latex], 3 is multiplied to both the real and imaginary parts. So we have\u00a0[latex](3)(6)+(3)(2i)[\/latex] =\u00a0[latex]18 + 6i[\/latex].\r\n<div id=\"fs-id1165137745292\" class=\"textbox shaded\">\r\n<h3 id=\"fs-id1165137426118\">How To: Given a complex number and a real number, multiply to find the product<\/h3>\r\n<ol id=\"fs-id1165137793647\">\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<\/h3>\r\nFind the product [latex]4\\left(2+5i\\right)[\/latex].\r\n[reveal-answer q=\"374377\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"374377\"]\r\n<p id=\"fs-id1165137804818\">Distribute the\u00a0[latex]4[\/latex].<\/p>\r\n\r\n<div id=\"eip-id1165135436310\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cc}4\\left(2+5i\\right)=\\left(4\\cdot 2\\right)+\\left(4\\cdot 5i\\right)\\hfill \\\\ =8+20i\\hfill \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>\u00a0Multiplying Complex Numbers Together<\/h2>\r\n<\/section><section id=\"fs-id1165137650841\">\r\n<p id=\"fs-id1165137832483\">Now, let us 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>\r\n\r\n<div id=\"eip-586\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci+bd{i}^{2}[\/latex]<\/div>\r\n<p id=\"fs-id1165137734803\">Because [latex]{i}^{2}=-1[\/latex], we have<\/p>\r\n\r\n<div id=\"eip-523\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci-bd[\/latex]<\/div>\r\n<p id=\"fs-id1165135186757\">To simplify, we combine the real parts, and we combine the imaginary parts.<\/p>\r\n\r\n<div id=\"eip-794\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=\\left(ac-bd\\right)+\\left(ad+bc\\right)i[\/latex]<\/div>\r\n<div id=\"fs-id1165137642817\" class=\"textbox\">\r\n<h3 id=\"fs-id1165137724898\">How To: Given two complex numbers, multiply to find the product<\/h3>\r\n<ol id=\"fs-id1165137561156\">\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<\/h3>\r\nMultiply [latex]\\left(4+3i\\right)\\left(2 - 5i\\right)[\/latex].\r\n[reveal-answer q=\"188458\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"188458\"]\r\n<p id=\"fs-id1165137459488\">Use FOIL.<\/p>\r\n\r\n<div id=\"eip-id1165137762412\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lll}\\left(4+3i\\right)\\left(2 - 5i\\right) &amp; =\\left(4\\cdot 2\\right)+\\left(4\\cdot \\left(-5i\\right)\\right)+\\left(3i\\cdot 2\\right)+\\left(\\left(3i\\right)\\cdot \\left(-5i\\right)\\right) &amp; \\\\ &amp; =8-20i+6i-15i^2 &amp; \\\\ &amp; = 8-20i+6i-15\\left(-1\\right) &amp; \\text{Substituting -1 for } {i}^{2} \\text{.} \\\\ &amp; = 8-20i+6i+15 &amp; \\\\ &amp; = \\left(8+15\\right)+\\left(-20+6\\right)i &amp; \\text{Grouping like terms together.} \\\\ &amp; =23-14i &amp; \\text{Simplifying.}\\end{array}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: center;\"><\/div>\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]3903[\/ohm_question]\r\n\r\n<\/div>\r\nIn the first video, we show more examples of multiplying complex numbers.\r\n\r\nhttps:\/\/youtu.be\/Fmr3o2zkwLM\r\n<h2>Simplifying Powers of <em>i<\/em><\/h2>\r\n<p id=\"fs-id1165132919554\">There is a pattern to evaluating powers of <em>i<\/em>. Let us look at what happens when we raise <em>i<\/em>\u00a0to increasing powers.<\/p>\r\n\r\n<div id=\"eip-783\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}{i}^{1}=i\\\\ {i}^{2}=-1\\\\ {i}^{3}={i}^{2}\\cdot i=-1\\cdot i=-i\\\\ {i}^{4}={i}^{3}\\cdot i=-i\\cdot i=-{i}^{2}=-\\left(-1\\right)=1\\\\ {i}^{5}={i}^{4}\\cdot i=1\\cdot i=i\\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137530297\">We can see that when we get to the fifth power of [latex]i[\/latex], it is equal to the first power. As we continue to multiply [latex]i[\/latex]\u00a0by itself for increasing powers, we will see a cycle of\u00a0[latex]4[\/latex]. Let us examine the next\u00a0[latex]4[\/latex] powers of [latex]i[\/latex].<\/p>\r\n\r\n<div id=\"eip-477\" class=\"equation unnumbered\">[latex]\\begin{array}{l}{i}^{6}={i}^{5}\\cdot i=i\\cdot i={i}^{2}=-1\\\\ {i}^{7}={i}^{6}\\cdot i={i}^{2}\\cdot i={i}^{3}=-i\\\\ {i}^{8}={i}^{7}\\cdot i={i}^{3}\\cdot i={i}^{4}=1\\\\ {i}^{9}={i}^{8}\\cdot i={i}^{4}\\cdot i={i}^{5}=i\\end{array}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\"><\/div>\r\n<div class=\"equation unnumbered\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex]{i}^{35}[\/latex].\r\n[reveal-answer q=\"295805\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"295805\"]\r\n<p id=\"fs-id1165137728290\">Since [latex]{i}^{4}=1[\/latex], we can simplify the problem by factoring out as many factors of [latex]{i}^{4}[\/latex] as possible. To do so, first determine how many times\u00a0[latex]4[\/latex] goes into\u00a0[latex]35[\/latex]: [latex]35=4\\cdot 8+3[\/latex].<\/p>\r\n\r\n<div id=\"eip-id1165134069265\" class=\"equation unnumbered\">[latex]{i}^{35}={i}^{4\\cdot 8+3}={i}^{4\\cdot 8}\\cdot {i}^{3}={\\left({i}^{4}\\right)}^{8}\\cdot {i}^{3}={1}^{8}\\cdot {i}^{3}={i}^{3}=-i[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137758921\" class=\"note precalculus qa textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<p id=\"fs-id1165135186727\"><strong>Can we write [latex]{i}^{35}[\/latex] in other helpful ways?<\/strong><\/p>\r\n<p id=\"fs-id1165135444053\"><em>As we saw in the previous example, we reduced [latex]{i}^{35}[\/latex] to [latex]{i}^{3}[\/latex] by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of [latex]{i}^{35}[\/latex] may be more useful. The table below\u00a0shows some other possible factorizations.<\/em><\/p>\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Factorization of [latex]{i}^{35}[\/latex]<\/strong><\/td>\r\n<td>[latex]{i}^{34}\\cdot i[\/latex]<\/td>\r\n<td>[latex]{i}^{33}\\cdot {i}^{2}[\/latex]<\/td>\r\n<td>[latex]{i}^{31}\\cdot {i}^{4}[\/latex]<\/td>\r\n<td>[latex]{i}^{19}\\cdot {i}^{16}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Reduced form<\/strong><\/td>\r\n<td>[latex]{\\left({i}^{2}\\right)}^{17}\\cdot i[\/latex]<\/td>\r\n<td>[latex]{i}^{33}\\cdot \\left(-1\\right)[\/latex]<\/td>\r\n<td>[latex]{i}^{31}\\cdot 1[\/latex]<\/td>\r\n<td>[latex]{i}^{19}\\cdot {\\left({i}^{4}\\right)}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Simplified form<\/strong><\/td>\r\n<td>[latex]{\\left(-1\\right)}^{17}\\cdot i[\/latex]<\/td>\r\n<td>[latex]-{i}^{33}[\/latex]<\/td>\r\n<td>[latex]{i}^{31}[\/latex]<\/td>\r\n<td>[latex]{i}^{19}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135255472\"><em>Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.<\/em><\/p>\r\n\r\n<\/div>\r\nIn the following video, you will see more examples of how to simplify powers of\u00a0[latex]i[\/latex].\r\n\r\nhttps:\/\/youtu.be\/sfP6SmEYHRw\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]15563[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Dividing Complex Numbers<\/h2>\r\n<p id=\"fs-id1165137612241\">Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. This idea is similar to rationalizing the denominator of a fraction that contains a radical, which <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-rationalizing-denominators\/\">we did earlier by multiplying the numerator and denominator by the conjugate of the denominator<\/a>. In the same way, to eliminate the complex or imaginary number in the denominator, you multiply both the numerator and denominator by\u00a0the <strong>complex conjugate<\/strong> of the denominator which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of [latex]a+bi[\/latex] is [latex]a-bi[\/latex].<\/p>\r\n<p id=\"fs-id1165137435064\">Note that complex conjugates have a reciprocal relationship: The complex conjugate of [latex]a+bi[\/latex] is [latex]a-bi[\/latex], and the complex conjugate of [latex]a-bi[\/latex] is [latex]a+bi[\/latex]. Furthermore, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.<\/p>\r\n<p id=\"fs-id1165137611741\">Suppose we want to divide [latex]c+di[\/latex] by [latex]a+bi[\/latex], where neither <em>a<\/em>\u00a0nor <em>b<\/em>\u00a0equals zero. We first write the division as a fraction then find the complex conjugate of the denominator and multiply.<\/p>\r\n\r\n<div id=\"eip-225\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{c+di}{a+bi}\\text{ where }a\\ne 0\\text{ and }b\\ne 0[\/latex]<\/div>\r\n<p id=\"fs-id1165134148263\">Multiply the numerator and denominator by the complex conjugate of the denominator.<\/p>\r\n\r\n<div id=\"eip-32\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\left(c+di\\right)}{\\left(a+bi\\right)}\\cdot \\frac{\\left(a-bi\\right)}{\\left(a-bi\\right)}=\\frac{\\left(c+di\\right)\\left(a-bi\\right)}{\\left(a+bi\\right)\\left(a-bi\\right)}[\/latex]<\/div>\r\n<p id=\"fs-id1165135260723\">Apply the distributive property.<\/p>\r\n\r\n<div id=\"eip-736\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]=\\frac{ca-cbi+adi-bd{i}^{2}}{{a}^{2}-abi+abi-{b}^{2}{i}^{2}}[\/latex]<\/div>\r\n<p id=\"fs-id1165137871668\">Simplify, remembering that [latex]{i}^{2}=-1[\/latex].<\/p>\r\n\r\n<div id=\"eip-64\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}=\\frac{ca-cbi+adi-bd\\left(-1\\right)}{{a}^{2}-abi+abi-{b}^{2}\\left(-1\\right)}\\hfill \\\\ =\\frac{\\left(ca+bd\\right)+\\left(ad-cb\\right)i}{{a}^{2}+{b}^{2}}\\hfill \\end{array}[\/latex]<\/div>\r\n<div id=\"fs-id1165135203870\" class=\"textbox shaded\">\r\n<h3 class=\"title\">A General Note: The Complex Conjugate<\/h3>\r\n<p id=\"fs-id1165137793758\">The <strong>complex conjugate<\/strong> of a complex number [latex]a+bi[\/latex] is [latex]a-bi[\/latex]. It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.<\/p>\r\n\r\n<ul id=\"fs-id1165135487089\">\r\n \t<li>When a complex number is multiplied by its complex conjugate, the result is a real number.<\/li>\r\n \t<li>When a complex number is added to its complex conjugate, the result is a real number.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p id=\"fs-id1165137896182\">Find the complex conjugate of each number.<\/p>\r\n\r\n<ol id=\"fs-id1165137896185\">\r\n \t<li>[latex]2+i\\sqrt{5}[\/latex]<\/li>\r\n \t<li>[latex]-\\frac{1}{2}i[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"426623\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"426623\"]\r\n<ol id=\"fs-id1165137742669\">\r\n \t<li>The number is already in the form [latex]a+bi[\/latex]. The complex conjugate is [latex]a-bi[\/latex], or [latex]2-i\\sqrt{5}[\/latex].<\/li>\r\n \t<li>We can rewrite this number in the form [latex]a-bi[\/latex] as [latex]0-\\frac{1}{2}i[\/latex]. The complex conjugate is [latex]a+bi[\/latex], or [latex]0+\\frac{1}{2}i[\/latex]. This can be written simply as [latex]\\frac{1}{2}i[\/latex].<\/li>\r\n<\/ol>\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]71454[\/ohm_question]\r\n\r\n<\/div>\r\n<h3>Analysis of the Solution<\/h3>\r\n<div id=\"Example_03_01_06\" class=\"example\">\r\n<div id=\"fs-id1165134032261\" class=\"exercise\">\r\n<div id=\"fs-id1165137772480\" class=\"commentary\">\r\n<p id=\"fs-id1165137762415\">Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. When dividing by a purely imaginary number, we can simply multiply the numerator and denominator by [latex]i[\/latex] instead of the complex conjugate.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137409413\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135471100\">How To: Given two complex numbers, divide one by the other.<\/h3>\r\n<ol id=\"fs-id1165135471104\">\r\n \t<li>Write the division problem as a fraction.<\/li>\r\n \t<li>Determine the complex conjugate of the denominator.<\/li>\r\n \t<li>Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide [latex]\\left(2+5i\\right)[\/latex] by [latex]\\left(4-i\\right)[\/latex].\r\n[reveal-answer q=\"665746\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"665746\"]\r\n<p id=\"fs-id1165137605861\">We begin by writing the problem as a fraction.<\/p>\r\n\r\n<div id=\"eip-id1165134234232\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\left(2+5i\\right)}{\\left(4-i\\right)}[\/latex]<\/div>\r\n<p id=\"fs-id1165137639613\">Then we multiply the numerator and denominator by the complex conjugate of the denominator.<\/p>\r\n\r\n<div id=\"eip-id1165137400110\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\left(2+5i\\right)}{\\left(4-i\\right)}\\cdot \\frac{\\left(4+i\\right)}{\\left(4+i\\right)}[\/latex]<\/div>\r\n<p id=\"fs-id1165137474228\">To multiply two complex numbers, we expand the product as we would with polynomials (the process commonly called FOIL).<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcll}\\Large{\\frac{(2+5i)}{(4-i)}}\\cdot\\frac{(4+i)}{4+i)}&amp;=&amp;\\Large{\\frac{8+2i+20i+5{i}^{2}}{16+4i - 4i-{i}^{2}}}&amp; \\\\&amp;=&amp;\\Large{\\frac{8+2i+20i+5(-1)}{16+4i - 4i-(-1)}}&amp;\\quad\\text{Because }i^2=-1\\\\&amp;=&amp;\\Large{\\frac{3+22i}{17}}&amp; \\\\&amp;=&amp;\\Large{\\frac{3}{17}+\\frac{22}{17}i}&amp;\\quad\\text{Separate real and imaginary parts}\\end{array}[\/latex]<\/p>\r\n&nbsp;\r\n<p id=\"fs-id1165137930346\">Note that this expresses the quotient in standard form.<\/p>\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]3157[\/ohm_question]\r\n\r\n<\/div>\r\nIn the last video, you will see more examples of dividing complex numbers.\r\n\r\nhttps:\/\/youtu.be\/XBJjbJAwM1c\r\n<h2>Summary<\/h2>\r\nMultiplying complex numbers is similar to multiplying polynomials. Remember that an imaginary number times another imaginary number gives a real result. When you divide complex numbers, you must first multiply the numerator and denominator by the complex conjugate to eliminate any imaginary parts, and then you can divide.\r\n\r\n<\/section>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply complex numbers<\/li>\n<li>Find conjugates of complex numbers<\/li>\n<li>Divide complex numbers<\/li>\n<li>Simplify\u00a0powers of [latex]i[\/latex]<\/li>\n<\/ul>\n<\/div>\n<section id=\"fs-id1165137417169\">\n<p id=\"fs-id1165137832911\">Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.<\/p>\n<h3>\u00a0Multiplying a Complex Number by a Real Number<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2535 size-full aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2016\/06\/23153423\/CNX_Precalc_Figure_03_01_0062.jpg\" alt=\"Showing how distribution works for complex numbers. For 3(6+2i), 3 is multiplied to both the real and imaginary parts. So we have (3)(6)+(3)(2i) = 18 + 6i.\" width=\"487\" height=\"87\" \/><\/p>\n<p>Let us begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for [latex]3(6+2i)[\/latex], 3 is multiplied to both the real and imaginary parts. So we have\u00a0[latex](3)(6)+(3)(2i)[\/latex] =\u00a0[latex]18 + 6i[\/latex].<\/p>\n<div id=\"fs-id1165137745292\" class=\"textbox shaded\">\n<h3 id=\"fs-id1165137426118\">How To: Given a complex number and a real number, multiply to find the product<\/h3>\n<ol id=\"fs-id1165137793647\">\n<li>Use the distributive property.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/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=\"q374377\">Show Solution<\/span><\/p>\n<div id=\"q374377\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137804818\">Distribute the\u00a0[latex]4[\/latex].<\/p>\n<div id=\"eip-id1165135436310\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cc}4\\left(2+5i\\right)=\\left(4\\cdot 2\\right)+\\left(4\\cdot 5i\\right)\\hfill \\\\ =8+20i\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Multiplying Complex Numbers Together<\/h2>\n<\/section>\n<section id=\"fs-id1165137650841\">\n<p id=\"fs-id1165137832483\">Now, let us 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<div id=\"eip-586\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci+bd{i}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165137734803\">Because [latex]{i}^{2}=-1[\/latex], we have<\/p>\n<div id=\"eip-523\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=ac+adi+bci-bd[\/latex]<\/div>\n<p id=\"fs-id1165135186757\">To simplify, we combine the real parts, and we combine the imaginary parts.<\/p>\n<div id=\"eip-794\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\left(a+bi\\right)\\left(c+di\\right)=\\left(ac-bd\\right)+\\left(ad+bc\\right)i[\/latex]<\/div>\n<div id=\"fs-id1165137642817\" class=\"textbox\">\n<h3 id=\"fs-id1165137724898\">How To: Given two complex numbers, multiply to find the product<\/h3>\n<ol id=\"fs-id1165137561156\">\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<\/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=\"q188458\">Show Solution<\/span><\/p>\n<div id=\"q188458\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137459488\">Use FOIL.<\/p>\n<div id=\"eip-id1165137762412\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{lll}\\left(4+3i\\right)\\left(2 - 5i\\right) & =\\left(4\\cdot 2\\right)+\\left(4\\cdot \\left(-5i\\right)\\right)+\\left(3i\\cdot 2\\right)+\\left(\\left(3i\\right)\\cdot \\left(-5i\\right)\\right) & \\\\ & =8-20i+6i-15i^2 & \\\\ & = 8-20i+6i-15\\left(-1\\right) & \\text{Substituting -1 for } {i}^{2} \\text{.} \\\\ & = 8-20i+6i+15 & \\\\ & = \\left(8+15\\right)+\\left(-20+6\\right)i & \\text{Grouping like terms together.} \\\\ & =23-14i & \\text{Simplifying.}\\end{array}[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\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>In the first video, we show more examples of multiplying complex numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 3:  Multiply Complex Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Fmr3o2zkwLM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Simplifying Powers of <em>i<\/em><\/h2>\n<p id=\"fs-id1165132919554\">There is a pattern to evaluating powers of <em>i<\/em>. Let us look at what happens when we raise <em>i<\/em>\u00a0to increasing powers.<\/p>\n<div id=\"eip-783\" class=\"equation unnumbered\">[latex]\\begin{array}{ll}{i}^{1}=i\\\\ {i}^{2}=-1\\\\ {i}^{3}={i}^{2}\\cdot i=-1\\cdot i=-i\\\\ {i}^{4}={i}^{3}\\cdot i=-i\\cdot i=-{i}^{2}=-\\left(-1\\right)=1\\\\ {i}^{5}={i}^{4}\\cdot i=1\\cdot i=i\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137530297\">We can see that when we get to the fifth power of [latex]i[\/latex], it is equal to the first power. As we continue to multiply [latex]i[\/latex]\u00a0by itself for increasing powers, we will see a cycle of\u00a0[latex]4[\/latex]. Let us examine the next\u00a0[latex]4[\/latex] powers of [latex]i[\/latex].<\/p>\n<div id=\"eip-477\" class=\"equation unnumbered\">[latex]\\begin{array}{l}{i}^{6}={i}^{5}\\cdot i=i\\cdot i={i}^{2}=-1\\\\ {i}^{7}={i}^{6}\\cdot i={i}^{2}\\cdot i={i}^{3}=-i\\\\ {i}^{8}={i}^{7}\\cdot i={i}^{3}\\cdot i={i}^{4}=1\\\\ {i}^{9}={i}^{8}\\cdot i={i}^{4}\\cdot i={i}^{5}=i\\end{array}[\/latex]<\/div>\n<div class=\"equation unnumbered\"><\/div>\n<div class=\"equation unnumbered\">\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]{i}^{35}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q295805\">Show Solution<\/span><\/p>\n<div id=\"q295805\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137728290\">Since [latex]{i}^{4}=1[\/latex], we can simplify the problem by factoring out as many factors of [latex]{i}^{4}[\/latex] as possible. To do so, first determine how many times\u00a0[latex]4[\/latex] goes into\u00a0[latex]35[\/latex]: [latex]35=4\\cdot 8+3[\/latex].<\/p>\n<div id=\"eip-id1165134069265\" class=\"equation unnumbered\">[latex]{i}^{35}={i}^{4\\cdot 8+3}={i}^{4\\cdot 8}\\cdot {i}^{3}={\\left({i}^{4}\\right)}^{8}\\cdot {i}^{3}={1}^{8}\\cdot {i}^{3}={i}^{3}=-i[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1165137758921\" class=\"note precalculus qa textbox\">\n<h3>Q &amp; A<\/h3>\n<p id=\"fs-id1165135186727\"><strong>Can we write [latex]{i}^{35}[\/latex] in other helpful ways?<\/strong><\/p>\n<p id=\"fs-id1165135444053\"><em>As we saw in the previous example, we reduced [latex]{i}^{35}[\/latex] to [latex]{i}^{3}[\/latex] by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of [latex]{i}^{35}[\/latex] may be more useful. The table below\u00a0shows some other possible factorizations.<\/em><\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>Factorization of [latex]{i}^{35}[\/latex]<\/strong><\/td>\n<td>[latex]{i}^{34}\\cdot i[\/latex]<\/td>\n<td>[latex]{i}^{33}\\cdot {i}^{2}[\/latex]<\/td>\n<td>[latex]{i}^{31}\\cdot {i}^{4}[\/latex]<\/td>\n<td>[latex]{i}^{19}\\cdot {i}^{16}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Reduced form<\/strong><\/td>\n<td>[latex]{\\left({i}^{2}\\right)}^{17}\\cdot i[\/latex]<\/td>\n<td>[latex]{i}^{33}\\cdot \\left(-1\\right)[\/latex]<\/td>\n<td>[latex]{i}^{31}\\cdot 1[\/latex]<\/td>\n<td>[latex]{i}^{19}\\cdot {\\left({i}^{4}\\right)}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Simplified form<\/strong><\/td>\n<td>[latex]{\\left(-1\\right)}^{17}\\cdot i[\/latex]<\/td>\n<td>[latex]-{i}^{33}[\/latex]<\/td>\n<td>[latex]{i}^{31}[\/latex]<\/td>\n<td>[latex]{i}^{19}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135255472\"><em>Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.<\/em><\/p>\n<\/div>\n<p>In the following video, you will see more examples of how to simplify powers of\u00a0[latex]i[\/latex].<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Raising the imaginary unit i to powers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/sfP6SmEYHRw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm15563\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15563&theme=oea&iframe_resize_id=ohm15563&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Dividing Complex Numbers<\/h2>\n<p id=\"fs-id1165137612241\">Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. This idea is similar to rationalizing the denominator of a fraction that contains a radical, which <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-or-watch-rationalizing-denominators\/\">we did earlier by multiplying the numerator and denominator by the conjugate of the denominator<\/a>. In the same way, to eliminate the complex or imaginary number in the denominator, you multiply both the numerator and denominator by\u00a0the <strong>complex conjugate<\/strong> of the denominator which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of [latex]a+bi[\/latex] is [latex]a-bi[\/latex].<\/p>\n<p id=\"fs-id1165137435064\">Note that complex conjugates have a reciprocal relationship: The complex conjugate of [latex]a+bi[\/latex] is [latex]a-bi[\/latex], and the complex conjugate of [latex]a-bi[\/latex] is [latex]a+bi[\/latex]. Furthermore, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.<\/p>\n<p id=\"fs-id1165137611741\">Suppose we want to divide [latex]c+di[\/latex] by [latex]a+bi[\/latex], where neither <em>a<\/em>\u00a0nor <em>b<\/em>\u00a0equals zero. We first write the division as a fraction then find the complex conjugate of the denominator and multiply.<\/p>\n<div id=\"eip-225\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{c+di}{a+bi}\\text{ where }a\\ne 0\\text{ and }b\\ne 0[\/latex]<\/div>\n<p id=\"fs-id1165134148263\">Multiply the numerator and denominator by the complex conjugate of the denominator.<\/p>\n<div id=\"eip-32\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\left(c+di\\right)}{\\left(a+bi\\right)}\\cdot \\frac{\\left(a-bi\\right)}{\\left(a-bi\\right)}=\\frac{\\left(c+di\\right)\\left(a-bi\\right)}{\\left(a+bi\\right)\\left(a-bi\\right)}[\/latex]<\/div>\n<p id=\"fs-id1165135260723\">Apply the distributive property.<\/p>\n<div id=\"eip-736\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]=\\frac{ca-cbi+adi-bd{i}^{2}}{{a}^{2}-abi+abi-{b}^{2}{i}^{2}}[\/latex]<\/div>\n<p id=\"fs-id1165137871668\">Simplify, remembering that [latex]{i}^{2}=-1[\/latex].<\/p>\n<div id=\"eip-64\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}=\\frac{ca-cbi+adi-bd\\left(-1\\right)}{{a}^{2}-abi+abi-{b}^{2}\\left(-1\\right)}\\hfill \\\\ =\\frac{\\left(ca+bd\\right)+\\left(ad-cb\\right)i}{{a}^{2}+{b}^{2}}\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1165135203870\" class=\"textbox shaded\">\n<h3 class=\"title\">A General Note: The Complex Conjugate<\/h3>\n<p id=\"fs-id1165137793758\">The <strong>complex conjugate<\/strong> of a complex number [latex]a+bi[\/latex] is [latex]a-bi[\/latex]. It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.<\/p>\n<ul id=\"fs-id1165135487089\">\n<li>When a complex number is multiplied by its complex conjugate, the result is a real number.<\/li>\n<li>When a complex number is added to its complex conjugate, the result is a real number.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165137896182\">Find the complex conjugate of each number.<\/p>\n<ol id=\"fs-id1165137896185\">\n<li>[latex]2+i\\sqrt{5}[\/latex]<\/li>\n<li>[latex]-\\frac{1}{2}i[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q426623\">Show Solution<\/span><\/p>\n<div id=\"q426623\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165137742669\">\n<li>The number is already in the form [latex]a+bi[\/latex]. The complex conjugate is [latex]a-bi[\/latex], or [latex]2-i\\sqrt{5}[\/latex].<\/li>\n<li>We can rewrite this number in the form [latex]a-bi[\/latex] as [latex]0-\\frac{1}{2}i[\/latex]. The complex conjugate is [latex]a+bi[\/latex], or [latex]0+\\frac{1}{2}i[\/latex]. This can be written simply as [latex]\\frac{1}{2}i[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm71454\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=71454&theme=oea&iframe_resize_id=ohm71454&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h3>Analysis of the Solution<\/h3>\n<div id=\"Example_03_01_06\" class=\"example\">\n<div id=\"fs-id1165134032261\" class=\"exercise\">\n<div id=\"fs-id1165137772480\" class=\"commentary\">\n<p id=\"fs-id1165137762415\">Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. When dividing by a purely imaginary number, we can simply multiply the numerator and denominator by [latex]i[\/latex] instead of the complex conjugate.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137409413\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135471100\">How To: Given two complex numbers, divide one by the other.<\/h3>\n<ol id=\"fs-id1165135471104\">\n<li>Write the division problem as a fraction.<\/li>\n<li>Determine the complex conjugate of the denominator.<\/li>\n<li>Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide [latex]\\left(2+5i\\right)[\/latex] by [latex]\\left(4-i\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q665746\">Show Solution<\/span><\/p>\n<div id=\"q665746\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137605861\">We begin by writing the problem as a fraction.<\/p>\n<div id=\"eip-id1165134234232\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\left(2+5i\\right)}{\\left(4-i\\right)}[\/latex]<\/div>\n<p id=\"fs-id1165137639613\">Then we multiply the numerator and denominator by the complex conjugate of the denominator.<\/p>\n<div id=\"eip-id1165137400110\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{\\left(2+5i\\right)}{\\left(4-i\\right)}\\cdot \\frac{\\left(4+i\\right)}{\\left(4+i\\right)}[\/latex]<\/div>\n<p id=\"fs-id1165137474228\">To multiply two complex numbers, we expand the product as we would with polynomials (the process commonly called FOIL).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcll}\\Large{\\frac{(2+5i)}{(4-i)}}\\cdot\\frac{(4+i)}{4+i)}&=&\\Large{\\frac{8+2i+20i+5{i}^{2}}{16+4i - 4i-{i}^{2}}}& \\\\&=&\\Large{\\frac{8+2i+20i+5(-1)}{16+4i - 4i-(-1)}}&\\quad\\text{Because }i^2=-1\\\\&=&\\Large{\\frac{3+22i}{17}}& \\\\&=&\\Large{\\frac{3}{17}+\\frac{22}{17}i}&\\quad\\text{Separate real and imaginary parts}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165137930346\">Note that this expresses the quotient in standard form.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3157\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3157&theme=oea&iframe_resize_id=ohm3157&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the last video, you will see more examples of dividing complex numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Dividing Complex Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/XBJjbJAwM1c?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>Multiplying complex numbers is similar to multiplying polynomials. Remember that an imaginary number times another imaginary number gives a real result. When you divide complex numbers, you must first multiply the numerator and denominator by the complex conjugate to eliminate any imaginary parts, and then you can divide.<\/p>\n<\/section>\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-16447\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Raising the imaginary unit i to powers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/sfP6SmEYHRw\">https:\/\/youtu.be\/sfP6SmEYHRw<\/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>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><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@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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@3.278:1\/Preface<\/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":169554,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Raising the imaginary unit i to powers\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/sfP6SmEYHRw\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Dividing Complex Numbers\",\"author\":\"James Sousa 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