document.write( "Question 1078164: Do the sum of the n nth roots of a complex number z always equals zero? Show this using the examples below.\r
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Algebra.Com's Answer #692634 by ikleyn(52790)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\"highlight%28cross%28Do%29%29\" Does the sum of the n-th roots of a complex number z always \"highlight%28cross%28equals%29%29\" equal zero? Show this using the examples below.\r
\n" ); document.write( "\n" ); document.write( "1. Find the cube roots of -i\r
\n" ); document.write( "\n" ); document.write( "2. Find the fourth roots of \"+-1%2F2-+%28sqrt+%283%29+%2F2%29i+\"
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\n" ); document.write( "\n" ); document.write( "0. Yes, this statement is TRUE.\r
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document.write( "    The simplest way to explain/to illustrate it is to use square roots of complex number.\r\n" );
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document.write( "    For any complex number z, if \"alpha\" = \"sqrt%28z%29\" is a square root of z, then \"-alpha\" is also square root of z  (the second value),   \r\n" );
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document.write( "    and \"alpha+%2B+%28-alpha%29\" = 0.\r\n" );
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\n" ); document.write( "\n" ); document.write( "1. Find the cube roots of -i\r
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document.write( "    -i = \"cos%283pi%2F4%29+%2B+i%2Asin%283pi%2F4%29\" = \"cis%283pi%2F4%29\".  \r\n" );
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document.write( "    Therefore, the three values of the cube root of -i are\r\n" );
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document.write( "        a)  \"cis%28pi%2F4%29\" = \"cos%28pi%2F4%29+%2B+i%2Asin%28pi%2F4%29\" = i;\r\n" );
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document.write( "        b)  \"cis%28pi%2F4+%2B+2pi%2F3%29\" = \"cis%28pi%2F4%29%2Acis%282pi%2F3%29\" = \"i%2Acis%282pi%2F3%29\";\r\n" );
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document.write( "        c)  \"cis%28pi%2F4+%2B+4pi%2F3%29\" = \"cis%28pi%2F4%29%2Acis%284pi%2F3%29\" = \"i%2Acis%284pi%2F3%29\".\r\n" );
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document.write( "        Therefore, the sum of the three cube roots is\r\n" );
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document.write( "        \"i+%2B+i%2Acis%282pi%2F3%29+%2B+i%2Acis%282pi%2F3%29\" = \"i%2A%281%2Bcis%282pi%2F3%29%2Bcis%284pi%2F3%29%29\" = 0,\r\n" );
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document.write( "        since \"1%2Bcis%282pi%2F3%29%2Bcis%284pi%2F3%29\" = 0,  as you can easily check.\r\n" );
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\n" ); document.write( "\n" ); document.write( "2. Same technique and the same logic works for the fourth roots of \"+-1%2F2-+%28sqrt+%283%29+%2F2%29i+\".\r
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document.write( "    You can check it on your own as an exercise for you.\r\n" );
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\n" ); document.write( "\n" ); document.write( "There is a bunch of lessons on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetical operations on them\r
\n" ); document.write( "\n" ); document.write( "    - Complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Addition and subtraction of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Multiplication and division of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Raising a complex number to an integer power\r
\n" ); document.write( "\n" ); document.write( "    - How to take a root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with real coefficients on complex domain\r
\n" ); document.write( "\n" ); document.write( "    - How to take a square root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with complex coefficients on complex domain\r
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\n" ); document.write( "\n" ); document.write( "    - Solved problems on taking roots of complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic operations on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problem on taking square root of complex number\r
\n" ); document.write( "\n" ); document.write( "    - Miscellaneous problems on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Advanced problem on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - A curious example of an equation in complex numbers which HAS NO a solution\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Complex numbers\".\r
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