document.write( "Question 1181402: Use De Moivre’s Theorem to solve for z and leave the answer in polar form with the angle in radians\r
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Algebra.Com's Answer #811322 by ikleyn(52776)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Your equation is\r\n" );
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document.write( "    z^3 = -8,             (1)\r\n" );
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document.write( "or, in polar form\r\n" );
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document.write( "    z^3 = \"8%2Acis%28pi%29\".     (2)\r\n" );
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document.write( "According to the DeMoivre's theorem, the solutions are these three complex numbers in polar form\r\n" );
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document.write( "    a)  \"z%5B1%5D\" = \"2%2Acis%28pi%2F3%29\"\r\n" );
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document.write( "    b)  \"z%5B2%5D\" = \"2%2Acis%28pi%2F3%2B2pi%2F3%29\" = \"2%2Acis%28pi%29\"\r\n" );
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document.write( "    c)  \"z%5B3%5D\" = \"2%2Acis%28pi%2F3%2B4pi%2F3%29\" = \"2%2Acis%285pi%2F3%29\"\r\n" );
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document.write( "The modulus is the cube root of 8, i.e. 2.\r\n" );
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document.write( "The argument of the first root is  \"1%2F3\"  of the argument of the right side equation (2).\r\n" );
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document.write( "The arguments of the second and third root have consecutive increments of  \"2pi%2F3\" from the argument of the first root.\r\n" );
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document.write( "Also notice that the complex root (b) is in reality the REAL number -2.\r\n" );
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\n" ); document.write( "\n" ); document.write( "There is a bunch of my 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( "    - Solved problems on de'Moivre formula\r
\n" ); document.write( "\n" ); document.write( "    - Calculating the sum 1*sin(1°) + 2*sin(2°) + 3*sin(3°) + . . . + 180*sin(180°) \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( "    - Solving one non-standard equation in complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Determine locus of points using complex numbers \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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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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