document.write( "Question 1129850: Use demoivre's theorem to find the three cube roots of -64i. Write your answers in both polar form and a + bi form. \n" ); document.write( "
Algebra.Com's Answer #746488 by ikleyn(52832)\"\" \"About 
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\n" ); document.write( "Use \"highlight%28cross%28demoivres%29%29\" de Moivre's theorem to find the three cube roots of -64i. Write your answers in both polar form and a + bi form.
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\n" ); document.write( "\n" ); document.write( "            de Moivre is the name of the French mathematician  (1667-1754).\r
\n" ); document.write( "\n" ); document.write( "            You can read about him from this Wikipedia article https://en.wikipedia.org/wiki/Abraham_de_Moivre \r
\n" ); document.write( "\n" ); document.write( "            As any/every person's name,  it should be written using first CAPITAL letter.\r
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document.write( "The trigonometric form (= the polar form) of the complex number  \"-64i\"  is  64*(cos(270°) + i*sin(270°)).\r\n" );
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document.write( "The modulus of  \"-64i\"  is  64,  the argument is  270° = \"3pi%2F2\".\r\n" );
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document.write( "According to the general theory, there are three complex cube roots of  \"-64i\".  They have the modulus of  \"root%283%2C64%29\" = 4. \r\n" );
document.write( "The first cube root has the argument of  90° = \"pi%2F2\",  one third of the argument of  \"-64i\". \r\n" );
document.write( "Each next cube root has the argument by  \"360%5Eo%2F3\" = 120° = \"2pi%2F3\"  more than the previous one. \r\n" );
document.write( "Thus the tree complex roots are\r\n" );
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document.write( "  1)  4*(cos(90°) + i*sin(90°))   = \"0+%2B+4%2Ai\"; \r\n" );
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document.write( "  2)  4*(cos(90°+120°) + i*sin(90° + 120°)) = 4*(cos(210°) + i*sin(210°) = \"4%2A%28-sqrt%283%29%2F2+-+%281%2F2%29%2Ai%29\" = \"-2%2Asqrt%283%29+-+2%2Ai\".\r\n" );
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document.write( "  3)  4*(cos(90°+240°) + i*sin(90° + 240°)) = 4*(cos(330°) + i*sin(330°)) = \"4%2A%28sqrt%283%29%2F2+-+%281%2F2%29%2Ai%29\" = \"2%2Asqrt%283%29+-+2%2Ai\".\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( "    - 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( "In this list,  I marked by  (*)  the lessons that are most relevant to this problem.\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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