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 #692772 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "She used trigonometry, but I think you can use just algebra,\r\n" );
document.write( "and no trig.  Think I'll try to do it that way, just for fun,,\r\n" );
document.write( "since she's already done it the trig way.\r\n" );
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document.write( "Let \r\n" );
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document.write( "(a + bi)³ = -i, where a, b are real.\r\n" );
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document.write( "a³ + 3a²bi + 3ab²i² + b³i³ = -i\r\n" );
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document.write( "a³ + 3a²bi - 3ab² - b³i = -i\r\n" );
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document.write( "Equating real parts:  | Equating imaginary parts:\r\n" );
document.write( "                      |\r\n" );
document.write( "a³ - 3ab² = 0         |  3a²b - b³ = -1\r\n" );
document.write( "a(a²-3b²) = 0         |\r\n" );
document.write( "a = 0; a²-3b² = 0     |\r\n" );
document.write( "           a² = 3b² ->  3(3b²)b - b² = -1\r\n" );
document.write( "                            9b³ - b³ = -1\r\n" );
document.write( "                                 8b³ = -1\r\n" );
document.write( "                                  b³ = -1/8\r\n" );
document.write( "       a² = 3(1/2)² <-             b = -1/2\r\n" );
document.write( "       a² = 3/4\r\n" );
document.write( "        a = ±√3/2\r\n" );
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document.write( "So the three cube roots of -i are:\r\n" );
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document.write( "a + bi\r\n" );
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document.write( "Using a = 0, b = -1/2\r\n" );
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document.write( "a + bi = (-1/2)i\r\n" );
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document.write( "Using a = ±√3/2, \r\n" );
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document.write( "a + bi = ±√3/2 - (1/2)i\r\n" );
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document.write( "(-1/2)i + [√3/2 - (1/2)i] + [-√3/2 - (1/2)i] =\r\n" );
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document.write( "(-1/2)i + √3/2 - (1/2)i -  √3/2 - (1/2)i = 0\r\n" );
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document.write( "----\r\n" );
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document.write( "Edwin
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