document.write( "Question 340848: Find the cube roots of -i. \n" ); document.write( "
Algebra.Com's Answer #244158 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
There are three cube roots of every number [except 0, which only has 0 as all of its roots of any kind.]
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document.write( "i written as a complex number of the form a+bi is 0+1i.\r\n" );
document.write( "The complex number a+bi is the vector that goes from the\r\n" );
document.write( "origin to the point (a,b).\r\n" );
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document.write( "Therefore the number 0+1i is the vector that goes from the\r\n" );
document.write( "origin to the point (0,1)\r\n" );
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document.write( "We want to put that number in trigonometric form.\r\n" );
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document.write( "First we draw the vector 0+1i which goes from the origin to \r\n" );
document.write( "the point (0,1)\r\n" );
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document.write( "and we see that its angle with the right hand side of the\r\n" );
document.write( "x-axis is 90°:\r\n" );
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document.write( "So we see that the vector's magnitude (length) is r=1, and\r\n" );
document.write( "its angle is \"theta=%2290%B0%22\"\r\n" );
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document.write( "Therefore the trigonometric form of i, or 0+1i, is\r\n" );
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document.write( "\"r%28cos%28theta%29%2Bi%2Asin%28theta%29%29\" or in this case\r\n" );
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document.write( "\"1%28cos%28%2290%B0%22%29%2Bi%2Asin%28%2290%B0%22%29%29\"\r\n" );
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document.write( "Now the cosine and sine will not change if we add or subtract 360° \r\n" );
document.write( "any whole number of times.  So we can write that as:\r\n" );
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document.write( "\"1%28cos%28%2290%B0%2B360%B0n%22%29%2Bi%2Asin%28%2290%B0%2B360%B0n%22%29%29\"\r\n" );
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document.write( "Next we use DeMoivre's theorem, which states that\r\n" );
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document.write( "with \"r=1\", \"theta=%2290%B0%2B360%B0n%22\", \"n=1%2F3\", since the cube root\r\n" );
document.write( "is the \"1%2F3\" power.\r\n" );
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document.write( "\"1%2A%28cos%28%2230%B0%2B120%B0n%22%29%2Bi%2Asin%28%2230%B0%2B120%B0n%22%29%29\"\r\n" );
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document.write( "\"cos%28%2230%B0%2B120%B0n%22%29%2Bi%2Asin%28%2230%B0%2B120%B0n%22%29\"\r\n" );
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document.write( "Now we substitute any three consecutive values of integer n\r\n" );
document.write( "We use the easiest ones, 0, 1 and 2\r\n" );
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document.write( "Substituting n=0\r\n" );
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document.write( "\"cos%28%2230%B0%2B120%B0n%22%29%2Bi%2Asin%28%2230%B0%2B120%B0n%22%29\"\r\n" );
document.write( "\"cos%28%2230%B0%2B120%B0%280%29%22%29%2Bi%2Asin%28%2230%B0%2B120%B0%280%29%22%29\"\r\n" );
document.write( "\"cos%28%2230%B0%22%29%2Bi%2Asin%28%2230%B0%22%29\"\r\n" );
document.write( "\"sqrt%283%29%2F2%2Bexpr%281%2F2%29i\"\r\n" );
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document.write( "Substituting n=1\r\n" );
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document.write( "\"cos%28%2230%B0%2B120%B0n%22%29%2Bi%2Asin%28%2230%B0%2B120%B0n%22%29\"\r\n" );
document.write( "\"cos%28%2230%B0%2B120%B0%281%29%22%29%2Bi%2Asin%28%2230%B0%2B120%B0%281%29%22%29\"\r\n" );
document.write( "\"cos%28%22150%B0%22%29%2Bi%2Asin%28%22150%B0%22%29\"\r\n" );
document.write( "\"-sqrt%283%29%2F2%2Bexpr%281%2F2%29i\"\r\n" );
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document.write( "Substituting n=2\r\n" );
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document.write( "\"cos%28%2230%B0%2B120%B0n%22%29%2Bi%2Asin%28%2230%B0%2B120%B0n%22%29\"\r\n" );
document.write( "\"cos%28%2230%B0%2B120%B0%282%29%22%29%2Bi%2Asin%28%2230%B0%2B120%B0%282%29%22%29\"\r\n" );
document.write( "\"cos%28%2230%B0%2B240%B0%22%29%2Bi%2Asin%28%2230%B0%2B240%B0%22%29\"\r\n" );
document.write( "\"cos%28%22270%B0%22%29%2Bi%2Asin%28%22270%B0%22%29\"\r\n" );
document.write( "\"0%2Bi%2A%28-1%29\"\r\n" );
document.write( "\"-i\"\r\n" );
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document.write( "So the three cube roots of i are\r\n" );
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document.write( "\"sqrt%283%29%2F2%2Bexpr%281%2F2%29i\", \"-sqrt%283%29%2F2%2Bexpr%281%2F2%29i\", \"-i\"\r\n" );
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document.write( "Edwin

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