document.write( "Question 200450: hi can you help me finding the square root of i and the cubic root of -i? if posible could u draw the answers in a circle?
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Algebra.Com's Answer #151101 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"i+=+0%2B1i\" so the number i 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( "That vector has an angle of \"pi%2F2\" or 90° as we see by the\r\n" );
document.write( "blue arrow.  I'll use this \"at\" symbol, \" @ \", for theta.\r\n" );
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document.write( "So @ = \"pi%2F2\" and r = 1.\r\n" );
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document.write( "Trig form for a complex number is \r\n" );
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document.write( "\"r%28cos%28%22@%22%29%2Bi%2Asin%28%22@%22%29%29\"\r\n" );
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document.write( "However \"%22@%22\" doesn't have to just be identified as \"pi%2F2\" because \r\n" );
document.write( "we may add any multiple of \"2pi\" to the angle and it will be in\r\n" );
document.write( "the exact same position. So we use \"%22@%22=pi%2F2%2B2npi\" where \"n\"\r\n" );
document.write( "represents any integer, positive negative or zero.  \r\n" );
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document.write( "So \"r%28cos%28%22@%22%29%2Bi%2Asin%28%22@%22%29%29\"\r\n" );
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document.write( "becomes:\r\n" );
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document.write( "\"r%28cos%28%22@%22%2B2pi%2An%29%2Bi%2Asin%28%22@%22%2B2pi%2An%29%29\"\r\n" );
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document.write( "And for both problems:\r\n" );
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document.write( "=\r\n" );
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document.write( "\"1%28cos%28pi%28%281%2B4n%29%2F2%29%29%2Bi%2Asin%28pi%28%281%2B4n%29%2F2%29%29%29\"\r\n" );
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document.write( "Now first we'll take the square root, we raise that to the \"1%2F2\"\r\n" );
document.write( "power:\r\n" );
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document.write( " \r\n" );
document.write( "Now the rule for simplifying a complex number in trig\r\n" );
document.write( "form is:\r\n" );
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document.write( "1. Raise the modulus (lenth of vector) to the power \r\n" );
document.write( "2. Multiply the argument (angle) by the power.\r\n" );
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document.write( "which becomes:\r\n" );
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document.write( "\"1%28cos%28pi%28%281%2B4n%29%2F4%29%29%2Bi%2Asin%28pi%28%281%2B4n%29%2F4%29%29%29\"\r\n" );
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document.write( "or\r\n" );
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document.write( "\"cos%28pi%28%281%2B4n%29%2F4%29%29%2Bi%2Asin%28pi%28%281%2B4n%29%2F4%29%29\"\r\n" );
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document.write( "Now we let n take on any two consecutive integer values, but\r\n" );
document.write( "the easiest are n=0 and n=1, so letting n=0:\r\n" );
document.write( " \r\n" );
document.write( "\"cos%28pi%28%281%2B4%2A0%29%2F4%29%29%2Bi%2Asin%28pi%28%281%2B4%2A0%29%2F4%29%29\"\r\n" );
document.write( " \r\n" );
document.write( "\"cos%28pi%2F4%29%2Bi%2Asin%28pi%2F4%29+=+sqrt%282%29%2F2%2Bi%2Asqrt%282%29%2F2\"\r\n" );
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document.write( "That is this vector:\r\n" );
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document.write( "Now letting n=1:\r\n" );
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document.write( "\"cos%28pi%28%281%2B4%2A1%29%2F4%29%29%2Bi%2Asin%28pi%28%281%2B4%2A1%29%2F4%29%29\"\r\n" );
document.write( " \r\n" );
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document.write( "That is this vector:\r\n" );
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document.write( "So the two square roots of i are \r\n" );
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document.write( "\"sqrt%282%29%2F2%2Bi%2Asqrt%282%29%2F2++\" and \"-sqrt%282%29%2F2-i%2Asqrt%282%29%2F2++\"\r\n" );
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document.write( "Putting them both on the same graph:\r\n" );
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document.write( "They form two equally spaced \"spokes of a wheel\":\r\n" );
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document.write( " -----------------------------------\r\n" );
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document.write( "Now we'll do the cube root of i:\r\n" );
document.write( "we raise that to the \"1%2F3\"\r\n" );
document.write( "power:\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "Now the rule for simplifying a complex number in trig\r\n" );
document.write( "form is:\r\n" );
document.write( " \r\n" );
document.write( "1. Raise the modulus (lenth of vector) to the power \r\n" );
document.write( "2. Multiply the argument (angle) by the power.\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "which becomes:\r\n" );
document.write( " \r\n" );
document.write( "\"1%28cos%28pi%28%281%2B4n%29%2F6%29%29%2Bi%2Asin%28pi%28%281%2B4n%29%2F6%29%29%29\"\r\n" );
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document.write( "or\r\n" );
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document.write( "\"cos%28pi%28%281%2B4n%29%2F6%29%29%2Bi%2Asin%28pi%28%281%2B4n%29%2F6%29%29\"\r\n" );
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document.write( "Now we let n take on any three consecutive integer values, but\r\n" );
document.write( "the easiest are n=0, n=1 and n=2, so letting n=0:\r\n" );
document.write( " \r\n" );
document.write( "\"cos%28pi%28%281%2B4%2A0%29%2F6%29%29%2Bi%2Asin%28pi%28%281%2B4%2A0%29%2F6%29%29\"\r\n" );
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document.write( "\"cos%28pi%2F6%29%2Bi%2Asin%28pi%2F6%29+=+sqrt%283%29%2F2%2Bi%2A1%2F2\"\r\n" );
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document.write( "That is this vector:\r\n" );
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document.write( "Now letting n=1:\r\n" );
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document.write( "\"cos%28pi%28%281%2B4%2A1%29%2F6%29%29%2Bi%2Asin%28pi%28%281%2B4%2A1%29%2F6%29%29\"\r\n" );
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document.write( "\"cos%28pi%285%2F6%29%29%2Bi%2Asin%28pi%285%2F6%29%29\"\r\n" );
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document.write( "\"cos%285pi%2F6%29%2Bi%2Asin%285pi%2F6%29=-sqrt%283%29%2F2%2B%281%2F2%29i\"\r\n" );
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document.write( "That is this vector:\r\n" );
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document.write( "Now letting n=2:\r\n" );
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document.write( "\"cos%28pi%28%281%2B4%2A2%29%2F6%29%29%2Bi%2Asin%28pi%28%281%2B4%2A2%29%2F6%29%29\"\r\n" );
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document.write( "\"cos%28pi%289%2F6%29%29%2Bi%2Asin%28pi%289%2F6%29%29\"\r\n" );
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document.write( "\"cos%28pi%283%2F2%29%29%2Bi%2Asin%28pi%283%2F2%29%29\"\r\n" );
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document.write( "\"cos%283pi%2F2%29%2Bi%2Asin%283pi%2F2%29=0%2B%28-1%29i+=+-i\"\r\n" );
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document.write( "That is this vector:\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%2B%281%2F2%29i\", \"-sqrt%283%29%2F2%2B%281%2F2%29i\", and -i\r\n" );
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document.write( "Putting all three on the same graph:\r\n" );
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document.write( "They form three equally spaced \"spokes of a wheel\":\r\n" );
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document.write( "Edwin
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