document.write( "Question 681027: Express the equation z^4=−2 + 2√3i in polar form. Hence solve the equation,
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Algebra.Com's Answer #422556 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"z%5E4\"\"%22%22=%22%22\"\"-2+%2B+2sqrt%283%29i\"\r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Cz%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C%282+%2B+2sqrt%283%29i%29%5E%281%2F4%29%29\"\r\n" );
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document.write( "\"x+=+-2\", \"y+=2sqrt%283%29\", \"r=sqrt%28x%5E2%2By%5E2%29=sqrt%28%28-2%29%5E2%2B%282sqrt%283%29%29%5E2%29\" = \"sqrt%284%2B4%2A3%29\" = \"sqrt%284%2B12%29\" = \"sqrt%2816%29\" = \"4\"\r\n" );
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document.write( "\"tan%28theta%29\"\"%22%22=%22%22\"\"y%2Fx\"\"%22%22=%22%22\"\"%282sqrt%283%29%29%2F%28-2%29\"\"%22%22=%22%22\"\"-sqrt%283%29\".\r\n" );
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document.write( "The reference angle for \"theta\" is 60° and since x is - and y is +,\r\n" );
document.write( "\"theta\" is in the 2nd quadrant, so we subtract 60° from 180° and get\r\n" );
document.write( "120°, which has the same trig functions as 120° + 360°·k, so we take:\r\n" );
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document.write( "\"theta\"\"%22%22=%22%22\"\"%22120%B0%22%2B%22360%B0%22k\".\r\n" );
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document.write( "Now we have\r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Cz%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C%282+%2B+2sqrt%283%29i%29%5E%281%2F4%29%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C%28r%28cos%28theta%29%2Bi%2Asin%28theta%29%29%29%5E%281%2F4%29%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\r\n" );
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document.write( "Now we use the formula: \"%28r%28cos%28theta%29%2Bi%2Asin%28theta%29%29%29%5En\"\"%22%22=%22%22\"\"r%5En%2A%28cos%28n%2Atheta%29%2Bi%2Asin%28n%2Atheta%29%29\"\r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Cz%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\r\n" );
document.write( "\"sqrt%282%29%28cos%28%2230%B0%22%2B%2290%22k%29%2Bi%2Asin%28%2230%B0%22%2B%2290%22k%29%29\"\r\n" );
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document.write( "Now we let k = 0,1,2,and 3 to get the 4 4th roots, the 4 possible\r\n" );
document.write( "answers for z:\r\n" );
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document.write( "For k = 0,\r\n" );
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document.write( "\"%22%22=%22%22\"\"sqrt%282%29%28cos%28%2230%B0%22%29%2Bi%2Asin%28%2230%B0%22%29%29\"\r\n" );
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document.write( "For k = 1,\r\n" );
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document.write( "\"%22%22=%22%22\"\"sqrt%282%29%28cos%28%22120%B0%22%29%2Bi%2Asin%28%22120%B0%22%29%29\"\r\n" );
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document.write( "For k = 2,\r\n" );
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document.write( "\"%22%22=%22%22\"\"sqrt%282%29%28cos%28%22210%B0%22%29%2Bi%2Asin%28%22210%B0%22%29%29\"\r\n" );
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document.write( "For k = 3,\r\n" );
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document.write( "\"%22%22=%22%22\"\"sqrt%282%29%28cos%28%22300%B0%22%29%2Bi%2Asin%28%22300%B0%22%29%29\"\r\n" );
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document.write( "Those are the four values for z.\r\n" );
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document.write( "Edwin
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