document.write( "Question 128835: Find the roots of z^6=1 for z complex. Show that the sum of the roots you found is zero. \n" ); document.write( "
Algebra.Com's Answer #94233 by Edwin McCravy(20060)\"\" \"About 
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Find the roots of z^6=1 for z complex. Show that the sum of the roots you found is zero.
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document.write( "\"z%5E6=1\"\r\n" );
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document.write( "Get 0 on the right:\r\n" );
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document.write( "\"z%5E6+-+1+=+0\"\r\n" );
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document.write( "Write as the difference of two squares:\r\n" );
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document.write( "\"%28z%5E3%29%5E2-1%5E2=0\"\r\n" );
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document.write( "Factor as the difference of squares:\r\n" );
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document.write( "\"%28z%5E3-1%29%28z%5E3%2B1%29=0\"\r\n" );
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document.write( "Write the first factor as the difference of two cubes\r\n" );
document.write( "and the second factor as the sum of two cubes:\r\n" );
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document.write( "\"%28z%5E3-1%5E3%29%28z%5E3%2B1%5E3%29=0\"\r\n" );
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document.write( "Factor each by the rule for factoring the difference\r\n" );
document.write( "(and sum) of two cubes\r\n" );
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document.write( "\"%28z-1%29%28z%5E2%2Bz%2B1%29%28z%2B1%29%28z%5E2-z%2B1%29=0\"\r\n" );
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document.write( "Set each factor = 0\r\n" );
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document.write( "\"z-1=0\" yields root \"z=1\"\r\n" );
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document.write( "\"z%2B1=0\" yields root \"z=-1\"\r\n" );
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document.write( "\"z%5E2%2Bz%2B1=0\", by the quadratic formula\r\n" );
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document.write( "yields roots \"z=%28-1+%2B+i%2Asqrt%283%29%29%2F2\" and \"z=%28-1+-+i%2Asqrt%283%29%29%2F2\"\r\n" );
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document.write( "\"z%5E2-z%2B1=0\", by the quadratic formula\r\n" );
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document.write( "yields roots \"z=%281+%2B+i%2Asqrt%283%29%29%2F2\" and \"z=%281+-+i%2Asqrt%283%29%29%2F2\".\r\n" );
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document.write( "So the 6 roots are:\r\n" );
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document.write( "\"1\", \"-1\", \"%28-1+%2B+i%2Asqrt%283%29%29%2F2\", \"%28-1+-+i%2Asqrt%283%29%29%2F2\", \"%281+%2B+i%2Asqrt%283%29%29%2F2\", \"%281+-+i%2Asqrt%283%29%29%2F2\"\r\n" );
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document.write( "Add them up:\r\n" );
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document.write( " =\r\n" );
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document.write( " = \"0\"\r\n" );
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document.write( "Edwin
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