document.write( "Question 1110604: how to find all the values of (1-sqrt3i)^1/3 \n" ); document.write( "

Algebra.Com's Answer #725583 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "Use deMoivre's Theorem to find roots (or powers) of complex numbers.

\n" ); document.write( "The complex number $\"1-i%2Asqrt%283%29\"$ on the Argand plane has a magnitude of 2 and an angle of 300 degrees:
\n" ); document.write( " $\"1-i%2Asqrt%283%29+=+2%28cos%28300%29%2Bi%2Asin%28300%29%29\"$ .

\n" ); document.write( "The theorem says that the magnitude of the cube root of the complex number is the cube root of the magnitude of the number, and the angle of the cube root is 1/3 of the angle of the number.

\n" ); document.write( "So one of the cube roots of (1-i*sqrt(3)) is
\n" ); document.write( " $\"2%5E%281%2F3%29%28cos%28100%29%2Bi%2Asin%28100%29%29\"$

\n" ); document.write( "The theorem also tells us that the n-th roots of a complex number all have the same magnitude and are located around the Argand plane at intervals of 360/n degrees. So the other two cube roots of (1-i*sqrt(3)) are
\n" ); document.write( " $\"2%5E%281%2F3%29%28cos%28220%29%2Bi%2Asin%28220%29%29\"$ and
\n" ); document.write( " $\"2%5E%281%2F3%29%28cos%28340%29%2Bi%2Asin%28340%29%29\"$ \n" ); document.write( "

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