**Unformatted text preview: **s Theorem
The complex number on the left hand side of the equation corresponds to the point with polar
coordinates |w|3 , 3α , while the complex number on the right hand side corresponds to the point
with polar coordinates (8, 0). Since |w| ≥ 0, so is |w|3 , which means |w|3 , 3α and (8, 0) are
two polar representations corresponding to the same complex number, both with positive r values.
From Section 11.4, we know |w|3 = 8 and 3α = 0+2πk for integers k . Since |w| is a real number, we
√
solve |w|3 = 8 by extracting the principal cube root to get |w| = 3 8 = 2. As for α, we get α = 2πk
3
for integers k . This produces three distinct points with polar coordinates corresponding to k = 0,
π
π
1, and 2: (2, 0), 2, 23 , and 2, 43 . These correspond to the complex numbers w1 = 2cis(0),
π
π
w2 = 2cis 23 and w3 = 2cis 43 , respectively. Writing these out in rectangular form yields
√
√
w0 = 2, w1 = −1 + i 3 and w2 = −1 − i 3. While this process seems a tad more involved than
our previous factoring approach, this procedure can be generalized to ﬁnd, for example, all of the
ﬁfth roots of 32.16 If we start with a generic comple...

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