Roots of Complex Number/Examples/5th Roots of -32

Example of Roots of Complex Number
Let $z^5 = -32$.

Then:
 * $z = 2 \paren {\map \cos {\dfrac {\pi + 2 k \pi} 5} + i \, \map \sin {\dfrac {\pi + 2 k \pi} 5} }$

for $k = 0, 1, 2, 3, 4$.

That is:

Proof

 * Complex 5th Roots of -32.png

In polar form:
 * $-32 = \polar {32, \pi + 2 k \pi}$

Let $z = r \cis \theta$.

Then: