Roots of Complex Number

Theorem
Let $z := \polar {r, \theta}$ be a complex number expressed in polar form, such that $z \ne 0$.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Then the $n$th roots of $z$ are given by:
 * $z^{1 / n} = \set {r^{1 / n} \paren {\map \cos {\dfrac {\theta + 2 \pi k} n} + i \, \map \sin {\dfrac {\theta + 2 \pi k} n} }: k \in \set {0, 1, 2, \ldots, n - 1} }$

There are $n$ distinct such $n$th roots.

Exponential Form
This result can also be expressed thus in exponential form:

Proof
Let:
 * $w := r^{1 / n} \paren {\map \cos {\dfrac {\theta + 2 \pi k} n} + i \, \map \sin {\dfrac {\theta + 2 \pi k} n} }$

for $k \in \Z_{>0}$.

Then:

Now let $m = k + n$.

Then:

exploiting the fact that Sine and Cosine are Periodic on Reals.

The result follows.