Roots of Complex Number

Theorem
Let $z := \left\langle{r, \theta}\right\rangle$ 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:
 * $z^{1 / n} = \left\{{r^{1 / n} \left({\cos \left({\dfrac {\theta + 2 \pi k} n}\right) + i \sin \left({\dfrac {\theta + 2 \pi k} n}\right)}\right): k \in \left\{{0, 1, 2, \ldots, n-1}\right\}}\right\}$

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

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

Proof
Let:
 * $w := r^{1 / n} \left({\cos \left({\dfrac {\theta + 2 \pi k} n}\right) + i \sin \left({\dfrac {\theta + 2 \pi k} n}\right)}\right)$

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.