Roots of Complex Number/Corollary

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:
 * $z^{1 / n} = \left\{{r^{1 / n} e^{i \theta / n} \omega^k: k \in \left\{{0, 1, 2, \ldots, n - 1}\right\}}\right\}$

where $\omega = e^{2 i \pi / n}$ is the first $n$th root of unity.

Proof
From Roots of Complex Number:Exponential Form, the $n$th roots of $z$ are given by:


 * $z^{1 / n} = \left\{{r^{1 / n} e^{i \left({\theta + 2 \pi k}\right) / n}: k \in \left\{{0, 1, 2, \ldots, n-1}\right\}}\right\}$

We have that:

Hence the result.