Roots of Complex Number/Examples/Cube Roots

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

Then the complex cube roots of $z$ are given by:
 * $z^{1 / 3} = \left\{{r^{1 / 3} \left({\cos \left({\dfrac {\theta + 2 \pi k} 3}\right) + i \sin \left({\dfrac {\theta + 2 \pi k} 3}\right)}\right): k \in \left\{{0, 1, 2}\right\}}\right\}$

There are $3$ distinct such complex cube roots.

These can also be expressed as:
 * $z^{1 / 3} = \left\{{r^{1 / 3} e^{i \left({\theta + 2 \pi k}\right) / 3}: k \in \left\{{0, 1, 2}\right\} }\right\}$

or:
 * $z^{1 / 3} = \left\{ {r^{1 / 3} e^{i \theta / 3} \omega^k: k \in \left\{{0, 1, 2}\right\} }\right\}$

where $\omega = e^{2 i \pi / 3} = -\dfrac 1 2 + \dfrac {i \sqrt 3} 2$ is the first cube root of unity.

Proof
An example of Roots of Complex Number.