Roots of Complex Number/Examples/Cube Roots

Theorem

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

Then the complex cube roots of $z$ are given by:

$z^{1 / 3} = \set {r^{1 / 3} \paren {\map \cos {\dfrac {\theta + 2 \pi k} 3} + i \, \map \sin {\dfrac {\theta + 2 \pi k} 3} }: k \in \set {0, 1, 2} }$

There are $3$ distinct such complex cube roots.

These can also be expressed as:

$z^{1 / 3} = \set {r^{1 / 3} e^{i \paren {\theta + 2 \pi k} / 3}: k \in \set {0, 1, 2} }$

or:

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

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.

$\blacksquare$