# Question #dd5ca

##### 1 Answer
May 1, 2017

Start with the Euler's Formula: $1 = \cos \left(2 n\right) \pi + i \sin \left(2 n\right) \pi = {e}^{i 2 n \pi}$

So ${1}^{\frac{1}{3}} = {\left({e}^{i 2 n \pi}\right)}^{\frac{1}{3}} = {e}^{\frac{i 2 n \pi}{3}}$

$n = 0 : = c i s 0 = 1$ This is the principal (real) root.

$n = 1 : = c i s \frac{2 \pi}{3} = - \frac{1}{2} + \frac{\sqrt{3}}{2} i$

$n = 2 : = c i s \frac{4 \pi}{3} = - \frac{1}{2} - \frac{\sqrt{3}}{2} i$

Thereafter the roots repeat. In the complex plane, each number has n distinct nth roots.