# Question #2e282

##### 1 Answer
Mar 16, 2017

$\frac{1}{2} \left(- 1 + i \sqrt{3}\right)$

#### Explanation:

We have that $z = - \frac{1}{2} \left(1 + i \sqrt{3}\right) = \left\mid z \right\mid {e}^{i \text{arg} \left(z\right)}$

where $\text{arg} \left(z\right) = \arctan \left(\frac{- \sqrt{3}}{- 1}\right)$

but $\left\mid z \right\mid = 1$ and $\text{arg} \left(z\right) = - \frac{2 \pi}{3}$

so ${z}^{83} = {\left({e}^{- i \frac{2 \pi}{3}}\right)}^{83} = {e}^{- i \frac{166 \pi}{3}} = {e}^{i \frac{2 \pi}{3}} = \frac{1}{2} \left(- 1 + i \sqrt{3}\right)$