# How do you find the fourth root of -8 + 8sqrt3i?

$- 8 + 8 \sqrt{3} i$ has four fourth roots. These are $\sqrt{3} + i$, $1 - \sqrt{3} i$, $- \sqrt{3} - i$ and $- 1 + \sqrt{3} i$.
Plot $- 8 + 8 \sqrt{3} i$ on the complex plane and note that it has an argument of 120° and a modulus of $\sqrt{{\left(- 8\right)}^{2} + {\left(8 \sqrt{3}\right)}^{2}} = 16$.
Consequently the argument and modulus of one root are (120°)/4 and ${16}^{\frac{1}{4}}$, that is, 30° and $2$, which is $\sqrt{3} + i$. The other three have the same modulus and arguments which are multiples of 90° added on.