Question

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

Answer 1

`-8+8sqrt(3)i` has four fourth roots. These are `sqrt(3)+i`, `1-sqrt(3)i`, `-sqrt(3)-i` and `-1+sqrt(3)i`.

Explanation:

Plot `-8+8 sqrt(3)i` on the complex plane and note that it has an argument of `120°` and a modulus of `sqrt((-8)^2+(8 sqrt(3))^2)=16`.
Consequently the argument and modulus of one root are `(120°)/4` and `16^(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.