complex numbers - Solve: $z^4 +2sqrt3 +2i = 0$

Solve: $z^4 +2\sqrt3 +2i = 0$

I'm already trying to solve this exercise for $20$ minutes, no luck. I got up to here:

$z^4 = -2i -2\sqrt3 = -2(\sqrt3 + i)$ but it's impossible to compute from here.

Also I tried: $z^8 = -2\sqrt3 -2i \rightarrow$ $z^8 = (4*3) -4 = 8 \rightarrow$ $z^8 = 8$

$z^8 = {{1}\over{8}}\text{Cis}(\pi)$ and from there to solve.

Answer

Hint:

$$z^{4}=4\left(-\frac{\sqrt{3}}{2}-\frac{1}{2}i\right)=4e^{-\frac{5}{6}\pi i}$$