$-8+i\cdot8\sqrt3$ converted to polar form is $16 \exp(\pi/3)$.
According to the theory,
fourth root of $16 = 2$, so all four roots should be calculated from:
$2\cdot\exp(i\cdot(\pi/12 + 2\pi k/4))$ where $k=0,1,2,3$.
I'm struggling to get the right answers from that. i.e. one of the answers should be: $\sqrt3 - i.$