Can someone help me solve $z^{3}=-i$ $z$ is a complex number. I tried to solve this question by setting $z=a+bi$, but when I calculated $(a+bi)^{3}$, I found that's a little bit complicated to compute, Can someone help me teach me some easier way to solve the problem? Thanks a lot.

$z^3=-i \leftrightarrow z^3+i=0 \leftrightarrow (z)^3+(-i)^3=0$.
The LHS factors into $(z-i)(z^2+iz-1)=0$.
Now you have $(z-i)=0 \rightarrow z=i$ and $(z^2+iz-1)=0 \rightarrow z=\frac{-i \pm \sqrt{3}}{2}$.
