Cube roots of a complex number

Exponential form. $z=8e^{i\pi/2}$.

Cube roots. $w_k=2e^{i(\pi/2+2k\pi)/3}$ for $k=0,1,2$:
- $k=0$: $w_0=2e^{i\pi/6}=2(\cos\frac\pi6+i\sin\frac\pi6)=2(\frac{\sqrt3}{2}+\frac i2)=\sqrt3+i$.
- $k=1$: $w_1=2e^{i5\pi/6}=2(-\frac{\sqrt3}{2}+\frac i2)=-\sqrt3+i$.
- $k=2$: $w_2=2e^{i3\pi/2}=2(-i)=-2i$.

Conclusion. The cube roots are $\sqrt3+i$, $-\sqrt3+i$, $-2i$.