All the values of $(8 i)^{\frac{1}{3}}$ are

Question

Aicharya · Accepted Answer

$\begin{aligned} & (8 i)^{1 / 3}=\left(-8 i^3ight)^{1 / 3}=-2 i \ & i=e^{i \frac{\pi}{2}}=e^{i \frac{5 \pi}{2}} \end{aligned}$ $(8 i)^{1 / 3}=2 e^{\frac{i \pi}{6}}=2\left(\frac{\sqrt{3}}{2}+\frac{i}{2}ight)=\sqrt{3}+i$ $(8 i)^{1 / 3}=2 e^{i \frac{5 \pi}{6}}=2\left(\frac{-\sqrt{3}}{2}+\frac{i}{2}ight)=-\sqrt{3}+i$ $(8 i)^{1 / 3}= \pm \sqrt{3}+i,-2 i$.

All the values of $(8 i)^{\frac{1}{3}}$ are

Solution