Answer
$Z_{0}=\;cos(0)+isin(0)=1\;, \;\;\;\;\; Z_{1}=\;cos(\frac{2\pi}{3})+isin(\frac{2\pi}{3})=-\frac{1}{2}+i\frac{\sqrt{3}}{2}\;,\;\;\;\;\; Z_{2}=\;cos(\frac{4\pi}{3})+isin(\frac{4\pi}{3})=-\frac{1}{2}-i\frac{\sqrt{3}}{2}$
Work Step by Step
$1= cos(0)+isin(0)=e^{i(0)}\\\\$
$1=cos(0+2n\pi)+isin(0+2n\pi)=\;e^{i(0+2n\pi)}\\\\$ $1^{\frac{1}{3}}\;=\;e^{i\frac{2n\pi}{3}}\\\\$
$\;\;\;\;\;\;\;\;\;\;\Rightarrow 1^{\frac{1}{3}}\;=\;e^{i(\frac{2n\pi}{3})}=\;cos(\frac{2n\pi}{3})+isin(\frac{2n\pi}{3})$
At $\;\;\;\;n=0$
$Z_{0}=\;cos(0)+isin(0)=1\\\\$
At $\;\;\;n=1$
$Z_{1}=\;cos(\frac{2\pi}{3})+isin(\frac{2\pi}{3})=-\frac{1}{2}+i\frac{\sqrt{3}}{2}\\\\$
At $\;\;\;\;n=2$
$Z_{2}=\;cos(\frac{4\pi}{3})+isin(\frac{4\pi}{3})=-\frac{1}{2}-i\frac{\sqrt{3}}{2}$
