Answer
$z_{0}=1\;\;\;\;\;\;,z_{1}=i\;\;\;\;\;\;\;z_{2}=-1\;\;\;\;\;\;\;\;z_{3}=-i$
Work Step by Step
$1= cos(0)+isin(0)=e^{i(0)}\\\\$
$1=cos(0+2n\pi)+isin(0+2n\pi)=\;e^{i(0+2n\pi)}\\\\$
$1^{\frac{1}{4}}\;=\;e^{i\frac{2n\pi}{4}}\\\\$
$\Rightarrow 1^{\frac{1}{4}}=\;e^{i(\frac{2n\pi}{4})}\;=\;cos(\frac{2n\pi}{4})+isin(\frac{2n\pi}{4})\\\\$
At: $\;\;\;n=0;$
$1^{\frac{1}{4}}=\;e^{0}\;=\;1\\\\$
At; $\;\;\;n=1;$
$1^{\frac{1}{4}}=\;e^{i(\frac{2\pi}{4})}\;=\;cos(\frac{2\pi}{4})+isin(\frac{2\pi}{4})=i\\\\$
At; $\;\;\;\;\;n=2;$
$1^{\frac{1}{4}}=\;e^{i(\frac{4\pi}{4})}\;=\;cos(\frac{4\pi}{4})+isin(\frac{4\pi}{4})=-1\\\\$
At; $\;\;\;\;n=3;$
$1^{\frac{1}{4}}=\;e^{i(\frac{6\pi}{4})}\;=\;cos(\frac{6\pi}{4})+isin(\frac{6\pi}{4})=-i\\\\$
