Answer
$z_{0}= 2^{\frac{1}{4}}\{\;cos(\frac{7\pi}{8})+isin(\frac{7\pi}{8})\}$
and
$z_{1}=2^{\frac{1}{4}}\{\;cos(\frac{15\pi}{8})+isin(\frac{15\pi }{8})\}$
Work Step by Step
$R=\sqrt{1^2+(-1)^2}=\sqrt{2}\\\\$
$tan\;\Theta \;=\frac{b}{a}=\;\frac{-1}{1}=-1\\$
$\Theta \;= tan^{-1}(-1)\;=\frac{7\pi}{4}\\\\$
$(1-i)=\sqrt{2}\{\;cos(\frac{7\pi}{4})+isin(\frac{7\pi}{4})\}\\\\$
$(1-i)^{\frac{1}{2}}=\;[\sqrt{2}\{\;cos(\frac{7\pi}{4})+isin(\frac{7\pi}{4})\}\;]^{\frac{1}{2}}\\\\$
$(1-i)^{\frac{1}{2}}=\;2^{\frac{1}{4}}\{\;cos(\frac{7\pi+8\pi n}{4})+isin(\frac{7\pi+8\pi n}{4})\}^{\frac{1}{2}}\\\\$
$(1-i)^{\frac{1}{2}}=\;2^{\frac{1}{4}}\{\;cos(\frac{7\pi+8\pi n}{8})+isin(\frac{7\pi+8\pi n}{8})\}\\\\$
At $\;\;\;\;\;\;\;\;\;n=0;$
$z_{0}= 2^{\frac{1}{4}}\{\;cos(\frac{7\pi}{8})+isin(\frac{7\pi}{8})\}$
At $\;\;\;\;\;\;n=1$
$z_{1}=2^{\frac{1}{4}}\{\;cos(\frac{15\pi}{8})+isin(\frac{15\pi }{8})\}$
