Answer
$w_0=\frac{\sqrt 2}{2}+\frac{\sqrt x}{2}i$
$w_1=-\frac{\sqrt 2}{2}+\frac{\sqrt x}{2}i$
$w_2=-\frac{\sqrt 2}{2}-\frac{\sqrt x}{2}i$
$w_3=\frac{\sqrt 2}{2}-\frac{\sqrt x}{2}i$
Work Step by Step
$r=|z|=1$
$θ=\pi~~$ (Negative real axis)
Polar form:
$z=1(cos~\pi+i~sin~\pi)$
$w_k=\sqrt[4] 1[cos(\frac{\pi+2k\pi}{4})+i~sin(\frac{\pi+2k\pi}{4})]$
$w_0=1(cos~\frac{\pi}{4}+i~sin~\frac{\pi}{4})=\frac{\sqrt 2}{2}+\frac{\sqrt x}{2}i$
$w_1=1(cos~\frac{3\pi}{4}+i~sin~\frac{3\pi}{4})=-\frac{\sqrt 2}{2}+\frac{\sqrt x}{2}i$
$w_2=1(cos~\frac{5\pi}{4}+i~sin~\frac{5\pi}{4})=-\frac{\sqrt 2}{2}-\frac{\sqrt x}{2}i$
$w_3=1(cos~\frac{7\pi}{4}+i~sin~\frac{7\pi}{4})=\frac{\sqrt 2}{2}-\frac{\sqrt x}{2}i$
