Answer
$w_0=1$
$w_1=\frac{\sqrt 2}{2}+\frac{\sqrt 2}{2}i$
$w_2=i$
$w_3=-\frac{\sqrt 2}{2}+\frac{\sqrt 2}{2}i$
$w_4=-1$
$w_5=-\frac{\sqrt 2}{2}-\frac{\sqrt 2}{2}i$
$w_6=-i$
$w_7=\frac{\sqrt 2}{2}-\frac{\sqrt 2}{2}i$
Work Step by Step
$r=|z|=1$
$θ=0~~$ (Positive real axis)
Polar form:
$z=1(cos~0+i~sin~0)$
$w_k=\sqrt[8] {1}[cos(\frac{0+2k\pi}{8})+i~sin(\frac{0+2k\pi}{8})]$
$w_0=cos~0+i~sin~0=1$
$w_1=cos(\frac{\pi}{4})+i~sin(\frac{\pi}{4})=\frac{\sqrt 2}{2}+\frac{\sqrt 2}{2}i$
$w_2=cos(\frac{\pi}{2})+i~sin(\frac{\pi}{2})=i$
$w_3=cos~\frac{3\pi}{4}+i~sin(\frac{3\pi}{4})=-\frac{\sqrt 2}{2}+\frac{\sqrt 2}{2}i$
$w_4=cos~\pi+i~sin~\pi=-1$
$w_5=cos(\frac{5\pi}{4})+i~sin(\frac{5\pi}{4})=-\frac{\sqrt 2}{2}-\frac{\sqrt 2}{2}i$
$w_6=cos(\frac{3\pi}{2})+i~sin(\frac{3\pi}{2})=-i$
$w_7=cos(\frac{7\pi}{4})+i~sin(\frac{7\pi}{4})=\frac{\sqrt 2}{2}-\frac{\sqrt 2}{2}i$
