Answer
$z_0=cos~\frac{\pi}{16}+i~sin~\frac{\pi}{16}$
$z_1=cos~\frac{3\pi}{16}+i~sin~\frac{3\pi}{16}$
$z_2=cos~\frac{5\pi}{16}+i~sin~\frac{5\pi}{16}$
$z_3=cos~\frac{7\pi}{16}+i~sin~\frac{7\pi}{16}$
$z_4=cos~\frac{9\pi}{16}+i~sin~\frac{9\pi}{16}$
$z_5=cos~\frac{11\pi}{16}+i~sin~\frac{11\pi}{16}$
$z_6=cos~\frac{13\pi}{16}+i~sin~\frac{13\pi}{16}$
$z_7=cos~\frac{15\pi}{16}+i~sin~\frac{15\pi}{16}$
Work Step by Step
$z^8-i=0$
$z^8=i$
$r=|i|=1$. Also $i$ lies in the positive real axis.
Polar form of $i$:
$1(cos~\frac{\pi}{2}+i~sin~\frac{\pi}{2})$
$z_k=\sqrt[8]{1}[cos(\frac{\frac{\pi}{2}+2k\pi}{8})+i~sin(\frac{\frac{\pi}{2}+2k\pi}{8})]$
$z_0=1(cos~\frac{\pi}{16}+i~sin~\frac{\pi}{16})$
$z_1=1(cos~\frac{3\pi}{16}+i~sin~\frac{3\pi}{16})$
$z_2=1(cos~\frac{5\pi}{16}+i~sin~\frac{5\pi}{16})$
$z_3=1(cos~\frac{7\pi}{16}+i~sin~\frac{7\pi}{16})$
$z_4=1(cos~\frac{9\pi}{16}+i~sin~\frac{9\pi}{16})$
$z_5=1(cos~\frac{11\pi}{16}+i~sin~\frac{11\pi}{16})$
$z_6=1(cos~\frac{13\pi}{16}+i~sin~\frac{13\pi}{16})$
$z_7=1(cos~\frac{15\pi}{16}+i~sin~\frac{15\pi}{16})$
