Answer
$z_0=\sqrt[6] 2(cos~\frac{5\pi}{12}+i~sin~\frac{5\pi}{12})$
$z_1=\sqrt[6] 2(cos~\frac{13\pi}{12}+i~sin~\frac{13\pi}{12})$
$z_2=\sqrt[6] 2(cos~\frac{21\pi}{12}+i~sin~\frac{21\pi}{12})$
Work Step by Step
$z^3+1=-i$
$z^3=-1-i$
Polar form of $-1-i$:
$r=|z|=\sqrt {a^2+b^2}=\sqrt {(-1)^2+(-1)^2}=\sqrt 2$
$tan~θ=\frac{b}{a}=\frac{-1}{-1}=1$
$θ=\frac{5\pi}{4}$
$-1-i=\sqrt 2(cos~\frac{5\pi}{4}+i~sin~\frac{5\pi}{4})$
$z_k=\sqrt[3] {\sqrt 2}[cos(\frac{\frac{5\pi}{4}+2k\pi}{3})+i~sin(\frac{\frac{5\pi}{4}+2k\pi}{3})]$
$z_0=\sqrt[6] 2(cos~\frac{5\pi}{12}+i~sin~\frac{5\pi}{12})$
$z_1=\sqrt[6] 2(cos~\frac{13\pi}{12}+i~sin~\frac{13\pi}{12})$
$z_2=\sqrt[6] 2(cos~\frac{21\pi}{12}+i~sin~\frac{21\pi}{12})$
