Answer
$z_0=2(cos~\frac{\pi}{18}+i~sin~\frac{\pi}{18})$
$z_1=2(cos~\frac{13\pi}{18}+i~sin~\frac{13\pi}{18})$
$z_2=2(cos~\frac{25\pi}{18}+i~sin~\frac{15\pi}{18})$
Work Step by Step
$z^3-4\sqrt 3-4i=0$
$z^3=4\sqrt 3+4i$
Polar form of $4\sqrt 3+4i$:
$r=|z|=\sqrt {a^2+b^2}=\sqrt {(4\sqrt 3)^2+4^2}=\sqrt {64}=8$
$tan~θ=\frac{b}{a}=\frac{4}{4\sqrt 3}=\frac{\sqrt 3}{3}$
$θ=\frac{\pi}{6}$
$4\sqrt 3+4i=8(cos~\frac{\pi}{6}+i~sin~\frac{\pi}{6})$
$z_k=\sqrt[3]{8}[cos(\frac{\frac{\pi}{6}+2k\pi}{3})+i~sin(\frac{\frac{\pi}{6}+2k\pi}{3})]$
$z_0=2(cos~\frac{\pi}{18}+i~sin~\frac{\pi}{18})$
$z_1=2(cos~\frac{13\pi}{18}+i~sin~\frac{13\pi}{18})$
$z_2=2(cos~\frac{25\pi}{18}+i~sin~\frac{15\pi}{18})$
