Answer
$w_0=\frac{3\sqrt 3}{2}+\frac{3}{2}i$
$w_1=-\frac{3\sqrt 3}{2}+\frac{3}{2}i$
$w_2=-3i$
Work Step by Step
$r=|z|=27$ ·
$θ=\frac{\pi}{2}~~$ (Positive imaginary axis)
Polar form:
$27i=27(cos~\frac{\pi}{2}+i~sin~\frac{\pi}{2})$
$w_k=\sqrt[3] {27}[cos(\frac{\frac{\pi}{2}+2k\pi}{3})+i~sin(\frac{\frac{\pi}{2}+2k\pi}{3})]$
$w_0=3(cos~\frac{\pi}{6}+i~sin~\frac{\pi}{6})=3(\frac{\sqrt 3}{2}+\frac{1}{2}i)=\frac{3\sqrt 3}{2}+\frac{3}{2}i$
$w_1=3(cos~\frac{5\pi}{6}+i~sin~\frac{5\pi}{6})=3(-\frac{\sqrt 3}{2}+\frac{1}{2}i)=-\frac{3\sqrt 3}{2}+\frac{3}{2}i$
$w_2=3(cos~\frac{3\pi}{2}+i~sin~\frac{3\pi}{2})=-3i$
