Could someone help me with this?
Find all solutions to z = (6 − 6√3)^(1/5).
I am guessing you mean $\displaystyle z=6(1-i\sqrt{3})$
we use De Moivre's formula
$\displaystyle z^{\frac{1}{n}}=r^{\frac{1}{n}}\left[ \cos(\frac{x+2\pi k}{n})+ i \sin(\frac{x+2\pi k}{n})\right]$
in the above formula x is the reference angle so ours is $\displaystyle -\frac{\pi}{3}$ r is the radius(=6) and k goes from 0 to n-1
$\displaystyle z^{\frac{1}{5}}=6^{\frac{1}{5}}\left[ \cos(\frac{-\frac{\pi}{3}+2\pi k}{6})+ i \sin(\frac{-\frac{\pi}{3}+2\pi k}{6})\right]$
if you plug in k=0,1,2,3,4,5 you will get all solutions