[tex]z=-3+3i=3(-1+i)=3\left(\sqrt2e^{i3\pi/4}\right)=3\sqrt2e^{i3\pi/4}[/tex]
Let [tex]z=w^8[/tex]. Then the eight roots of [tex]z[/tex] are
[tex]w=\left(3\sqrt2e^{i3\pi/4}\right)^{1/8}[/tex]
[tex]w=3^{1/8}2^{1/16}}e^{i(3\pi/4+2\pi k)/8}[/tex]
where [tex]k=0,1,\ldots,7[/tex]. So the eighth roots are
[tex]w=\begin{cases}3^{1/8}2^{1/16}e^{i3\pi/32}&\text{for }k=0\\\\3^{1/8}2^{1/16}e^{i11\pi/32}&\text{for }k=1\\\\3^{1/8}2^{1/16}e^{i19\pi/32}&\text{for }k=2\\\\3^{1/8}2^{1/16}e^{i27\pi/32}&\text{for }k=3\\\\3^{1/8}2^{1/16}e^{i35\pi/32}&\text{for }k=4\\\\3^{1/8}2^{1/16}e^{i43\pi/32}&\text{for }k=5\\\\3^{1/8}2^{1/16}e^{i51\pi/32}&\text{for }k=6\\\\3^{1/8}2^{1/16}e^{i59\pi/32}&\text{for }k=7\end{cases}[/tex]
