Aloha :)
Willkommen in der Mathelounge... \o/
$$z_1=\frac{(-\sqrt3+3i)^6}{(6\sqrt2+6i\sqrt2)^2}=\frac{\left((3i-\sqrt3)^3\right)^2}{(6\sqrt2)^2(1+i)^2}=\frac{((3i)^3-3(3i)^2\sqrt3+3(3i)(\sqrt3)^2-(\sqrt3)^3)^2}{36\cdot2\cdot(1^2+2i+i^2)}$$$$\phantom{z_1}=\frac{(27i^3-i^227\sqrt3+i27-3\sqrt3)^2}{36\cdot2\cdot(1^2+2i+i^2)}\stackrel{(i^2=-1)}{=}\frac{(-27i+27\sqrt3+i27-3\sqrt3)^2}{72\cdot(1+2i-1)}$$$$\phantom{z_1}=\frac{(24\sqrt3)^2}{72\cdot2i}=\frac{12}{i}\stackrel{(i^2=-1)}{=}=\frac{-12i^2}{i}=\pink{-12\,i}=\green{12e^{-i\,\pi/2}}$$
$$z_2=\frac{(-1+i\sqrt3)^4}{(-\sqrt2+i\sqrt2)^6}=\frac{(-1+i\sqrt3)^3}{\left((-\sqrt2+i\sqrt2)^2\right)^3}(-1+i\sqrt3)$$$$\phantom{z_2}=\frac{(-1)^3+3(-1)^2i\sqrt3+3(-1)(i\sqrt3)^2+(i\sqrt3)^3}{\left((-\sqrt2)^2-2\cdot\sqrt2\cdot i\sqrt2+(i\sqrt2)^2\right)^3}(-1+i\sqrt3)$$$$\phantom{z_2}=\frac{-1+i3\sqrt3-9i^2+i^33\sqrt3}{\left(2-4i+2i^2\right)^3}(-1+i\sqrt3)\stackrel{(i^2=-1)}{=}\frac{-1+i3\sqrt3+9-i3\sqrt3}{\left(2-4i-2\right)^3}(-1+i\sqrt3)$$$$\phantom{z_2}=\frac{8}{\left(-4i\right)^3}(-1+i\sqrt3)=\frac{1}{-8i^3}(-1+i\sqrt3)\stackrel{(1=i^4)}{=}\frac{-i}{8}(-1+i\sqrt3)=\pink{\frac{\sqrt3+i}{8}}$$$$\phantom{z_2}=\frac14\left(\frac{\sqrt3}{2}+i\,\frac12\right)=\frac14\left(\cos\frac\pi6+i\,\sin\frac\pi6\right)=\green{\frac14\,e^{i\,\pi/6}}$$
