$z^6=(1+i4\sqrt{3})^3$ I have found solutions of this equation: $z^2=1+i4\sqrt{3}$
they are the following: $z_0=\pm2\pm i\sqrt{3}$ Now, hence i want to find the remaining solutions of the equation. I the exercise i have a hint to multiply both sides of equation by $z_0^{-6}$ but still i do not understand how this could help in finding solutions of the equation.
Since $(zz_0^{-1})^6=1$ you know that $zz_0^{-1}$ is a sixth root of $1$, that is, $$zz_0^{-1}=e^{k\pi i/3}$$ or $$z=z_0e^{k\pi i/3}$$ for $k\in\{0,1,2,3,4,5\}$.
Now you can obtain $z$ for each $k$ in that set.
Note: the sixth roots of unity are given by: $$e^{2k\pi i/6}=e^{k\pi i/3}$$