Complex numbers exercise - homework

Question

We have to prove that $z_1^{24n}+z_2^{24n}=2^{12n+1}$ if we know that a)$z_1z_2=2$ b)$z_1^3+z_2^3=-4$

I have tried many things but nothing worked so far

Answer 1

It is basically along the lines of @labbhattacharya's answer, just a little different.

Notice that $$z_1^3+z_2^3=-4,\ z_1^3z_2^3=8\Rightarrow $$ $z_1^3,z_2^3$ are the roots of the equation $$t^2+4t+8=0\Rightarrow t=2(-1\pm i)=2\sqrt{2}e^{\displaystyle \pm i \frac{\pi}{4}}$$ Now, the answer follows.