If $x=1+i\sqrt 3$ and $y=1-i\sqrt 3$ and $z=2$, then for any prime $p>3, x^p+y^p=$?

Question

Let $x=-2w$ and $y=-2w^2$

Where $w$ and $w^2$ are imaginary cube roots of unity.

Then $$(-2w)^p +(-2w^2)^p$$ $$-2^p w^p + -2^pw^{2p}$$ what should do next?

Answer 1

Hint: Write $p=3k\pm 1$ to simplify $w^p$. Recall that $1+w+w^2=0$.