Complex number equations I cannot solve two problems regarding complex equations.
1)Let $z^2+w^2=0$, prove that $$z^{4n+2}+w^{4n+2}=0, n \in \mathbb{N^{*}}$$
What I tried;
$$z^2 \cdot z^{4n}+w^2 \cdot w^{4n}=0 \iff w^2(w^{4n}-z^{4n})=0$$
but it doesn't really prove anything.
2) Let $z=\frac{1+i\sqrt{3}}{2}$, evaluate $1+z^{1997}-z^{1998}+z^{1999}$
I can think of a way turning $z=e^{\frac{i\pi}{3}}$ but I have to solve it without anything but complex number properties(and algebra of course).

1) Rearrange your equation, conversely just plug in $z^2 = -w^2$.
2) You are right, use your $z = e^{i\frac{\pi}{3}}$
You just need to think about what the exponents are mod 6. That's all that matters here. Do you see why?
