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).

It's rather simple. $$z^2+w^2=0$$ Then $$z^2=-w^2$$ Which means, $$z^4=w^4$$ and for all $n$ : $$z^{4n}=w^{4n}$$ Therefore $$4z^{4n}z^2=-w^{4n}w^2$$ as a result, for all $n$, $$z^{4n+2}+w^{4n+2}=0$$
