Complex Numbers $x,y,z$ Find $x^{2007}+y^{2007}+z^{2007}$ Let $x,y,z$ be complex numbers such that
$$x+y+z = x^{5}+y^{5}+z^{5} = 0, \hspace{10pt}

x^3+y^3+z^3=2$$
Find all possible values of 
$$x^{2007}+y^{2007}+z^{2007}$$

It $x+y+z$ = 0, then $x,y,z$ are roots of $t^3 + at-b = 0$.
Then, we can show that, $x^5 + y^5 + z^5 = -5ab$ and $x^3 + y^3 + z^3 = 3b$, either using Newton's Identities or as in my answer here: https://math.stackexchange.com/a/115534/1102
Since $b \ne 0$, we must have that $a = 0$.
Thus $x,y,z$ are roots of $t^3 = b$. We know that $b = \frac{2}{3}$ and thus can compute the expression you need easily as $3b^{669} = \dfrac{2^{669}}{3^{668}}$.
