If and are the roots of x^2-x+1=0, then ^2009+^2009=

Question

If $\alpha$ and $\beta$ are the roots of $x^2-x+1=0$, then $\alpha^{2009}+\beta^{2009}=$

Accepted Answer

$\alpha=-\omega^2,\ \beta=-\omega$ (roots of $x^2-x+1=0$).

$$\alpha^{2009}+\beta^{2009}=(-\omega^2)^{2009}+(-\omega)^{2009}=-\omega^{4018}-\omega^{2009}=-\omega^2-\omega=1.$$

Solutions · 1