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

The roots of $x^2-x+1=0$ are $-\omega$ and $-{\omega }^2$, where $\omega$ is a complex cube root of unity.$\alpha =-\omega ,\beta =-{\omega }^2$.${\alpha }^{2009}+{\beta }^{2009}=(-\omega {)}^{2009}+(-{\omega }^2{)}^{2009}=-({\omega }^{2009}+{\omega }^{4018})$.$2009=3\times 669+2⟹{\omega }^{2009}={\omega }^2$.$4018=3\times 1339+1⟹{\omega }^{4018}=\omega$.${\alpha }^{2009}+{\beta }^{2009}=-({\omega }^2+\omega )$.Since $1+\omega +{\omega }^2=0⟹\omega +{\omega }^2=-1$.Result $=-(-1)=1$.