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

$\because x^2-x+1=0$

$\therefore$             $x=\frac{1\pm \sqrt{\left(1-4\right)}}{2}=\frac{1\pm i\sqrt{3}}{2}$

                   $=\frac{1+i\sqrt{3}}{2},\frac{1-i\sqrt{3}}{2}$

$\therefore$             $x=-{\omega }^2,-\omega$

$\therefore$            $\alpha =-{\omega }^2,\beta =-\omega$

$\Rightarrow {\alpha }^{2009}+{\beta }^{2009}=-{\omega }^{4018}-{\omega }^{2009}$

                         $=-\omega -{\omega }^2=-\left(\omega +{\omega }^2\right)=-\left(-1\right)=1$