Question

$\alpha , \beta$are roots of x² - x + 1 = 0
$\therefore \alpha =-\omega and \beta =-{\omega }^2$
where $\omega$ is non-real cube root of unity
so, ${\alpha }^{101} + {\beta }^{107}$
$\Rightarrow (-\omega {)}^{101}+(-\omega {}^2{)}^{107}$
$\Rightarrow -[{\omega }^2+\omega ]$
$\Rightarrow$ -[-1] = 1 
(As 1 +$\omega$ + $\omega$² = 0 & $\omega$³ = 1)