If α, β ∈ C are the distinct roots of the equation x ^ 2 - x + 1 =...

${x}^{2} - x + 1 = 0$

Roots are $- \omega, - {\omega}^{2}$.

${\alpha}^{101} + {\beta}^{107} = {\left(- \omega\right)}^{101} + {\left(- {\omega}^{2}\right)}^{107}$

$= - {\omega}^{2} - \omega\left(∵ {\omega}^{3} = 1\right)$

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

$= 1\left(∵ 1 + \omega + {\omega}^{2} = 0\right)$