If $$\alpha ,\beta \in C$$   are the distinct roots, of the equation $${x^2} - x + 1 = 0,$$    then $${\alpha ^{101}} + {\beta ^{107}}$$   is equal to:

Solution :
$$\eqalign{ & \alpha ,\beta \,\,{\text{are roots of }}{x^2} - x + 1 = 0 \cr & \therefore \,\,\alpha = - \omega \,\,{\text{and }}\beta = - {\omega ^2} \cr} $$
where $$\omega $$ is cube root of unity
$$\eqalign{ & \therefore \,\,{\alpha ^{101}} + {\beta ^{107}} = {\left( { - \omega } \right)^{101}} + {\left( { - \omega } \right)^{107}} \cr & = - \left[ {{\omega ^2} + \omega } \right] = - \left[ { - 1} \right] = 1 \cr} $$