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

$\begin{array}{rcl}{\mathrm{x}}^2-\mathrm{x}+1& =& 0\end{array}$

$\begin{array}{rcl} & \Rightarrow & \mathrm{x}=\frac{1\pm \mathrm{i}\sqrt{3}}{2} \Rightarrow \mathrm{x}=-\mathrm{w} \mathrm{or} -{\mathrm{w}}^2\end{array}$

$\begin{array}{rcl}\mathrm{\alpha }& =& -\mathrm{w}, \mathrm{\beta }=-{\mathrm{w}}^2\end{array}$

$\begin{array}{rcl}{\mathrm{\alpha }}^{101}& =& {(-\mathrm{w})}^{101}=-{\left({\mathrm{w}}^3\right)}^{33} {\mathrm{w}}^2=-{\mathrm{w}}^2\end{array}$

$\begin{array}{rcl}{\mathrm{\beta }}^{107}& =& {(-{\mathrm{w}}^2)}^{107}=-{\mathrm{w}}^{214}=-{\left({\mathrm{w}}^3\right)}^{71}\mathrm{w}=-\mathrm{w}\end{array}$

$\begin{array}{rcl}{\mathrm{\alpha }}^{101}+{\mathrm{\beta }}^{107}& =& -{\mathrm{w}}^2-\mathrm{w}=1\end{array}$