$\mathrm{If} \mathrm{a}=\cos \left[\frac{8\mathrm{\pi }}{11}\right]+\mathrm{isin}\left[\frac{8\mathrm{\pi }}{11}\right], \mathrm{then} \mathrm{Re}\left(\mathrm{a}+{\mathrm{a}}^2+{\mathrm{a}}^3+{\mathrm{a}}^4+{\mathrm{a}}^5\right)=$

$\begin{array}{rcl}\mathrm{a}& =& \cos \frac{8\mathrm{\pi }}{11}+\mathrm{i} \sin \frac{8\mathrm{\pi }}{11}\end{array}$

$\begin{array}{rcl} & & \boxed{\mathrm{Rez}=\frac{\mathrm{z}+\overline{\mathrm{z}}}{2}}\end{array}$

$\begin{array}{rcl} & & \\ & & \\ & & \mathrm{Re} \left(\mathrm{a}+{\mathrm{a}}^2+{\mathrm{a}}^3+{\mathrm{a}}^4+{\mathrm{a}}^5\right)\\ & =& \frac{\mathrm{a}+{\mathrm{a}}^2+{\mathrm{a}}^3+{\mathrm{a}}^4+{\mathrm{a}}^5+\left(\overline{\mathrm{a}+{\mathrm{a}}^2+{\mathrm{a}}^3+{\mathrm{a}}^4+{\mathrm{a}}^5}\right)}{2}\\ & =& \frac{1}{2}\left(\mathrm{a}+{\mathrm{a}}^2+{\mathrm{a}}^3+{\mathrm{a}}^4+{\mathrm{a}}^5+\frac{1}{\mathrm{a}}+\frac{1}{{\mathrm{a}}^2}+\frac{1}{{\mathrm{a}}^3}+\frac{1}{{\mathrm{a}}^4}+\frac{1}{{\mathrm{a}}^5}\right)\\ & =& \frac{1}{2{\mathrm{a}}^5}\left({\mathrm{a}}^6+{\mathrm{a}}^7+{\mathrm{a}}^8+{\mathrm{a}}^9+{\mathrm{a}}^{10}+{\mathrm{a}}^4+{\mathrm{a}}^3+{\mathrm{a}}^2+\mathrm{a}+1\right)\\ & =& \frac{1}{2{\mathrm{a}}^5}\left(1+\mathrm{a}+{\mathrm{a}}^2+....+{\mathrm{a}}^{10}-{\mathrm{a}}^5\right)\\ & =& \frac{1}{2{\mathrm{a}}^5}\left(0-{\mathrm{a}}^5\right)=\frac{-1}{2}\end{array}$