$\mathrm{If} \mathrm{Z}=\frac{\sqrt{3}+\mathrm{i}}{2}, \mathrm{then} {\left({\mathrm{z}}^{101}+{\mathrm{i}}^{103}\right)}^{105}=$

$\begin{array}{rcl}\mathrm{z}& =& \frac{\sqrt{3}+\mathrm{i}}{2} \mathrm{then} {\left(z^{101}+{\mathrm{i}}^{103}\right)}^{105}\end{array}$

$\begin{array}{rcl}\mathrm{z}& =& \frac{-{\mathrm{i}}^2\sqrt{3}+\mathrm{i}}{2}=-\mathrm{i}\left(\frac{-1+\mathrm{i}\sqrt{3}}{2}\right)=-\mathrm{iw}\end{array}$

$\begin{array}{rcl}{\mathrm{z}}^{101}& =& -{\mathrm{i}}^{101} {\mathrm{w}}^{101}=-{\left({\mathrm{i}}^4\right)}^{25} \mathrm{i} {\left({\mathrm{w}}^3\right)}^{33} {\mathrm{w}}^2\end{array}$

$\begin{array}{rcl}{\mathrm{z}}^{101}& =& -\mathrm{i} {\mathrm{w}}^2\end{array}$

$\begin{array}{rcl}{\mathrm{i}}^{103}& =& {\left({\mathrm{i}}^4\right)}^{25} {\mathrm{i}}^2.\mathrm{i}=-\mathrm{i}\end{array}$

$\begin{array}{rcl}{\left(z^{101}+{\mathrm{i}}^{103}\right)}^{105}& =& {\left(-{\mathrm{iw}}^2-\mathrm{i}\right)}^{105}\end{array}$

$\begin{array}{rcl} & =& {(-\mathrm{i})}^{1052}{(1+{\mathrm{w}}^2)}^{105} =-{\left({\mathrm{i}}^4\right)}^{26} \mathrm{i} {(-\mathrm{w})}^{105} =-\mathrm{i}{\left({\mathrm{w}}^3\right)}^{35}=-\mathrm{i}={\mathrm{z}}^3\end{array}$