If $\mathrm{z}=2+3\mathrm{i}\text{, then }{\mathrm{z}}^5+(\overline{\mathrm{z}}{)}^5$ is equal to:

$\mathrm{z}=2+3\mathrm{i}\Rightarrow \overline{\mathrm{z}}=2-3\mathrm{i}$
$\begin{array}{l}{\mathrm{z}}^5+(\overline{\mathrm{z}}{)}^5=(2+3\mathrm{i}{)}^5+(2-3\mathrm{i}{)}^5\\ =2\left[{ }^5{\mathrm{C}}_0{2}^5{+}^5{\mathrm{C}}_2{2}^3\left(3\mathrm{i}{)}^2{+}^5{\mathrm{C}}_4{2}^1\right(3\mathrm{i}{)}^4\right]\\ =2[32+10\times 8\times (-9)+5\times 2\times 81]\\ =2[32-720+810]=2\times 122=244\end{array}$