$$z = (2 + 3i)$$

$$ \Rightarrow {z^5} = (2 + 3i){\left( {{{(2 + 3i)}^2}} \right)^2}$$

$$ = (2 + 3i){( - 5 + 12i)^2}$$

$$ = (2 + 3i)( - 119 - 120i)$$

$$ = - 238 - 240i - 357i + 360$$

$$ = 122 - 597i$$

$${\overline z ^5} = 122 + 597i$$

$${z^5} + {\overline z ^5} = 244$$