If z = 2 + 3 i , then z 5 + z ¯ 5 is equal to:

Given $z=2+3i$,

So, $z^5+{\left(\overline{z}\right)}^5={\left(2+3i\right)}^5+{\left(2-3i\right)}^5$

Applying binomial expansion we get,

$=2\left({}^{5}C_0{2}^5+{}^{5}C_2{2}^3{\left(3i\right)}^2+{}^{5}C_4{2}^1{\left(3i\right)}^4\right)$

$=2\left(32+10\times 8\left(-9\right)+5\times 2\times 81\right)=244$