Answer
$\frac{16}{13}-\frac{11}{13} i$
Work Step by Step
Given
\begin{equation}
\frac{5+2 i}{2+3 i}.
\end{equation} Multiply both numerator and denominator by conjugate of the denominator and simplify.
\begin{equation}
\begin{aligned}
\frac{5+2 i}{2+3 i}&=\frac{5+2 i}{2+3 i}\cdot \frac{(2-3 i)}{(2-3 i)}\\
& =\frac{5(2-3 i)+2i(2-3i)}{4+9}\\
&=\frac{10-15 i+4i+6}{13}\\
&= \frac{16}{13}-\frac{11}{13} i. \\
\end{aligned}
\end{equation} The solution is $$\frac{5+2 i}{2+3 i}= \frac{16}{13}-\frac{11}{13} i.$$
