Answer
$\dfrac{10}{13}-\dfrac{15}{13}i$
Work Step by Step
Multiplying the numerator and the denominator by the conjugate of the denominator, the given expression, $
\dfrac{5}{2+3i}
,$ is equivalent to
\begin{align*}\require {cancel}
&
\dfrac{5}{2+3i}\cdot\dfrac{2-3i}{2-3i}
\\\\&=
\dfrac{5(2-3i)}{(2)^2-(3i)^2}
&\left( \text{use }(a+b)(a-b)=a^2-b^2 \right)
\\\\&=
\dfrac{5(2-3i)}{4-9i^2}
\\\\&=
\dfrac{10-15i}{4-9i^2}
&\left( \text{use Distributive Property } \right)
\\\\&=
\dfrac{10-15i}{4-9(-1)}
&\left( \text{use }i^2=-1 \right)
\\\\&=
\dfrac{10-15i}{4+9}
\\\\&=
\dfrac{10-15i}{13}
\\\\&=
\dfrac{10}{13}-\dfrac{15}{13}i
.\end{align*}
Hence, in the form $a\pm bi,$ the given expression is equivalent to $
\dfrac{10}{13}-\dfrac{15}{13}i
$.
