Answer
$\dfrac{3+5i}{17}$
Work Step by Step
Multiplying by the conjugate of the denominator, the given expression, $
\dfrac{2i}{5+3i}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{2i}{5+3i}\cdot\dfrac{5-3i}{5-3i}
\\\\=
\dfrac{2i(5)+2i(-3i)}{(5)^2-(3i)^2}
\\\\=
\dfrac{10i-6i^2}{25-9i^2}
\\\\=
\dfrac{10i-6(-1)}{25-9(-1)}
\\\\=
\dfrac{10i+6}{25+9}
\\\\=
\dfrac{6+10i}{34}
\\\\=
\dfrac{\cancel{2}(3+5i)}{\cancel{2}\cdot 17}
\\\\=
\dfrac{3+5i}{17}
.\end{array}
