Answer
$30a^{5}\sqrt{3}$
Work Step by Step
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression, $
3\sqrt{5a^7}\cdot2\sqrt{15a^3}
,$ is equivalent to
\begin{array}{l}\require{cancel}
3\sqrt{5a^7}\cdot2\sqrt{15a^3}
\\\\=
3(2)\sqrt{5a^7(15a^3)}
\\\\=
6\sqrt{75a^{10}}
.\end{array}
Extracting the factor that is a perfect power of the index (all radicands are assumed positive), then
\begin{array}{l}\require{cancel}
6\sqrt{75a^{10}}
\\\\=
6\sqrt{25a^{10}\cdot3}
\\\\=
6\sqrt{(5a^{5})^2\cdot3}
\\\\=
6\cdot5a^{5}\sqrt{3}
\\\\=
30a^{5}\sqrt{3}
.\end{array}