3\sqrt{5c}\sqrt{15^3}
=3\sqrt{5}\sqrt{15^3}\sqrt{c} Reason: (\sqrt{5c}=\sqrt{5}\sqrt{c})
=3\cdot \15\sqrt{5}\sqrt{15}\sqrt{c} Reason: (\sqrt{15^3}=15\sqrt{15})
=3\sqrt{5}\sqrt{c}\cdot \5\cdot \3\sqrt{15} Reason:(15=3\cdot \5)
=3\sqrt{5}\sqrt{c}\cdot \5\cdot \3\sqrt{5\cdot \3} Reason:(15=3\cdot \5)
=3\sqrt{5}\sqrt{c}\cdot \5\cdot \3\sqrt{5}\sqrt{3} Reason:(\sqrt[n]^{ab}=\sqrt[n]^{a}\sqrt[n]^{b})
=5\cdot \3^{\frac{5}{2}}\sqrt{5}\sqrt{5}\sqrt{c} Reason:(a^b\cdot \a^c=a^{b+c})
=3^2\cdot \5^2\sqrt{3}\sqrt{c} Reason:(\:a^b\cdot \:a^c=a^{b+c})
=225\sqrt{3}\sqrt{c} Refined answer