\[\left( 10a^{2}y \right)^{2}\left( 3ay^{2} \right)^{3} =\]
\[= 10^{2}a^{4}y^{2} \cdot 3^{3}a^{3}y^{6} =\]
\[= 100 \cdot 27a^{4 + 3}y^{2 + 6} =\]
\[= 2700a^{7}y^{8}.\]