\[\left( a \cdot a^{4} \right)^{2}\ :\ *\ = a^{2}\]
\[\left( a^{1 + 4} \right)^{2}\ :\ *\ = a^{2}\]
\[\left( a^{5} \right)^{2}\ :\ *\ = a^{2}\]
\[a^{10}\ :\ *\ = a^{2}\]
\[*\ = a^{10}\ :a^{2}\]
\[*\ = a^{10 - 2}\]
\[*\ = a^{8}.\]