\[\frac{a^{- 3} \cdot \left( a^{4} \right)^{2}}{a^{- 6}} = \frac{\left( a^{- 3} \cdot a^{8} \right)}{a^{- 6}} =\]
\[= \frac{a^{5}}{a^{- 6}} = a^{11}.\]