5) \( \frac{4 u^{-2} v^{-4}}{3 u^{-2} v^{3} \cdot 4 u^{-2} v^{3}} \)

To simplify the expression \( \frac{4 u^{-2} v^{-4}}{3 u^{-2} v^{3} \cdot 4 u^{-2} v^{3}} \), we first simplify the denominator. The denominator becomes: \[ 3 u^{-2} v^{3} \cdot 4 u^{-2} v^{3} = 12 u^{-4} v^{6} \] Now, we can rewrite the entire expression: \[ \frac{4 u^{-2} v^{-4}}{12 u^{-4} v^{6}} \] Next, we simplify this fraction: \[ = \frac{4}{12} \cdot \frac{u^{-2}}{u^{-4}} \cdot \frac{v^{-4}}{v^{6}} = \frac{1}{3} \cdot u^{2} \cdot v^{-10} \] This can be written as: \[ \frac{u^{2}}{3 v^{10}} \] Thus, the simplified result is: \[ \frac{u^{2}}{3 v^{10}} \]