Simplify completely and write your answer in terms of positive exponents where applicable: \[ \frac{\left(a b^{3}\right)^{-2} \sqrt{a^{4} b^{10}}}{\left(a^{2} b\right)^{2} b^{-3}} \]

To simplify the expression \(\frac{\left(a b^{3}\right)^{-2} \sqrt{a^{4} b^{10}}}{\left(a^{2} b\right)^{2} b^{-3}}\), we will follow these steps: 1. **Handle the numerator:** - Start with \(\left(a b^{3}\right)^{-2}\): \[ \left(a b^{3}\right)^{-2} = a^{-2} b^{-6} \] - Now simplify \(\sqrt{a^{4} b^{10}}\): \[ \sqrt{a^{4} b^{10}} = a^{4/2} b^{10/2} = a^{2} b^{5} \] - Combining these together in the numerator: \[ a^{-2} b^{-6} \cdot a^{2} b^{5} = a^{-2 + 2} b^{-6 + 5} = a^{0} b^{-1} = \frac{1}{b} \] 2. **Handle the denominator:** - Begin with \(\left(a^{2} b\right)^{2}\): \[ \left(a^{2} b\right)^{2} = a^{4} b^{2} \] - Now we include \(b^{-3}\) in the denominator: \[ a^{4} b^{2} b^{-3} = a^{4} b^{2 - 3} = a^{4} b^{-1} \] 3. **Put it all together:** Now our expression becomes: \[ \frac{\frac{1}{b}}{a^{4} b^{-1}} = \frac{1}{b} \cdot \frac{b}{a^{4}} = \frac{1}{a^{4}} \] 4. **Final answer:** The completely simplified form is: \[ \frac{1}{a^{4}} \] So, the final answer is: \[ a^{-4} \]