To simplify the expression
\[ \frac{(a^{-2}b^{3})^{-4}}{(a^{-3}b^{2})^{-2}(ab)^{-4}}, \]
we will start by applying the power of a power property \((x^m)^n = x^{m \cdot n}\) to both the numerator and the denominator.
Step 1: Simplify the numerator
\[ (a^{-2}b^{3})^{-4} = a^{-2 \cdot (-4)} b^{3 \cdot (-4)} = a^{8} b^{-12}. \]
Step 2: Simplify the denominator
First, simplify \((a^{-3}b^{2})^{-2}\): \[ (a^{-3}b^{2})^{-2} = a^{-3 \cdot (-2)} b^{2 \cdot (-2)} = a^{6} b^{-4}. \]
Now simplify \((ab)^{-4}\): \[ (ab)^{-4} = a^{-4} b^{-4}. \]
Now, we multiply the results from these two parts of the denominator: \[ (a^{-3}b^{2})^{-2} (ab)^{-4} = (a^{6} b^{-4})(a^{-4} b^{-4}) = a^{6 + (-4)} b^{-4 + (-4)} = a^{2} b^{-8}. \]
Step 3: Combine the simplified parts
Now we can rewrite our expression: \[ \frac{a^{8} b^{-12}}{a^{2} b^{-8}}. \]
Step 4: Use the property of division
When dividing like bases, we subtract the exponents: \[ \frac{a^{8}}{a^{2}} = a^{8-2} = a^{6}, \] \[ \frac{b^{-12}}{b^{-8}} = b^{-12 - (-8)} = b^{-12 + 8} = b^{-4}. \]
Step 5: Rewrite using positive exponents
Now our expression is: \[ a^{6} b^{-4}. \]
To rewrite it with positive exponents, we can express \(b^{-4}\) as \(\frac{1}{b^{4}}\): \[ \frac{a^{6}}{b^{4}}. \]
Thus, the final simplified expression using positive exponents is: \[ \frac{a^{6}}{b^{4}}. \]