Which expression is equivalent to \( \left(\frac{\left(2 a^{-3} b^{4}\right)^{2}}{\left(3 a^{5} b\right)^{-2}}\right)^{-1} \) ? Assume \( a \neq 0, b \neq 0 \)

To simplify the expression \( \left(\frac{\left(2 a^{-3} b^{4}\right)^{2}}{\left(3 a^{5} b\right)^{-2}}\right)^{-1} \), we start with the numerator and denominator separately. 1. **Numerator:** \[ \left(2 a^{-3} b^{4}\right)^{2} = 2^{2} (a^{-3})^{2} (b^{4})^{2} = 4 a^{-6} b^{8} \] 2. **Denominator:** \[ \left(3 a^{5} b\right)^{-2} = \frac{1}{(3 a^{5} b)^{2}} = \frac{1}{3^{2} (a^{5})^{2} (b)^{2}} = \frac{1}{9 a^{10} b^{2}} = 9^{-1} a^{-10} b^{-2} \] Therefore, the entire expression becomes: \[ \frac{4 a^{-6} b^{8}}{\frac{1}{9 a^{10} b^{2}}} = 4 a^{-6} b^{8} \cdot 9 a^{10} b^{2} = 36 a^{4} b^{10} \] 3. Now applying the outer negative exponent: \[ \left(36 a^{4} b^{10}\right)^{-1} = \frac{1}{36} a^{-4} b^{-10} \] Thus, the expression simplifies to: \[ \frac{b^{-10}}{36 a^{4}} \] The equivalent expression is: \[ \frac{1}{36 a^{4} b^{10}} \]