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 \( a \neq 0, b \neq 0 \)

1. First, compute the numerator: \[ \left(2a^{-3}b^4\right)^2 = 2^2 \cdot \left(a^{-3}\right)^2 \cdot \left(b^4\right)^2 = 4a^{-6}b^8. \] 2. Next, compute the denominator: \[ \left(3a^5b\right)^{-2} = 3^{-2} \cdot \left(a^5\right)^{-2} \cdot b^{-2} = \frac{1}{9a^{10}b^2}. \] 3. Now, write the fraction: \[ \frac{\left(2a^{-3}b^4\right)^2}{\left(3a^5b\right)^{-2}} = \frac{4a^{-6}b^8}{\frac{1}{9a^{10}b^2}} = 4a^{-6}b^8 \cdot 9a^{10}b^2. \] 4. Multiply the constants and add the exponents: \[ 4 \cdot 9 = 36, \] \[ a^{-6} \cdot a^{10} = a^{4}, \] \[ b^8 \cdot b^2 = b^{10}. \] Therefore, \[ 4a^{-6}b^8 \cdot 9a^{10}b^2 = 36a^4b^{10}. \] 5. Finally, apply the outer \((-1)\) exponent (which takes the reciprocal): \[ \left(36a^4b^{10}\right)^{-1} = \frac{1}{36a^4b^{10}}. \] The expression equivalent to \[ \left(\frac{\left(2a^{-3}b^4\right)^2}{\left(3a^5b\right)^{-2}}\right)^{-1} \] is \[ \frac{1}{36a^4b^{10}}. \]