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 \( a \neq 0, b \neq 0 \), \( 0 \frac{2}{3 a^{4} b^{10}} \) \( 0 \frac{4}{9 a^{4} b^{10}} \) \( 0 \frac{1}{36 a^{4} b^{10}} \) \( 0 \frac{36 a^{4} b^{10}}{2} \)

To simplify the given expression \( \left(\frac{\left(2 a^{-3} b^{4}\right)^{2}}{\left(3 a^{5} b\right)^{-2}}\right)^{-1} \), we start by calculating the numerator and the denominator separately. 1. **Numerator**: \( \left(2 a^{-3} b^{4}\right)^{2} = 4 a^{-6} b^{8} \). 2. **Denominator**: By applying the rule \( (x^{-n})^{-1} = x^{n} \), \( \left(3 a^{5} b\right)^{-2} = \frac{1}{(3 a^{5} b)^{2}} = \frac{1}{9 a^{10} b^{2}} \). Now, we get the fraction: \[ \frac{4 a^{-6} b^{8}}{\frac{1}{9 a^{10} b^{2}}} = 4 a^{-6} b^{8} \cdot \frac{9 a^{10} b^{2}}{1} = 36 a^{4} b^{10}. \] Next, we have to take the reciprocal because of the negative exponent: \[ \left(36 a^{4} b^{10}\right)^{-1} = \frac{1}{36 a^{4} b^{10}}. \] Therefore, the expression simplifies to \( \frac{1}{36 a^{4} b^{10}} \), which matches one of the provided options! Let’s consider some contextual tips to make this clearer. When tackling expressions with variable exponents, remember that multiplying or dividing bases means you can simply add or subtract their exponents. This is akin to simplifying fractions – the more you can reduce, the easier it is to manage your calculations! Also, watch out for negative exponents! They can be tricky, but just remember that they mean "reciprocal". Just flip the base and change the sign on the exponent, and you’ll avoid common mistakes!