Which expression is equivalent to \( \left(\frac{x^{-4} y}{x^{-9} y^{5}}\right)^{-2} \) ? Assume \( x \neq 0, y \neq 0 \)

1. Start with the given expression: \[ \left(\frac{x^{-4} y}{x^{-9} y^{5}}\right)^{-2} \] 2. Simplify the fraction inside the parentheses. Using the rule \(\frac{a^m}{a^n} = a^{m-n}\): - For \(x\): \[ x^{-4} \div x^{-9} = x^{-4 - (-9)} = x^{5} \] - For \(y\): \[ y^{1} \div y^{5} = y^{1-5} = y^{-4} \] So, the fraction simplifies to: \[ x^{5} y^{-4} \] 3. Now raise the result to the power of \(-2\) using the power rule \((a^m)^n = a^{mn}\): \[ \left(x^{5} y^{-4}\right)^{-2} = x^{5(-2)} y^{-4(-2)} = x^{-10} y^{8} \] 4. Therefore, the equivalent expression is: \[ \frac{y^{8}}{x^{10}} \]