Which expression is equivalent to [(3xy^-5)^3 / (x^-2y^2)^-4]^-2

The answer is (x¹⁰y¹⁴)/729.

Explanation:
We can begin simplifying inside the innermost parentheses using the properties of exponents. The power of a power property says when you raise a power to a power, you multiply the exponents. This gives us

[(3³x³y⁻¹⁵)/(x⁸y⁻⁸)]⁻².

Negative exponents tell us to "flip" sides of the fraction, so within the parentheses we have
[(3³x³y⁸)/(x⁸y¹⁵)]⁻².

Using the quotient property, we subtract exponents when dividing powers, which gives us
(3³/x⁵y⁷)⁻².

Evaluating 3³, we have
(27/x⁵y⁷)⁻².

Using the power of a power property again, we have
27⁻²/x⁻¹⁰y⁻¹⁴.

Flipping the negative exponents again gives us x¹⁰y¹⁴/729.

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