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

The answer is (x ¹⁰ y ¹⁴) / 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

[ (3 ³ x ³ y ⁻¹⁵) / (x ⁸ y ⁻⁸) ] ⁻².

Negative exponents tell us to "flip" sides of the fraction, so within the parentheses we have

[ (3 ³ x ³ y ⁸) / (x ⁸ y ¹⁵) ] ⁻².

Using the quotient property, we subtract exponents when dividing powers, which gives us

(3 ³ / x ⁵ y ⁷) ⁻².

Evaluating 3 ³, we have

(27/x ⁵ y ⁷) ⁻².

Using the power of a power property again, we have

27 ⁻² / x ⁻¹⁰ y ⁻¹⁴.

Flipping the negative exponents again gives us x ¹⁰ y ¹⁴ / 729.