Question

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

Answer 1

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.

Answer 2

Answer with explanation:

The given expression is

$=[\frac{(3 x y^{-5})^3}{(x^{-2}y^2)^{-4}}]^{-2}\\\\=\frac{(3 x y^{-5})^{-6}}{(x^{-2}y^2)^{8}}\\\\=\frac{(3^{-6} x^{-6} y^{30})}{(x^{-16}y^{16})}\\\\=3^{-6}\times x^{-6+16}\times y^{30-16}\\\\=\frac{x^{10}\times y^{14}}{729}$

Used the following law of indices

$1.[\frac{x}{y}]^m=\frac{x^m}{y^m}\\\\ 2. \frac{x^m}{x^n}=x^{m-n}\\\\ 3. \frac{1}{x^{-m}}=x^m$

The given expression is equivalent to

$\frac{x^{10}\times y^{14}}{729}$