Question

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

Answer 1

Answer with explanation:

The given expression is

`=[((3 x y^(-5))^3)/((x^(-2)y^2)^(-4))]^(-2)\\\\=((3 x y^(-5))^(-6))/((x^(-2)y^2)^(8))\\\\=((3^(-6) x^(-6) y^(30)))/((x^(-16)y^(16)))\\\\=3^(-6)* x^(-6+16)* y^(30-16)\\\\=(x^(10)* y^(14))/(729)`

Used the following law of indices

`1.[(x)/(y)]^m=(x^m)/(y^m)\\\\ 2. (x^m)/(x^n)=x^(m-n)\\\\ 3. (1)/(x^(-m))=x^m`

The given expression is equivalent to

`(x^(10)* y^(14))/(729)`

Answer 2

The answer is (x¹⁰y¹⁴)/729.

Step-by-step 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.