Simplify ((x^-2)^2)/((3y^-2)^3)

$\frac{{(x^{- 2})}^{2}}{{(3 y^{- 2})}^{3}}$

Step 1

Simplify the numerator.

Multiply the exponents in ${(x^{- 2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{- 2 \cdot 2}}{{(3 y^{- 2})}^{3}}$

Multiply $- 2$ by $2$ .

$\frac{x^{- 4}}{{(3 y^{- 2})}^{3}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{1}{x^{4}}}{{(3 y^{- 2})}^{3}}$

Step 2

Simplify the denominator.

Apply the product rule to $3 y^{- 2}$ .

$\frac{\frac{1}{x^{4}}}{3^{3} {(y^{- 2})}^{3}}$

Raise $3$ to the power of $3$ .

$\frac{\frac{1}{x^{4}}}{27 {(y^{- 2})}^{3}}$

Multiply the exponents in ${(y^{- 2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{1}{x^{4}}}{27 y^{- 2 \cdot 3}}$

Multiply $- 2$ by $3$ .

$\frac{\frac{1}{x^{4}}}{27 y^{- 6}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{1}{x^{4}}}{27 \frac{1}{y^{6}}}$

Step 3

Combine $27$ and $\frac{1}{y^{6}}$ .

$\frac{\frac{1}{x^{4}}}{\frac{27}{y^{6}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{x^{4}} \cdot \frac{y^{6}}{27}$

Step 5

Combine.

$\frac{1 y^{6}}{x^{4} \cdot 27}$

Step 6

Simplify the expression.

Multiply $y^{6}$ by $1$ .

$\frac{y^{6}}{x^{4} \cdot 27}$

Move $27$ to the left of $x^{4}$ .

$\frac{y^{6}}{27 x^{4}}$