Simplify ((2x^-3y^4)/(3x^-4y^2))^-2

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

Move $x^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Simplify the numerator.

Multiply $x^{- 3}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Subtract $3$ from $4$ .

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

Simplify $2 x^{1} y^{4}$ .

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

Cancel the common factor of $y^{4}$ and $y^{2}$ .

Factor $y^{2}$ out of $2 x y^{4}$ .

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

Cancel the common factors.

Factor $y^{2}$ out of $3 y^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{3}{2 x y^{2}}$ .

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

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

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

Apply the product rule to $2 x$ .

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

Raise $3$ to the power of $2$ .

$\frac{9}{2^{2} x^{2} {(y^{2})}^{2}}$

Simplify the denominator.

Raise $2$ to the power of $2$ .

$\frac{9}{4 x^{2} {(y^{2})}^{2}}$

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

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

$\frac{9}{4 x^{2} y^{2 \cdot 2}}$

Multiply $2$ by $2$ .

$\frac{9}{4 x^{2} y^{4}}$