Simplify ((3x^-1)/(9y^2))^-2

${(\frac{3 x^{- 1}}{9 y^{2}})}^{- 2}$

Step 1

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

${(\frac{3}{9 y^{2} x})}^{- 2}$

Step 2

Cancel the common factor of $3$ and $9$ .

Factor $3$ out of $3$ .

${(\frac{3 (1)}{9 y^{2} x})}^{- 2}$

Cancel the common factors.

Factor $3$ out of $9 y^{2} x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

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

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

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

Step 5

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

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

Step 6

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

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

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

Multiply $2$ by $2$ .

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