Simplify ((3x^-1y^-6)/(9x^5y^-2))^-1

${(\frac{3 x^{- 1} y^{- 6}}{9 x^{5} y^{- 2}})}^{- 1}$

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

${(\frac{3 y^{- 6}}{9 x^{5} y^{- 2} x})}^{- 1}$

Move $y^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{3}{9 x^{5} y^{- 2} x y^{6}})}^{- 1}$

Simplify the denominator.

Multiply $x^{5}$ by $x$ by adding the exponents.

Move $x$ .

${(\frac{3}{9 (x \cdot x^{5}) y^{- 2} y^{6}})}^{- 1}$

Multiply $x$ by $x^{5}$ .

Raise $x$ to the power of $1$ .

${(\frac{3}{9 (x^{1} x^{5}) y^{- 2} y^{6}})}^{- 1}$

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

${(\frac{3}{9 x^{1 + 5} y^{- 2} y^{6}})}^{- 1}$

Add $1$ and $5$ .

${(\frac{3}{9 x^{6} y^{- 2} y^{6}})}^{- 1}$

Multiply $y^{- 2}$ by $y^{6}$ by adding the exponents.

Move $y^{6}$ .

${(\frac{3}{9 x^{6} (y^{6} y^{- 2})})}^{- 1}$

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

${(\frac{3}{9 x^{6} y^{6 - 2}})}^{- 1}$

Subtract $2$ from $6$ .

${(\frac{3}{9 x^{6} y^{4}})}^{- 1}$

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

Factor $3$ out of $3$ .

${(\frac{3 (1)}{9 x^{6} y^{4}})}^{- 1}$

Cancel the common factors.

Factor $3$ out of $9 x^{6} y^{4}$ .

${(\frac{3 (1)}{3 (3 x^{6} y^{4})})}^{- 1}$

Cancel the common factor.

${(\frac{3 \cdot 1}{3 (3 x^{6} y^{4})})}^{- 1}$

Rewrite the expression.

${(\frac{1}{3 x^{6} y^{4}})}^{- 1}$

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

$3 x^{6} y^{4}$