Simplify ((x^-4y)/(x^-9y^5))^-2

${(\frac{x^{- 4} y}{x^{- 9} y^{5}})}^{- 2}$

Step 1

Move $x^{- 9}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{x^{- 4} y x^{9}}{y^{5}})}^{- 2}$

Step 2

Multiply $x^{- 4}$ by $x^{9}$ by adding the exponents.

Move $x^{9}$ .

${(\frac{x^{9} x^{- 4} y}{y^{5}})}^{- 2}$

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

${(\frac{x^{9 - 4} y}{y^{5}})}^{- 2}$

Subtract $4$ from $9$ .

${(\frac{x^{5} y}{y^{5}})}^{- 2}$

Step 3

Cancel the common factor of $y$ and $y^{5}$ .

Factor $y$ out of $x^{5} y$ .

${(\frac{y x^{5}}{y^{5}})}^{- 2}$

Cancel the common factors.

Factor $y$ out of $y^{5}$ .

${(\frac{y x^{5}}{y \cdot y^{4}})}^{- 2}$

Cancel the common factor.

${(\frac{y x^{5}}{y \cdot y^{4}})}^{- 2}$

Rewrite the expression.

${(\frac{x^{5}}{y^{4}})}^{- 2}$

Step 4

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

${(\frac{y^{4}}{x^{5}})}^{2}$

Step 5

Apply the product rule to $\frac{y^{4}}{x^{5}}$ .

$\frac{{(y^{4})}^{2}}{{(x^{5})}^{2}}$

Step 6

Multiply the exponents in ${(y^{4})}^{2}$ .

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

$\frac{y^{4 \cdot 2}}{{(x^{5})}^{2}}$

Multiply $4$ by $2$ .

$\frac{y^{8}}{{(x^{5})}^{2}}$

Step 7

Multiply the exponents in ${(x^{5})}^{2}$ .

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

$\frac{y^{8}}{x^{5 \cdot 2}}$

Multiply $5$ by $2$ .

$\frac{y^{8}}{x^{10}}$