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

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

Step 1

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

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

Step 2

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

Move $y$ .

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

Multiply $y$ by $y^{2}$ .

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

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

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

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

Add $1$ and $2$ .

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

Step 3

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

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

Step 4

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

Apply the product rule to $\frac{6 y^{3}}{x^{5}}$ .

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

Apply the product rule to $6 y^{3}$ .

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

Step 5

Simplify the numerator.

Raise $6$ to the power of $2$ .

$\frac{36 {(y^{3})}^{2}}{{(x^{5})}^{2}} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

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

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

$\frac{36 y^{3 \cdot 2}}{{(x^{5})}^{2}} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Multiply $3$ by $2$ .

$\frac{36 y^{6}}{{(x^{5})}^{2}} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Step 6

Simplify the expression.

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

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

$\frac{36 y^{6}}{x^{5 \cdot 2}} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Multiply $5$ by $2$ .

$\frac{36 y^{6}}{x^{10}} \cdot {(\frac{x^{2} \cdot y^{- 2}}{9 y})}^{2}$

Move $y^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{36 y^{6}}{x^{10}} \cdot {(\frac{x^{2}}{9 y \cdot y^{2}})}^{2}$

Step 7

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

Move $y^{2}$ .

$\frac{36 y^{6}}{x^{10}} \cdot {(\frac{x^{2}}{9 (y^{2} y)})}^{2}$

Multiply $y^{2}$ by $y$ .

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

$\frac{36 y^{6}}{x^{10}} \cdot {(\frac{x^{2}}{9 (y^{2} y^{1})})}^{2}$

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

$\frac{36 y^{6}}{x^{10}} \cdot {(\frac{x^{2}}{9 y^{2 + 1}})}^{2}$

Add $2$ and $1$ .

$\frac{36 y^{6}}{x^{10}} \cdot {(\frac{x^{2}}{9 y^{3}})}^{2}$

Step 8

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

Apply the product rule to $\frac{x^{2}}{9 y^{3}}$ .

$\frac{36 y^{6}}{x^{10}} \cdot \frac{{(x^{2})}^{2}}{{(9 y^{3})}^{2}}$

Apply the product rule to $9 y^{3}$ .

$\frac{36 y^{6}}{x^{10}} \cdot \frac{{(x^{2})}^{2}}{9^{2} {(y^{3})}^{2}}$

Step 9

Combine fractions.

Combine.

$\frac{36 y^{6} {(x^{2})}^{2}}{x^{10} (9^{2} {(y^{3})}^{2})}$

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

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

$\frac{36 y^{6} x^{2 \cdot 2}}{x^{10} \cdot 9^{2} {(y^{3})}^{2}}$

Multiply $2$ by $2$ .

$\frac{36 y^{6} x^{4}}{x^{10} \cdot 9^{2} {(y^{3})}^{2}}$

Step 10

Simplify the denominator.

Raise $9$ to the power of $2$ .

$\frac{36 y^{6} x^{4}}{x^{10} \cdot 81 {(y^{3})}^{2}}$

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

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

$\frac{36 y^{6} x^{4}}{x^{10} \cdot 81 y^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$\frac{36 y^{6} x^{4}}{x^{10} \cdot 81 y^{6}}$

Step 11

Reduce the expression by cancelling the common factors.

Cancel the common factor of $36$ and $81$ .

Factor $9$ out of $36 y^{6} x^{4}$ .

$\frac{9 (4 y^{6} x^{4})}{x^{10} \cdot 81 y^{6}}$

Cancel the common factors.

Factor $9$ out of $x^{10} \cdot 81 y^{6}$ .

$\frac{9 (4 y^{6} x^{4})}{9 (x^{10} \cdot 9 y^{6})}$

Cancel the common factor.

$\frac{9 (4 y^{6} x^{4})}{9 (x^{10} \cdot 9 y^{6})}$

Rewrite the expression.

$\frac{4 y^{6} x^{4}}{x^{10} \cdot 9 y^{6}}$

Cancel the common factor of $y^{6}$ .

Cancel the common factor.

$\frac{4 y^{6} x^{4}}{x^{10} \cdot 9 y^{6}}$

Rewrite the expression.

$\frac{4 x^{4}}{x^{10} \cdot 9}$

Cancel the common factor of $x^{4}$ and $x^{10}$ .

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

$\frac{x^{4} \cdot 4}{x^{10} \cdot 9}$

Cancel the common factors.

Factor $x^{4}$ out of $x^{10} \cdot 9$ .

$\frac{x^{4} \cdot 4}{x^{4} (x^{6} \cdot 9)}$

Cancel the common factor.

$\frac{x^{4} \cdot 4}{x^{4} (x^{6} \cdot 9)}$

Rewrite the expression.

$\frac{4}{x^{6} \cdot 9}$

Move $9$ to the left of $x^{6}$ .

$\frac{4}{9 x^{6}}$

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

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