Simplify ((4x^-3y^2)/(6xy^-3))^-2*(y^4)/(x^6y^-5)

${(\frac{4 x^{- 3} y^{2}}{6 x y^{- 3}})}^{- 2} \cdot \frac{y^{4}}{x^{6} y^{- 5}}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{4 x^{- 3} y^{2}}{6 x y^{- 3}})}^{- 2} \cdot \frac{y^{4}}{x^{6} \frac{1}{y^{5}}}$

Step 2

Combine $x^{6}$ and $\frac{1}{y^{5}}$ .

${(\frac{4 x^{- 3} y^{2}}{6 x y^{- 3}})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Step 3

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{4 y^{2}}{6 x y^{- 3} x^{3}})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Step 4

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

Move $x^{3}$ .

${(\frac{4 y^{2}}{6 (x^{3} x) y^{- 3}})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Multiply $x^{3}$ by $x$ .

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

${(\frac{4 y^{2}}{6 (x^{3} x^{1}) y^{- 3}})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

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

${(\frac{4 y^{2}}{6 x^{3 + 1} y^{- 3}})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Add $3$ and $1$ .

${(\frac{4 y^{2}}{6 x^{4} y^{- 3}})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Step 5

Move $y^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{4 y^{2} y^{3}}{6 x^{4}})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Step 6

Multiply $y^{2}$ by $y^{3}$ by adding the exponents.

Move $y^{3}$ .

${(\frac{4 (y^{3} y^{2})}{6 x^{4}})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

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

${(\frac{4 y^{3 + 2}}{6 x^{4}})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Add $3$ and $2$ .

${(\frac{4 y^{5}}{6 x^{4}})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Step 7

Cancel the common factor of $4$ and $6$ .

Factor $2$ out of $4 y^{5}$ .

${(\frac{2 (2 y^{5})}{6 x^{4}})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Cancel the common factors.

Factor $2$ out of $6 x^{4}$ .

${(\frac{2 (2 y^{5})}{2 (3 x^{4})})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Cancel the common factor.

${(\frac{2 (2 y^{5})}{2 (3 x^{4})})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Rewrite the expression.

${(\frac{2 y^{5}}{3 x^{4}})}^{- 2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Step 8

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

${(\frac{3 x^{4}}{2 y^{5}})}^{2} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Step 9

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

Apply the product rule to $\frac{3 x^{4}}{2 y^{5}}$ .

$\frac{{(3 x^{4})}^{2}}{{(2 y^{5})}^{2}} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Apply the product rule to $3 x^{4}$ .

$\frac{3^{2} {(x^{4})}^{2}}{{(2 y^{5})}^{2}} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Apply the product rule to $2 y^{5}$ .

$\frac{3^{2} {(x^{4})}^{2}}{2^{2} {(y^{5})}^{2}} \cdot \frac{y^{4}}{\frac{x^{6}}{y^{5}}}$

Step 10

Combine.

$\frac{3^{2} {(x^{4})}^{2} y^{4}}{2^{2} {(y^{5})}^{2} \frac{x^{6}}{y^{5}}}$

Step 11

Simplify the numerator.

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

$\frac{9 {(x^{4})}^{2} y^{4}}{2^{2} {(y^{5})}^{2} \frac{x^{6}}{y^{5}}}$

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

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

$\frac{9 x^{4 \cdot 2} y^{4}}{2^{2} {(y^{5})}^{2} \frac{x^{6}}{y^{5}}}$

Multiply $4$ by $2$ .

$\frac{9 x^{8} y^{4}}{2^{2} {(y^{5})}^{2} \frac{x^{6}}{y^{5}}}$

Step 12

Simplify the denominator.

Combine $2^{2}$ and $\frac{x^{6}}{y^{5}}$ .

$\frac{9 x^{8} y^{4}}{{(y^{5})}^{2} \frac{2^{2} x^{6}}{y^{5}}}$

Combine ${(y^{5})}^{2}$ and $\frac{2^{2} x^{6}}{y^{5}}$ .

$\frac{9 x^{8} y^{4}}{\frac{{(y^{5})}^{2} (2^{2} x^{6})}{y^{5}}}$

Step 13

Simplify the numerator.

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

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

$\frac{9 x^{8} y^{4}}{\frac{y^{5 \cdot 2} \cdot 2^{2} x^{6}}{y^{5}}}$

Multiply $5$ by $2$ .

$\frac{9 x^{8} y^{4}}{\frac{y^{10} \cdot 2^{2} x^{6}}{y^{5}}}$

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

$\frac{9 x^{8} y^{4}}{\frac{y^{10} \cdot 4 x^{6}}{y^{5}}}$

Step 14

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{y^{10} \cdot 4 x^{6}}{y^{5}}$ by cancelling the common factors.

Factor $y^{5}$ out of $y^{10} \cdot 4 x^{6}$ .

$\frac{9 x^{8} y^{4}}{\frac{y^{5} (y^{5} \cdot 4 x^{6})}{y^{5}}}$

Multiply by $1$ .

$\frac{9 x^{8} y^{4}}{\frac{y^{5} (y^{5} \cdot 4 x^{6})}{y^{5} \cdot 1}}$

Cancel the common factor.

$\frac{9 x^{8} y^{4}}{\frac{y^{5} (y^{5} \cdot 4 x^{6})}{y^{5} \cdot 1}}$

Rewrite the expression.

$\frac{9 x^{8} y^{4}}{\frac{y^{5} \cdot 4 x^{6}}{1}}$

Divide $y^{5} \cdot 4 x^{6}$ by $1$ .

$\frac{9 x^{8} y^{4}}{y^{5} \cdot 4 x^{6}}$

Step 15

Cancel the common factor of $x^{8}$ and $x^{6}$ .

Factor $x^{6}$ out of $9 x^{8} y^{4}$ .

$\frac{x^{6} (9 x^{2} y^{4})}{y^{5} \cdot 4 x^{6}}$

Cancel the common factors.

Factor $x^{6}$ out of $y^{5} \cdot 4 x^{6}$ .

$\frac{x^{6} (9 x^{2} y^{4})}{x^{6} (y^{5} \cdot 4)}$

Cancel the common factor.

$\frac{x^{6} (9 x^{2} y^{4})}{x^{6} (y^{5} \cdot 4)}$

Rewrite the expression.

$\frac{9 x^{2} y^{4}}{y^{5} \cdot 4}$

Step 16

Cancel the common factor of $y^{4}$ and $y^{5}$ .

Factor $y^{4}$ out of $9 x^{2} y^{4}$ .

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

Cancel the common factors.

Factor $y^{4}$ out of $y^{5} \cdot 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 17

Move $4$ to the left of $y$ .

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

Simplify ((4x^-3y^2)/(6xy^-3))^-2*(y^4)/(x^6y^-5)

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