Simplify ((2x^3y^-2)^-3)/((3x^4y^2)^-2)

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

Simplify the expression.

Move ${(2 x^{3} y^{- 2})}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Move ${(3 x^{4} y^{2})}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Simplify the numerator.

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

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

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

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

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

$\frac{9 {(x^{4})}^{2} {(y^{2})}^{2}}{{(2 x^{3} y^{- 2})}^{3}}$

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

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

$\frac{9 x^{4 \cdot 2} {(y^{2})}^{2}}{{(2 x^{3} y^{- 2})}^{3}}$

Multiply $4$ by $2$ .

$\frac{9 x^{8} {(y^{2})}^{2}}{{(2 x^{3} y^{- 2})}^{3}}$

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

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

$\frac{9 x^{8} y^{2 \cdot 2}}{{(2 x^{3} y^{- 2})}^{3}}$

Multiply $2$ by $2$ .

$\frac{9 x^{8} y^{4}}{{(2 x^{3} y^{- 2})}^{3}}$

Simplify the denominator.

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

$\frac{9 x^{8} y^{4}}{{(2 x^{3})}^{3} {(y^{- 2})}^{3}}$

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

$\frac{9 x^{8} y^{4}}{2^{3} {(x^{3})}^{3} {(y^{- 2})}^{3}}$

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

$\frac{9 x^{8} y^{4}}{8 {(x^{3})}^{3} {(y^{- 2})}^{3}}$

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

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

$\frac{9 x^{8} y^{4}}{8 x^{3 \cdot 3} {(y^{- 2})}^{3}}$

Multiply $3$ by $3$ .

$\frac{9 x^{8} y^{4}}{8 x^{9} {(y^{- 2})}^{3}}$

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

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

$\frac{9 x^{8} y^{4}}{8 x^{9} y^{- 2 \cdot 3}}$

Multiply $- 2$ by $3$ .

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

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

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

Combine exponents.

Combine $8$ and $\frac{1}{y^{6}}$ .

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

Combine $x^{9}$ and $\frac{8}{y^{6}}$ .

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

Move $8$ to the left of $x^{9}$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Factor $x^{8}$ out of $8 x^{9}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine $9$ and $\frac{y^{6}}{8 x}$ .

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

Combine $y^{4}$ and $\frac{9 y^{6}}{8 x}$ .

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

Multiply $y^{4}$ by $y^{6}$ by adding the exponents.

Move $y^{6}$ .

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

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

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

Add $6$ and $4$ .

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

Move $9$ to the left of $y^{10}$ .

$\frac{9 y^{10}}{8 x}$

Simplify ((2x^3y^-2)^-3)/((3x^4y^2)^-2)

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