Evaluate ((3x^2y^-3)/(2x^-3y^2))^3

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

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

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

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

Move $y^{3}$ .

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

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

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

Add $3$ and $2$ .

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

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

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

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

Move $x^{3}$ .

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

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

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

Add $3$ and $2$ .

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

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

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

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

Apply the product rule to $3 x^{5}$ .

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

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

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

Simplify the numerator.

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

$\frac{27 {(x^{5})}^{3}}{2^{3} {(y^{5})}^{3}}$

Multiply the exponents in ${(x^{5})}^{3}$ .

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

$\frac{27 x^{5 \cdot 3}}{2^{3} {(y^{5})}^{3}}$

Multiply $5$ by $3$ .

$\frac{27 x^{15}}{2^{3} {(y^{5})}^{3}}$

Simplify the denominator.

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

$\frac{27 x^{15}}{8 {(y^{5})}^{3}}$

Multiply the exponents in ${(y^{5})}^{3}$ .

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

$\frac{27 x^{15}}{8 y^{5 \cdot 3}}$

Multiply $5$ by $3$ .

$\frac{27 x^{15}}{8 y^{15}}$