Evaluate ((2x^-2y^3)/(3xy^-4))^4

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

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

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

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

Multiply $y^{3}$ by $y^{4}$ by adding the exponents.

Move $y^{4}$ .

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

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

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

Add $4$ and $3$ .

${(\frac{2 y^{7}}{3 x^{3}})}^{4}$

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

Apply the product rule to $\frac{2 y^{7}}{3 x^{3}}$ .

$\frac{{(2 y^{7})}^{4}}{{(3 x^{3})}^{4}}$

Apply the product rule to $2 y^{7}$ .

$\frac{2^{4} {(y^{7})}^{4}}{{(3 x^{3})}^{4}}$

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

$\frac{2^{4} {(y^{7})}^{4}}{3^{4} {(x^{3})}^{4}}$

Simplify the numerator.

Raise $2$ to the power of $4$ .

$\frac{16 {(y^{7})}^{4}}{3^{4} {(x^{3})}^{4}}$

Multiply the exponents in ${(y^{7})}^{4}$ .

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

$\frac{16 y^{7 \cdot 4}}{3^{4} {(x^{3})}^{4}}$

Multiply $7$ by $4$ .

$\frac{16 y^{28}}{3^{4} {(x^{3})}^{4}}$

Simplify the denominator.

Raise $3$ to the power of $4$ .

$\frac{16 y^{28}}{81 {(x^{3})}^{4}}$

Multiply the exponents in ${(x^{3})}^{4}$ .

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

$\frac{16 y^{28}}{81 x^{3 \cdot 4}}$

Multiply $3$ by $4$ .

$\frac{16 y^{28}}{81 x^{12}}$