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

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

Cancel the common factor of $x$ and $x^{2}$ .

Factor $x$ out of $2 x z^{3}$ .

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

Cancel the common factors.

Factor $x$ out of $3 x^{2} y^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

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

Apply the product rule to $3 x$ .

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

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

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

Simplify the numerator.

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

$\frac{81 x^{4} {(y^{3})}^{4}}{2^{4} {(z^{3})}^{4}}$

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

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

$\frac{81 x^{4} y^{3 \cdot 4}}{2^{4} {(z^{3})}^{4}}$

Multiply $3$ by $4$ .

$\frac{81 x^{4} y^{12}}{2^{4} {(z^{3})}^{4}}$

Simplify the denominator.

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

$\frac{81 x^{4} y^{12}}{16 {(z^{3})}^{4}}$

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

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

$\frac{81 x^{4} y^{12}}{16 z^{3 \cdot 4}}$

Multiply $3$ by $4$ .

$\frac{81 x^{4} y^{12}}{16 z^{12}}$