Simplify ((2x^3y^0z^4)/(x^4z^-1))^3

${(\frac{2 x^{3} y^{0} z^{4}}{x^{4} z^{- 1}})}^{3}$

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

${(\frac{2 x^{3} y^{0} z^{4} z}{x^{4}})}^{3}$

Simplify the numerator.

Multiply $z^{4}$ by $z$ by adding the exponents.

Move $z$ .

${(\frac{2 x^{3} y^{0} (z \cdot z^{4})}{x^{4}})}^{3}$

Multiply $z$ by $z^{4}$ .

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

${(\frac{2 x^{3} y^{0} (z^{1} z^{4})}{x^{4}})}^{3}$

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

${(\frac{2 x^{3} y^{0} z^{1 + 4}}{x^{4}})}^{3}$

Add $1$ and $4$ .

${(\frac{2 x^{3} y^{0} z^{5}}{x^{4}})}^{3}$

Simplify $2 x^{3} y^{0} z^{5}$ .

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

Cancel the common factor of $x^{3}$ and $x^{4}$ .

Factor $x^{3}$ out of $2 x^{3} z^{5}$ .

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

Cancel the common factors.

Factor $x^{3}$ out of $x^{4}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

Simplify the numerator.

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

$\frac{8 {(z^{5})}^{3}}{x^{3}}$

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

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

$\frac{8 z^{5 \cdot 3}}{x^{3}}$

Multiply $5$ by $3$ .

$\frac{8 z^{15}}{x^{3}}$