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

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

Step 1

Multiply $x^{2}$ by $x^{- 4}$ by adding the exponents.

Move $x^{- 4}$ .

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

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

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

Add $- 4$ and $2$ .

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

Step 2

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

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

Step 3

Simplify the denominator.

Multiply $x^{- 1}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

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

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

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

Subtract $1$ from $2$ .

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

Simplify $x^{1} z^{- 2}$ .

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

Step 4

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

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

Step 5

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

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

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

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

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

Step 6

Simplify the numerator.

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

$\frac{y^{2} z^{2 \cdot 3}}{x^{3}}$

Multiply $2$ by $3$ .

$\frac{y^{2} z^{6}}{x^{3}}$

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

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