Simplify ((-3x^4yz^-2)^2)/(3x^5y^2z)

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

Simplify the numerator.

Apply the product rule to $- 3 x^{4} y z^{- 2}$ .

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

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

Apply the product rule to $- 3 x^{4} y$ .

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

Apply the product rule to $- 3 x^{4}$ .

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

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

$\frac{9 {(x^{4})}^{2} y^{2} {(z^{- 2})}^{2}}{3 x^{5} y^{2} z}$

Multiply the exponents in ${(x^{4})}^{2}$ .

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

$\frac{9 x^{4 \cdot 2} y^{2} {(z^{- 2})}^{2}}{3 x^{5} y^{2} z}$

Multiply $4$ by $2$ .

$\frac{9 x^{8} y^{2} {(z^{- 2})}^{2}}{3 x^{5} y^{2} z}$

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

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

$\frac{9 x^{8} y^{2} z^{- 2 \cdot 2}}{3 x^{5} y^{2} z}$

Multiply $- 2$ by $2$ .

$\frac{9 x^{8} y^{2} z^{- 4}}{3 x^{5} y^{2} z}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{9 x^{8} y^{2} \frac{1}{z^{4}}}{3 x^{5} y^{2} z}$

Combine exponents.

Combine $9$ and $\frac{1}{z^{4}}$ .

$\frac{x^{8} y^{2} \frac{9}{z^{4}}}{3 x^{5} y^{2} z}$

Combine $x^{8}$ and $\frac{9}{z^{4}}$ .

$\frac{y^{2} \frac{x^{8} \cdot 9}{z^{4}}}{3 x^{5} y^{2} z}$

Combine $y^{2}$ and $\frac{x^{8} \cdot 9}{z^{4}}$ .

$\frac{\frac{y^{2} (x^{8} \cdot 9)}{z^{4}}}{3 x^{5} y^{2} z}$

Remove unnecessary parentheses.

$\frac{\frac{y^{2} x^{8} \cdot 9}{z^{4}}}{3 x^{5} y^{2} z}$

Move $9$ to the left of $y^{2} x^{8}$ .

$\frac{\frac{9 y^{2} x^{8}}{z^{4}}}{3 x^{5} y^{2} z}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{9 y^{2} x^{8}}{z^{4}} \cdot \frac{1}{3 x^{5} y^{2} z}$

Combine.

$\frac{9 y^{2} x^{8} \cdot 1}{z^{4} (3 x^{5} y^{2} z)}$

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

Move $z$ .

$\frac{9 y^{2} x^{8} \cdot 1}{z \cdot z^{4} (3 x^{5} y^{2})}$

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

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

$\frac{9 y^{2} x^{8} \cdot 1}{z^{1} z^{4} (3 x^{5} y^{2})}$

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

$\frac{9 y^{2} x^{8} \cdot 1}{z^{1 + 4} (3 x^{5} y^{2})}$

Add $1$ and $4$ .

$\frac{9 y^{2} x^{8} \cdot 1}{z^{5} (3 x^{5} y^{2})}$

Multiply $9 y^{2} x^{8}$ by $1$ .

$\frac{9 y^{2} x^{8}}{z^{5} (3 x^{5} y^{2})}$

Cancel the common factor of $9$ and $3$ .

Factor $3$ out of $9 y^{2} x^{8}$ .

$\frac{3 (3 y^{2} x^{8})}{z^{5} (3 x^{5} y^{2})}$

Cancel the common factors.

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

$\frac{3 (3 y^{2} x^{8})}{3 (z^{5} (x^{5} y^{2}))}$

Cancel the common factor.

$\frac{3 (3 y^{2} x^{8})}{3 (z^{5} (x^{5} y^{2}))}$

Rewrite the expression.

$\frac{3 y^{2} x^{8}}{z^{5} (x^{5} y^{2})}$

Cancel the common factor of $y^{2}$ .

Cancel the common factor.

$\frac{3 y^{2} x^{8}}{z^{5} (x^{5} y^{2})}$

Rewrite the expression.

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

Cancel the common factor of $x^{8}$ and $x^{5}$ .

Factor $x^{5}$ out of $3 x^{8}$ .

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

Cancel the common factors.

Factor $x^{5}$ out of $z^{5} (x^{5})$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify ((-3x^4yz^-2)^2)/(3x^5y^2z)

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