Simplify ((24x^5yz^-2)/(3x^2y^3z^4))^2

${(\frac{24 x^{5} y z^{- 2}}{3 x^{2} y^{3} z^{4}})}^{2}$

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

${(\frac{24 x^{5} y}{3 x^{2} y^{3} z^{4} z^{2}})}^{2}$

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

Move $z^{2}$ .

${(\frac{24 x^{5} y}{3 x^{2} y^{3} (z^{2} z^{4})})}^{2}$

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

${(\frac{24 x^{5} y}{3 x^{2} y^{3} z^{2 + 4}})}^{2}$

Add $2$ and $4$ .

${(\frac{24 x^{5} y}{3 x^{2} y^{3} z^{6}})}^{2}$

Cancel the common factor of $24$ and $3$ .

Factor $3$ out of $24 x^{5} y$ .

${(\frac{3 (8 x^{5} y)}{3 x^{2} y^{3} z^{6}})}^{2}$

Cancel the common factors.

Factor $3$ out of $3 x^{2} y^{3} z^{6}$ .

${(\frac{3 (8 x^{5} y)}{3 (x^{2} y^{3} z^{6})})}^{2}$

Cancel the common factor.

${(\frac{3 (8 x^{5} y)}{3 (x^{2} y^{3} z^{6})})}^{2}$

Rewrite the expression.

${(\frac{8 x^{5} y}{x^{2} y^{3} z^{6}})}^{2}$

Cancel the common factor of $x^{5}$ and $x^{2}$ .

Factor $x^{2}$ out of $8 x^{5} y$ .

${(\frac{x^{2} (8 x^{3} y)}{x^{2} y^{3} z^{6}})}^{2}$

Cancel the common factors.

Factor $x^{2}$ out of $x^{2} y^{3} z^{6}$ .

${(\frac{x^{2} (8 x^{3} y)}{x^{2} (y^{3} z^{6})})}^{2}$

Cancel the common factor.

${(\frac{x^{2} (8 x^{3} y)}{x^{2} (y^{3} z^{6})})}^{2}$

Rewrite the expression.

${(\frac{8 x^{3} y}{y^{3} z^{6}})}^{2}$

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

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

${(\frac{y (8 x^{3})}{y^{3} z^{6}})}^{2}$

Cancel the common factors.

Factor $y$ out of $y^{3} z^{6}$ .

${(\frac{y (8 x^{3})}{y (y^{2} z^{6})})}^{2}$

Cancel the common factor.

${(\frac{y (8 x^{3})}{y (y^{2} z^{6})})}^{2}$

Rewrite the expression.

${(\frac{8 x^{3}}{y^{2} z^{6}})}^{2}$

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

Apply the product rule to $\frac{8 x^{3}}{y^{2} z^{6}}$ .

$\frac{{(8 x^{3})}^{2}}{{(y^{2} z^{6})}^{2}}$

Apply the product rule to $8 x^{3}$ .

$\frac{8^{2} {(x^{3})}^{2}}{{(y^{2} z^{6})}^{2}}$

Apply the product rule to $y^{2} z^{6}$ .

$\frac{8^{2} {(x^{3})}^{2}}{{(y^{2})}^{2} {(z^{6})}^{2}}$

Simplify the numerator.

Raise $8$ to the power of $2$ .

$\frac{64 {(x^{3})}^{2}}{{(y^{2})}^{2} {(z^{6})}^{2}}$

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

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

$\frac{64 x^{3 \cdot 2}}{{(y^{2})}^{2} {(z^{6})}^{2}}$

Multiply $3$ by $2$ .

$\frac{64 x^{6}}{{(y^{2})}^{2} {(z^{6})}^{2}}$

Simplify the denominator.

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

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

$\frac{64 x^{6}}{y^{2 \cdot 2} {(z^{6})}^{2}}$

Multiply $2$ by $2$ .

$\frac{64 x^{6}}{y^{4} {(z^{6})}^{2}}$

Multiply the exponents in ${(z^{6})}^{2}$ .

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

$\frac{64 x^{6}}{y^{4} z^{6 \cdot 2}}$

Multiply $6$ by $2$ .

$\frac{64 x^{6}}{y^{4} z^{12}}$

Simplify ((24x^5yz^-2)/(3x^2y^3z^4))^2

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