Simplify ((12x^5yz^-2)/(4x^3y^5z^6))^-2

${(\frac{12 x^{5} y z^{- 2}}{4 x^{3} y^{5} z^{6}})}^{- 2}$

Step 1

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

${(\frac{12 x^{5} y}{4 x^{3} y^{5} z^{6} z^{2}})}^{- 2}$

Step 2

Multiply $z^{6}$ by $z^{2}$ by adding the exponents.

Move $z^{2}$ .

${(\frac{12 x^{5} y}{4 x^{3} y^{5} (z^{2} z^{6})})}^{- 2}$

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

${(\frac{12 x^{5} y}{4 x^{3} y^{5} z^{2 + 6}})}^{- 2}$

Add $2$ and $6$ .

${(\frac{12 x^{5} y}{4 x^{3} y^{5} z^{8}})}^{- 2}$

Step 3

Cancel the common factor of $12$ and $4$ .

Factor $4$ out of $12 x^{5} y$ .

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

Cancel the common factors.

Factor $4$ out of $4 x^{3} y^{5} z^{8}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Cancel the common factor of $x^{5}$ and $x^{3}$ .

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

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

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

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

Cancel the common factors.

Factor $y$ out of $y^{5} z^{8}$ .

${(\frac{y (3 x^{2})}{y (y^{4} z^{8})})}^{- 2}$

Cancel the common factor.

${(\frac{y (3 x^{2})}{y (y^{4} z^{8})})}^{- 2}$

Rewrite the expression.

${(\frac{3 x^{2}}{y^{4} z^{8}})}^{- 2}$

Step 6

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

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

Step 7

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

Apply the product rule to $\frac{y^{4} z^{8}}{3 x^{2}}$ .

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

Apply the product rule to $y^{4} z^{8}$ .

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

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

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

Step 8

Simplify the numerator.

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

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

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

Multiply $4$ by $2$ .

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

Multiply the exponents in ${(z^{8})}^{2}$ .

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

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

Multiply $8$ by $2$ .

$\frac{y^{8} z^{16}}{3^{2} {(x^{2})}^{2}}$

Step 9

Simplify the denominator.

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

$\frac{y^{8} z^{16}}{9 {(x^{2})}^{2}}$

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

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

$\frac{y^{8} z^{16}}{9 x^{2 \cdot 2}}$

Multiply $2$ by $2$ .

$\frac{y^{8} z^{16}}{9 x^{4}}$

Simplify ((12x^5yz^-2)/(4x^3y^5z^6))^-2

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