Simplify ((3x^-4y^-5z^-2)/(9x^2y^-8z^-9))^-2

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

Step 1

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

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

Move $x^{4}$ .

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

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

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

Add $4$ and $2$ .

${(\frac{3 y^{- 5} z^{- 2}}{9 x^{6} y^{- 8} z^{- 9}})}^{- 2}$

Step 3

Move $y^{- 8}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{3 y^{- 5} z^{- 2} y^{8}}{9 x^{6} z^{- 9}})}^{- 2}$

Step 4

Move $z^{- 9}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{3 y^{- 5} z^{- 2} y^{8} z^{9}}{9 x^{6}})}^{- 2}$

Step 5

Simplify the numerator.

Multiply $y^{- 5}$ by $y^{8}$ by adding the exponents.

Move $y^{8}$ .

${(\frac{3 (y^{8} y^{- 5}) z^{- 2} z^{9}}{9 x^{6}})}^{- 2}$

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

${(\frac{3 y^{8 - 5} z^{- 2} z^{9}}{9 x^{6}})}^{- 2}$

Subtract $5$ from $8$ .

${(\frac{3 y^{3} z^{- 2} z^{9}}{9 x^{6}})}^{- 2}$

Multiply $z^{- 2}$ by $z^{9}$ by adding the exponents.

Move $z^{9}$ .

${(\frac{3 y^{3} (z^{9} z^{- 2})}{9 x^{6}})}^{- 2}$

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

${(\frac{3 y^{3} z^{9 - 2}}{9 x^{6}})}^{- 2}$

Subtract $2$ from $9$ .

${(\frac{3 y^{3} z^{7}}{9 x^{6}})}^{- 2}$

Step 6

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

Factor $3$ out of $3 y^{3} z^{7}$ .

${(\frac{3 (y^{3} z^{7})}{9 x^{6}})}^{- 2}$

Cancel the common factors.

Factor $3$ out of $9 x^{6}$ .

${(\frac{3 (y^{3} z^{7})}{3 (3 x^{6})})}^{- 2}$

Cancel the common factor.

${(\frac{3 (y^{3} z^{7})}{3 (3 x^{6})})}^{- 2}$

Rewrite the expression.

${(\frac{y^{3} z^{7}}{3 x^{6}})}^{- 2}$

Step 7

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

${(\frac{3 x^{6}}{y^{3} z^{7}})}^{2}$

Step 8

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

Apply the product rule to $\frac{3 x^{6}}{y^{3} z^{7}}$ .

$\frac{{(3 x^{6})}^{2}}{{(y^{3} z^{7})}^{2}}$

Apply the product rule to $3 x^{6}$ .

$\frac{3^{2} {(x^{6})}^{2}}{{(y^{3} z^{7})}^{2}}$

Apply the product rule to $y^{3} z^{7}$ .

$\frac{3^{2} {(x^{6})}^{2}}{{(y^{3})}^{2} {(z^{7})}^{2}}$

Step 9

Simplify the numerator.

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

$\frac{9 {(x^{6})}^{2}}{{(y^{3})}^{2} {(z^{7})}^{2}}$

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

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

$\frac{9 x^{6 \cdot 2}}{{(y^{3})}^{2} {(z^{7})}^{2}}$

Multiply $6$ by $2$ .

$\frac{9 x^{12}}{{(y^{3})}^{2} {(z^{7})}^{2}}$

Step 10

Simplify the denominator.

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

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

$\frac{9 x^{12}}{y^{3 \cdot 2} {(z^{7})}^{2}}$

Multiply $3$ by $2$ .

$\frac{9 x^{12}}{y^{6} {(z^{7})}^{2}}$

Multiply the exponents in ${(z^{7})}^{2}$ .

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

$\frac{9 x^{12}}{y^{6} z^{7 \cdot 2}}$

Multiply $7$ by $2$ .

$\frac{9 x^{12}}{y^{6} z^{14}}$

Simplify ((3x^-4y^-5z^-2)/(9x^2y^-8z^-9))^-2

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