Simplify ((27a^-5b^-3)/(3^5a^-7b^-5))^-1

${(\frac{27 a^{- 5} b^{- 3}}{3^{5} a^{- 7} b^{- 5}})}^{- 1}$

Step 1

Move $a^{- 7}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{27 a^{- 5} b^{- 3} a^{7}}{3^{5} b^{- 5}})}^{- 1}$

Step 2

Move $b^{- 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{27 a^{- 5} b^{- 3} a^{7} b^{5}}{3^{5}})}^{- 1}$

Step 3

Simplify the numerator.

Multiply $a^{- 5}$ by $a^{7}$ by adding the exponents.

Move $a^{7}$ .

${(\frac{27 (a^{7} a^{- 5}) b^{- 3} b^{5}}{3^{5}})}^{- 1}$

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

${(\frac{27 a^{7 - 5} b^{- 3} b^{5}}{3^{5}})}^{- 1}$

Subtract $5$ from $7$ .

${(\frac{27 a^{2} b^{- 3} b^{5}}{3^{5}})}^{- 1}$

Multiply $b^{- 3}$ by $b^{5}$ by adding the exponents.

Move $b^{5}$ .

${(\frac{27 a^{2} (b^{5} b^{- 3})}{3^{5}})}^{- 1}$

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

${(\frac{27 a^{2} b^{5 - 3}}{3^{5}})}^{- 1}$

Subtract $3$ from $5$ .

${(\frac{27 a^{2} b^{2}}{3^{5}})}^{- 1}$

Step 4

Raise $3$ to the power of $5$ .

${(\frac{27 a^{2} b^{2}}{243})}^{- 1}$

Step 5

Cancel the common factor of $27$ and $243$ .

Factor $27$ out of $27 a^{2} b^{2}$ .

${(\frac{27 (a^{2} b^{2})}{243})}^{- 1}$

Cancel the common factors.

Factor $27$ out of $243$ .

${(\frac{27 (a^{2} b^{2})}{27 \cdot 9})}^{- 1}$

Cancel the common factor.

${(\frac{27 (a^{2} b^{2})}{27 \cdot 9})}^{- 1}$

Rewrite the expression.

${(\frac{a^{2} b^{2}}{9})}^{- 1}$

Step 6

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

$\frac{9}{a^{2} b^{2}}$