Simplify ((36a^-3b)^-1)/((9a^-2b)^-1)

$\frac{{(36 a^{- 3} b)}^{- 1}}{{(9 a^{- 2} b)}^{- 1}}$

Step 1

Move ${(36 a^{- 3} b)}^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(9 a^{- 2} b)}^{- 1} (36 a^{- 3} b)}$

Step 2

Move ${(9 a^{- 2} b)}^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{9 a^{- 2} b}{36 a^{- 3} b}$

Step 3

Move $a^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{9 a^{- 2} b a^{3}}{36 b}$

Step 4

Simplify the numerator.

Multiply $a^{- 2}$ by $a^{3}$ by adding the exponents.

Move $a^{3}$ .

$\frac{9 (a^{3} a^{- 2}) b}{36 b}$

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

$\frac{9 a^{3 - 2} b}{36 b}$

Subtract $2$ from $3$ .

$\frac{9 a^{1} b}{36 b}$

Simplify $9 a^{1} b$ .

$\frac{9 a b}{36 b}$

Step 5

Cancel the common factor of $9$ and $36$ .

Factor $9$ out of $9 a b$ .

$\frac{9 (a b)}{36 b}$

Cancel the common factors.

Factor $9$ out of $36 b$ .

$\frac{9 (a b)}{9 (4 b)}$

Cancel the common factor.

$\frac{9 (a b)}{9 (4 b)}$

Rewrite the expression.

$\frac{a b}{4 b}$

Step 6

Cancel the common factor of $b$ .

Cancel the common factor.

$\frac{a b}{4 b}$

Rewrite the expression.

$\frac{a}{4}$