Simplify ((36a^5)/(4a^4b^5))^-2

${(\frac{36 a^{5}}{4 a^{4} b^{5}})}^{- 2}$

Step 1

Cancel the common factor of $36$ and $4$ .

Factor $4$ out of $36 a^{5}$ .

${(\frac{4 (9 a^{5})}{4 a^{4} b^{5}})}^{- 2}$

Cancel the common factors.

Factor $4$ out of $4 a^{4} b^{5}$ .

${(\frac{4 (9 a^{5})}{4 (a^{4} b^{5})})}^{- 2}$

Cancel the common factor.

${(\frac{4 (9 a^{5})}{4 (a^{4} b^{5})})}^{- 2}$

Rewrite the expression.

${(\frac{9 a^{5}}{a^{4} b^{5}})}^{- 2}$

Step 2

Cancel the common factor of $a^{5}$ and $a^{4}$ .

Factor $a^{4}$ out of $9 a^{5}$ .

${(\frac{a^{4} (9 a)}{a^{4} b^{5}})}^{- 2}$

Cancel the common factors.

Factor $a^{4}$ out of $a^{4} b^{5}$ .

${(\frac{a^{4} (9 a)}{a^{4} (b^{5})})}^{- 2}$

Cancel the common factor.

${(\frac{a^{4} (9 a)}{a^{4} b^{5}})}^{- 2}$

Rewrite the expression.

${(\frac{9 a}{b^{5}})}^{- 2}$

Step 3

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

${(\frac{b^{5}}{9 a})}^{2}$

Step 4

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

Apply the product rule to $\frac{b^{5}}{9 a}$ .

$\frac{{(b^{5})}^{2}}{{(9 a)}^{2}}$

Apply the product rule to $9 a$ .

$\frac{{(b^{5})}^{2}}{9^{2} a^{2}}$

Step 5

Multiply the exponents in ${(b^{5})}^{2}$ .

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

$\frac{b^{5 \cdot 2}}{9^{2} a^{2}}$

Multiply $5$ by $2$ .

$\frac{b^{10}}{9^{2} a^{2}}$

Step 6

Raise $9$ to the power of $2$ .

$\frac{b^{10}}{81 a^{2}}$