Simplify ((ab^3)/(a^5b))^-2

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

Step 1

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

Factor $a$ out of $a b^{3}$ .

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

Cancel the common factors.

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

${(\frac{a (b^{3})}{a (a^{4} b)})}^{- 2}$

Cancel the common factor.

${(\frac{a b^{3}}{a (a^{4} b)})}^{- 2}$

Rewrite the expression.

${(\frac{b^{3}}{a^{4} b})}^{- 2}$

Step 2

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

Factor $b$ out of $b^{3}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

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

Step 4

Apply the product rule to $\frac{a^{4}}{b^{2}}$ .

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

Step 5

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

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

$\frac{a^{4 \cdot 2}}{{(b^{2})}^{2}}$

Multiply $4$ by $2$ .

$\frac{a^{8}}{{(b^{2})}^{2}}$

Step 6

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

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

$\frac{a^{8}}{b^{2 \cdot 2}}$

Multiply $2$ by $2$ .

$\frac{a^{8}}{b^{4}}$