Simplify ((a^-1b)^2)/(ab^2)

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

Step 1

Simplify the numerator.

Apply the product rule to $a^{- 1} b$ .

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

Multiply the exponents in ${(a^{- 1})}^{2}$ .

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

$\frac{a^{- 1 \cdot 2} b^{2}}{a b^{2}}$

Multiply $- 1$ by $2$ .

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{1}{a^{2}} b^{2}}{a b^{2}}$

Step 2

Combine $\frac{1}{a^{2}}$ and $b^{2}$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{b^{2}}{a^{2}} \cdot \frac{1}{a b^{2}}$

Step 4

Combine.

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

Step 5

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

Move $a$ .

$\frac{b^{2} \cdot 1}{a \cdot a^{2} b^{2}}$

Multiply $a$ by $a^{2}$ .

Raise $a$ to the power of $1$ .

$\frac{b^{2} \cdot 1}{a^{1} a^{2} b^{2}}$

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

$\frac{b^{2} \cdot 1}{a^{1 + 2} b^{2}}$

Add $1$ and $2$ .

$\frac{b^{2} \cdot 1}{a^{3} b^{2}}$

Step 6

Multiply $b^{2}$ by $1$ .

$\frac{b^{2}}{a^{3} b^{2}}$

Step 7

Cancel the common factor of $b^{2}$ .

Cancel the common factor.

$\frac{b^{2}}{a^{3} b^{2}}$

Rewrite the expression.

$\frac{1}{a^{3}}$