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

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

Step 1

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

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

Step 2

Combine $a^{2}$ and $\frac{1}{b^{3}}$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Multiply $\frac{b^{3}}{a^{2}}$ by $1$ .

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

Step 5

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

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

Step 6

Combine $a^{2}$ and $\frac{1}{b^{3}}$ .

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

Step 7

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

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

Step 8

Combine.

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

Step 9

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

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

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

Add $3$ and $3$ .

$\frac{b^{6}}{a^{2} a^{2}}$

Step 10

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

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

$\frac{b^{6}}{a^{2 + 2}}$

Add $2$ and $2$ .

$\frac{b^{6}}{a^{4}}$