Simplify (a^4b^-1)^-2(a^-3b)^2

${(a^{4} b^{- 1})}^{- 2} {(a^{- 3} b)}^{2}$

Step 1

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

${(a^{4} \frac{1}{b})}^{- 2} {(a^{- 3} b)}^{2}$

Step 2

Combine $a^{4}$ and $\frac{1}{b}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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

Multiply $4$ by $2$ .

$\frac{b^{2}}{a^{8}} {(a^{- 3} b)}^{2}$

Step 6

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

$\frac{b^{2}}{a^{8}} {(\frac{1}{a^{3}} b)}^{2}$

Step 7

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

$\frac{b^{2}}{a^{8}} {(\frac{b}{a^{3}})}^{2}$

Step 8

Apply the product rule to $\frac{b}{a^{3}}$ .

$\frac{b^{2}}{a^{8}} \cdot \frac{b^{2}}{{(a^{3})}^{2}}$

Step 9

Combine.

$\frac{b^{2} b^{2}}{a^{8} {(a^{3})}^{2}}$

Step 10

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

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

$\frac{b^{2 + 2}}{a^{8} {(a^{3})}^{2}}$

Add $2$ and $2$ .

$\frac{b^{4}}{a^{8} {(a^{3})}^{2}}$

Step 11

Multiply the exponents in ${(a^{3})}^{2}$ .

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

$\frac{b^{4}}{a^{8} a^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$\frac{b^{4}}{a^{8} a^{6}}$

Step 12

Multiply $a^{8}$ by $a^{6}$ by adding the exponents.

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

$\frac{b^{4}}{a^{8 + 6}}$

Add $8$ and $6$ .

$\frac{b^{4}}{a^{14}}$

Simplify (a^4b^-1)^-2(a^-3b)^2

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