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

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

Step 1

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

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

Step 2

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

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

Step 3

Combine.

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

Step 4

Multiply $1$ by $1$ .

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

Step 5

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

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

Step 6

Apply the product rule to $a^{3} b^{4}$ .

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

Step 7

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

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

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

Multiply $3$ by $2$ .

$a (a^{6} {(b^{4})}^{2}) b^{- 3}$

Step 8

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

Move $a^{6}$ .

$a^{6} a {(b^{4})}^{2} b^{- 3}$

Multiply $a^{6}$ by $a$ .

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

$a^{6} a^{1} {(b^{4})}^{2} b^{- 3}$

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

$a^{6 + 1} {(b^{4})}^{2} b^{- 3}$

Add $6$ and $1$ .

$a^{7} {(b^{4})}^{2} b^{- 3}$

Step 9

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

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

$a^{7} b^{4 \cdot 2} b^{- 3}$

Multiply $4$ by $2$ .

$a^{7} b^{8} b^{- 3}$

Step 10

Multiply $b^{8}$ by $b^{- 3}$ by adding the exponents.

Move $b^{- 3}$ .

$a^{7} (b^{- 3} b^{8})$

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

$a^{7} b^{- 3 + 8}$

Add $- 3$ and $8$ .

$a^{7} b^{5}$

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

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