Simplify ((a^-8b)/(a^-5b^3))^-3

${(\frac{a^{- 8} b}{a^{- 5} b^{3}})}^{- 3}$

Step 1

Move $a^{- 8}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{b}{a^{- 5} b^{3} a^{8}})}^{- 3}$

Step 2

Multiply $a^{- 5}$ by $a^{8}$ by adding the exponents.

Move $a^{8}$ .

${(\frac{b}{a^{8} a^{- 5} b^{3}})}^{- 3}$

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

${(\frac{b}{a^{8 - 5} b^{3}})}^{- 3}$

Subtract $5$ from $8$ .

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

Step 3

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

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

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

Factor $b$ out of $b^{1}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

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

Step 5

Apply the product rule to $a^{3} b^{2}$ .

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

Step 6

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

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

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

Multiply $3$ by $3$ .

$a^{9} {(b^{2})}^{3}$

Step 7

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

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

$a^{9} b^{2 \cdot 3}$

Multiply $2$ by $3$ .

$a^{9} b^{6}$