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

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

Step 1

Move $a^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

Simplify the numerator.

Multiply $a^{- 2}$ by $a^{3}$ by adding the exponents.

Move $a^{3}$ .

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

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

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

Subtract $2$ from $3$ .

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

Simplify $2 a^{1} b^{4}$ .

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

Step 3

Cancel the common factor of $b^{4}$ and $b^{3}$ .

Factor $b^{3}$ out of $2 a b^{4}$ .

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

Cancel the common factors.

Multiply by $1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 a b$ by $1$ .

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

Step 4

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

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

Step 5

Simplify the denominator.

Apply the product rule to $2 a b$ .

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

Apply the product rule to $2 a$ .

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

Raise $2$ to the power of $3$ .

$\frac{1}{8 a^{3} b^{3}}$