Simplify ((2a^-2b)^-3)/(5a^2b^4)

$\frac{{(2 a^{- 2} b)}^{- 3}}{5 a^{2} b^{4}}$

Move ${(2 a^{- 2} b)}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{5 a^{2} b^{4} {(2 a^{- 2} b)}^{3}}$

Simplify the denominator.

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

$\frac{1}{5 a^{2} b^{4} {(2 a^{- 2})}^{3} b^{3}}$

Multiply $b^{4}$ by $b^{3}$ by adding the exponents.

Move $b^{3}$ .

$\frac{1}{5 a^{2} (b^{3} b^{4}) {(2 a^{- 2})}^{3}}$

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

$\frac{1}{5 a^{2} b^{3 + 4} {(2 a^{- 2})}^{3}}$

Add $3$ and $4$ .

$\frac{1}{5 a^{2} b^{7} {(2 a^{- 2})}^{3}}$

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

$\frac{1}{5 a^{2} b^{7} {(2 \frac{1}{a^{2}})}^{3}}$

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

$\frac{1}{5 a^{2} b^{7} {(\frac{2}{a^{2}})}^{3}}$

Apply the product rule to $\frac{2}{a^{2}}$ .

$\frac{1}{5 a^{2} b^{7} \frac{2^{3}}{{(a^{2})}^{3}}}$

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

$\frac{1}{5 a^{2} b^{7} \frac{8}{{(a^{2})}^{3}}}$

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

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

$\frac{1}{5 a^{2} b^{7} \frac{8}{a^{2 \cdot 3}}}$

Multiply $2$ by $3$ .

$\frac{1}{5 a^{2} b^{7} \frac{8}{a^{6}}}$

Combine exponents.

Combine $5$ and $\frac{8}{a^{6}}$ .

$\frac{1}{a^{2} b^{7} \frac{5 \cdot 8}{a^{6}}}$

Multiply $5$ by $8$ .

$\frac{1}{a^{2} b^{7} \frac{40}{a^{6}}}$

Combine $a^{2}$ and $\frac{40}{a^{6}}$ .

$\frac{1}{b^{7} \frac{a^{2} \cdot 40}{a^{6}}}$

Combine $b^{7}$ and $\frac{a^{2} \cdot 40}{a^{6}}$ .

$\frac{1}{\frac{b^{7} (a^{2} \cdot 40)}{a^{6}}}$

Remove unnecessary parentheses.

$\frac{1}{\frac{b^{7} a^{2} \cdot 40}{a^{6}}}$

Reduce the expression $\frac{b^{7} a^{2} \cdot 40}{a^{6}}$ by cancelling the common factors.

Factor $a^{2}$ out of $b^{7} a^{2} \cdot 40$ .

$\frac{1}{\frac{a^{2} (b^{7} \cdot 40)}{a^{6}}}$

Factor $a^{2}$ out of $a^{6}$ .

$\frac{1}{\frac{a^{2} (b^{7} \cdot 40)}{a^{2} a^{4}}}$

Cancel the common factor.

$\frac{1}{\frac{a^{2} (b^{7} \cdot 40)}{a^{2} a^{4}}}$

Rewrite the expression.

$\frac{1}{\frac{b^{7} \cdot 40}{a^{4}}}$

Move $40$ to the left of $b^{7}$ .

$\frac{1}{\frac{40 b^{7}}{a^{4}}}$

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{a^{4}}{40 b^{7}}$

Multiply $\frac{a^{4}}{40 b^{7}}$ by $1$ .

$\frac{a^{4}}{40 b^{7}}$

Simplify ((2a^-2b)^-3)/(5a^2b^4)

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