Simplify ((a^-2b^4)^-6)/((a^4b^-8)^3)

$\frac{{(a^{- 2} b^{4})}^{- 6}}{{(a^{4} b^{- 8})}^{3}}$

Step 1

Move ${(a^{- 2} b^{4})}^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(a^{4} b^{- 8})}^{3} {(a^{- 2} b^{4})}^{6}}$

Step 2

Simplify the denominator.

Apply the product rule to $a^{4} b^{- 8}$ .

$\frac{1}{{(a^{4})}^{3} {(b^{- 8})}^{3} {(a^{- 2} b^{4})}^{6}}$

Apply the product rule to $a^{- 2} b^{4}$ .

$\frac{1}{{(a^{4})}^{3} {(b^{- 8})}^{3} {(a^{- 2})}^{6} {(b^{4})}^{6}}$

Multiply the exponents in ${(a^{4})}^{3}$ .

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

$\frac{1}{a^{4 \cdot 3} {(b^{- 8})}^{3} {(a^{- 2})}^{6} {(b^{4})}^{6}}$

Multiply $4$ by $3$ .

$\frac{1}{a^{12} {(b^{- 8})}^{3} {(a^{- 2})}^{6} {(b^{4})}^{6}}$

Multiply the exponents in ${(b^{- 8})}^{3}$ .

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

$\frac{1}{a^{12} b^{- 8 \cdot 3} {(a^{- 2})}^{6} {(b^{4})}^{6}}$

Multiply $- 8$ by $3$ .

$\frac{1}{a^{12} b^{- 24} {(a^{- 2})}^{6} {(b^{4})}^{6}}$

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

$\frac{1}{a^{12} \frac{1}{b^{24}} {(a^{- 2})}^{6} {(b^{4})}^{6}}$

Multiply the exponents in ${(a^{- 2})}^{6}$ .

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

$\frac{1}{a^{12} \frac{1}{b^{24}} a^{- 2 \cdot 6} {(b^{4})}^{6}}$

Multiply $- 2$ by $6$ .

$\frac{1}{a^{12} \frac{1}{b^{24}} a^{- 12} {(b^{4})}^{6}}$

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

$\frac{1}{a^{12} \frac{1}{b^{24}} \frac{1}{a^{12}} {(b^{4})}^{6}}$

Multiply the exponents in ${(b^{4})}^{6}$ .

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

$\frac{1}{a^{12} \frac{1}{b^{24}} \frac{1}{a^{12}} b^{4 \cdot 6}}$

Multiply $4$ by $6$ .

$\frac{1}{a^{12} \frac{1}{b^{24}} \frac{1}{a^{12}} b^{24}}$

Combine exponents.

Combine $a^{12}$ and $\frac{1}{b^{24}}$ .

$\frac{1}{\frac{a^{12}}{b^{24}} \cdot \frac{1}{a^{12}} b^{24}}$

Multiply $\frac{a^{12}}{b^{24}}$ by $\frac{1}{a^{12}}$ .

$\frac{1}{\frac{a^{12}}{b^{24} a^{12}} b^{24}}$

Combine $\frac{a^{12}}{b^{24} a^{12}}$ and $b^{24}$ .

$\frac{1}{\frac{a^{12} b^{24}}{b^{24} a^{12}}}$

Reduce the expression $\frac{a^{12} b^{24}}{b^{24} a^{12}}$ by cancelling the common factors.

Cancel the common factor.

$\frac{1}{\frac{a^{12} b^{24}}{b^{24} a^{12}}}$

Rewrite the expression.

$\frac{1}{\frac{b^{24}}{b^{24}}}$

Cancel the common factor of $b^{24}$ .

Cancel the common factor.

$\frac{1}{\frac{b^{24}}{b^{24}}}$

Rewrite the expression.

$\frac{1}{1}$

Step 3

Divide $1$ by $1$ .

$1$

Simplify ((a^-2b^4)^-6)/((a^4b^-8)^3)

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