Simplify ((a^-6b^4)/(2a^-9b^-2))^-2*(ab^-3)

${(\frac{a^{- 6} b^{4}}{2 a^{- 9} b^{- 2}})}^{- 2} \cdot (a b^{- 3})$

Step 1

Move $a^{- 9}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{a^{- 6} b^{4} a^{9}}{2 b^{- 2}})}^{- 2} \cdot (a b^{- 3})$

Step 2

Move $b^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{a^{- 6} b^{4} a^{9} b^{2}}{2})}^{- 2} \cdot (a b^{- 3})$

Step 3

Simplify the numerator.

Multiply $a^{- 6}$ by $a^{9}$ by adding the exponents.

Move $a^{9}$ .

${(\frac{a^{9} a^{- 6} b^{4} b^{2}}{2})}^{- 2} \cdot (a b^{- 3})$

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

${(\frac{a^{9 - 6} b^{4} b^{2}}{2})}^{- 2} \cdot (a b^{- 3})$

Subtract $6$ from $9$ .

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

Multiply $b^{4}$ by $b^{2}$ by adding the exponents.

Move $b^{2}$ .

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

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

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

Add $2$ and $4$ .

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

Step 4

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

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

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{2}{a^{3} b^{6}}$ .

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

Apply the product rule to $a^{3} b^{6}$ .

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

Step 6

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

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

Step 7

Simplify the denominator.

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

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

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

Multiply $3$ by $2$ .

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

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

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

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

Multiply $6$ by $2$ .

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

Step 8

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

Factor $a b^{- 3}$ out of $a^{6} b^{12}$ .

$\frac{4}{a b^{- 3} (a^{5} b^{15})} \cdot (a b^{- 3})$

Cancel the common factor.

$\frac{4}{a b^{- 3} (a^{5} b^{15})} \cdot (a b^{- 3})$

Rewrite the expression.

$\frac{4}{a^{5} b^{15}}$

Simplify ((a^-6b^4)/(2a^-9b^-2))^-2*(ab^-3)

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