Simplify ((2a^-2b^4c^2)/(-4a^-2b^-5c^-7))^-1

${(\frac{2 a^{- 2} b^{4} c^{2}}{- 4 a^{- 2} b^{- 5} c^{- 7}})}^{- 1}$

Step 1

Move $b^{- 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{2 a^{- 2} b^{4} c^{2} b^{5}}{- 4 a^{- 2} c^{- 7}})}^{- 1}$

Step 2

Move $c^{- 7}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{2 a^{- 2} b^{4} c^{2} b^{5} c^{7}}{- 4 a^{- 2}})}^{- 1}$

Step 3

Simplify the numerator.

Multiply $b^{4}$ by $b^{5}$ by adding the exponents.

Move $b^{5}$ .

${(\frac{2 a^{- 2} (b^{5} b^{4}) c^{2} c^{7}}{- 4 a^{- 2}})}^{- 1}$

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

${(\frac{2 a^{- 2} b^{5 + 4} c^{2} c^{7}}{- 4 a^{- 2}})}^{- 1}$

Add $5$ and $4$ .

${(\frac{2 a^{- 2} b^{9} c^{2} c^{7}}{- 4 a^{- 2}})}^{- 1}$

Multiply $c^{2}$ by $c^{7}$ by adding the exponents.

Move $c^{7}$ .

${(\frac{2 a^{- 2} b^{9} (c^{7} c^{2})}{- 4 a^{- 2}})}^{- 1}$

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

${(\frac{2 a^{- 2} b^{9} c^{7 + 2}}{- 4 a^{- 2}})}^{- 1}$

Add $7$ and $2$ .

${(\frac{2 a^{- 2} b^{9} c^{9}}{- 4 a^{- 2}})}^{- 1}$

Step 4

Cancel the common factor of $2$ and $- 4$ .

Factor $2$ out of $2 a^{- 2} b^{9} c^{9}$ .

${(\frac{2 (a^{- 2} b^{9} c^{9})}{- 4 a^{- 2}})}^{- 1}$

Cancel the common factors.

Factor $2$ out of $- 4 a^{- 2}$ .

${(\frac{2 (a^{- 2} b^{9} c^{9})}{2 (- 2 a^{- 2})})}^{- 1}$

Cancel the common factor.

${(\frac{2 (a^{- 2} b^{9} c^{9})}{2 (- 2 a^{- 2})})}^{- 1}$

Rewrite the expression.

${(\frac{a^{- 2} b^{9} c^{9}}{- 2 a^{- 2}})}^{- 1}$

Step 5

Cancel the common factor of $a^{- 2}$ .

Cancel the common factor.

${(\frac{a^{- 2} b^{9} c^{9}}{- 2 a^{- 2}})}^{- 1}$

Rewrite the expression.

${(\frac{b^{9} c^{9}}{- 2})}^{- 1}$

Step 6

Move the negative in front of the fraction.

${(- \frac{b^{9} c^{9}}{2})}^{- 1}$

Step 7

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

$- \frac{2}{b^{9} c^{9}}$

Simplify ((2a^-2b^4c^2)/(-4a^-2b^-5c^-7))^-1

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