Simplify ((2a^-4b^-9c^-1)/(10a^-7bc^-5))^-1

${(\frac{2 a^{- 4} b^{- 9} c^{- 1}}{10 a^{- 7} b c^{- 5}})}^{- 1}$

Step 1

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

${(\frac{2 a^{- 4} c^{- 1}}{10 a^{- 7} b c^{- 5} b^{9}})}^{- 1}$

Step 2

Multiply $b$ by $b^{9}$ by adding the exponents.

Move $b^{9}$ .

${(\frac{2 a^{- 4} c^{- 1}}{10 a^{- 7} (b^{9} b) c^{- 5}})}^{- 1}$

Multiply $b^{9}$ by $b$ .

Raise $b$ to the power of $1$ .

${(\frac{2 a^{- 4} c^{- 1}}{10 a^{- 7} (b^{9} b^{1}) c^{- 5}})}^{- 1}$

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

${(\frac{2 a^{- 4} c^{- 1}}{10 a^{- 7} b^{9 + 1} c^{- 5}})}^{- 1}$

Add $9$ and $1$ .

${(\frac{2 a^{- 4} c^{- 1}}{10 a^{- 7} b^{10} c^{- 5}})}^{- 1}$

Step 3

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

${(\frac{2 a^{- 4} c^{- 1} a^{7}}{10 b^{10} c^{- 5}})}^{- 1}$

Step 4

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

${(\frac{2 a^{- 4} c^{- 1} a^{7} c^{5}}{10 b^{10}})}^{- 1}$

Step 5

Simplify the numerator.

Multiply $a^{- 4}$ by $a^{7}$ by adding the exponents.

Move $a^{7}$ .

${(\frac{2 (a^{7} a^{- 4}) c^{- 1} c^{5}}{10 b^{10}})}^{- 1}$

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

${(\frac{2 a^{7 - 4} c^{- 1} c^{5}}{10 b^{10}})}^{- 1}$

Subtract $4$ from $7$ .

${(\frac{2 a^{3} c^{- 1} c^{5}}{10 b^{10}})}^{- 1}$

Multiply $c^{- 1}$ by $c^{5}$ by adding the exponents.

Move $c^{5}$ .

${(\frac{2 a^{3} (c^{5} c^{- 1})}{10 b^{10}})}^{- 1}$

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

${(\frac{2 a^{3} c^{5 - 1}}{10 b^{10}})}^{- 1}$

Subtract $1$ from $5$ .

${(\frac{2 a^{3} c^{4}}{10 b^{10}})}^{- 1}$

Step 6

Cancel the common factor of $2$ and $10$ .

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

${(\frac{2 (a^{3} c^{4})}{10 b^{10}})}^{- 1}$

Cancel the common factors.

Factor $2$ out of $10 b^{10}$ .

${(\frac{2 (a^{3} c^{4})}{2 (5 b^{10})})}^{- 1}$

Cancel the common factor.

${(\frac{2 (a^{3} c^{4})}{2 (5 b^{10})})}^{- 1}$

Rewrite the expression.

${(\frac{a^{3} c^{4}}{5 b^{10}})}^{- 1}$

Step 7

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

$\frac{5 b^{10}}{a^{3} c^{4}}$

Simplify ((2a^-4b^-9c^-1)/(10a^-7bc^-5))^-1

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