Simplify ((12a^-3b^8c^-4)/(3a^7b^-3c^7))^-2

${(\frac{12 a^{- 3} b^{8} c^{- 4}}{3 a^{7} b^{- 3} c^{7}})}^{- 2}$

Step 1

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

${(\frac{12 b^{8} c^{- 4}}{3 a^{7} b^{- 3} c^{7} a^{3}})}^{- 2}$

Step 2

Move $c^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{12 b^{8}}{3 a^{7} b^{- 3} c^{7} a^{3} c^{4}})}^{- 2}$

Step 3

Simplify the denominator.

Multiply $a^{7}$ by $a^{3}$ by adding the exponents.

Move $a^{3}$ .

${(\frac{12 b^{8}}{3 (a^{3} a^{7}) b^{- 3} c^{7} c^{4}})}^{- 2}$

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

${(\frac{12 b^{8}}{3 a^{3 + 7} b^{- 3} c^{7} c^{4}})}^{- 2}$

Add $3$ and $7$ .

${(\frac{12 b^{8}}{3 a^{10} b^{- 3} c^{7} c^{4}})}^{- 2}$

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

Move $c^{4}$ .

${(\frac{12 b^{8}}{3 a^{10} b^{- 3} (c^{4} c^{7})})}^{- 2}$

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

${(\frac{12 b^{8}}{3 a^{10} b^{- 3} c^{4 + 7}})}^{- 2}$

Add $4$ and $7$ .

${(\frac{12 b^{8}}{3 a^{10} b^{- 3} c^{11}})}^{- 2}$

Step 4

Move $b^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{12 b^{8} b^{3}}{3 a^{10} c^{11}})}^{- 2}$

Step 5

Multiply $b^{8}$ by $b^{3}$ by adding the exponents.

Move $b^{3}$ .

${(\frac{12 (b^{3} b^{8})}{3 a^{10} c^{11}})}^{- 2}$

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

${(\frac{12 b^{3 + 8}}{3 a^{10} c^{11}})}^{- 2}$

Add $3$ and $8$ .

${(\frac{12 b^{11}}{3 a^{10} c^{11}})}^{- 2}$

Step 6

Cancel the common factor of $12$ and $3$ .

Factor $3$ out of $12 b^{11}$ .

${(\frac{3 (4 b^{11})}{3 a^{10} c^{11}})}^{- 2}$

Cancel the common factors.

Factor $3$ out of $3 a^{10} c^{11}$ .

${(\frac{3 (4 b^{11})}{3 (a^{10} c^{11})})}^{- 2}$

Cancel the common factor.

${(\frac{3 (4 b^{11})}{3 (a^{10} c^{11})})}^{- 2}$

Rewrite the expression.

${(\frac{4 b^{11}}{a^{10} c^{11}})}^{- 2}$

Step 7

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

${(\frac{a^{10} c^{11}}{4 b^{11}})}^{2}$

Step 8

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

Apply the product rule to $\frac{a^{10} c^{11}}{4 b^{11}}$ .

$\frac{{(a^{10} c^{11})}^{2}}{{(4 b^{11})}^{2}}$

Apply the product rule to $a^{10} c^{11}$ .

$\frac{{(a^{10})}^{2} {(c^{11})}^{2}}{{(4 b^{11})}^{2}}$

Apply the product rule to $4 b^{11}$ .

$\frac{{(a^{10})}^{2} {(c^{11})}^{2}}{4^{2} {(b^{11})}^{2}}$

Step 9

Simplify the numerator.

Multiply the exponents in ${(a^{10})}^{2}$ .

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

$\frac{a^{10 \cdot 2} {(c^{11})}^{2}}{4^{2} {(b^{11})}^{2}}$

Multiply $10$ by $2$ .

$\frac{a^{20} {(c^{11})}^{2}}{4^{2} {(b^{11})}^{2}}$

Multiply the exponents in ${(c^{11})}^{2}$ .

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

$\frac{a^{20} c^{11 \cdot 2}}{4^{2} {(b^{11})}^{2}}$

Multiply $11$ by $2$ .

$\frac{a^{20} c^{22}}{4^{2} {(b^{11})}^{2}}$

Step 10

Simplify the denominator.

Raise $4$ to the power of $2$ .

$\frac{a^{20} c^{22}}{16 {(b^{11})}^{2}}$

Multiply the exponents in ${(b^{11})}^{2}$ .

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

$\frac{a^{20} c^{22}}{16 b^{11 \cdot 2}}$

Multiply $11$ by $2$ .

$\frac{a^{20} c^{22}}{16 b^{22}}$

Simplify ((12a^-3b^8c^-4)/(3a^7b^-3c^7))^-2

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