Simplify ((c^-3d^-1)/(5c^7d^-4))^-2

${(\frac{c^{- 3} d^{- 1}}{5 c^{7} d^{- 4}})}^{- 2}$

Step 1

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

${(\frac{d^{- 1}}{5 c^{7} d^{- 4} c^{3}})}^{- 2}$

Step 2

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

Move $c^{3}$ .

${(\frac{d^{- 1}}{5 (c^{3} c^{7}) d^{- 4}})}^{- 2}$

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

${(\frac{d^{- 1}}{5 c^{3 + 7} d^{- 4}})}^{- 2}$

Add $3$ and $7$ .

${(\frac{d^{- 1}}{5 c^{10} d^{- 4}})}^{- 2}$

Step 3

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

${(\frac{d^{- 1} d^{4}}{5 c^{10}})}^{- 2}$

Step 4

Multiply $d^{- 1}$ by $d^{4}$ by adding the exponents.

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

${(\frac{d^{- 1 + 4}}{5 c^{10}})}^{- 2}$

Add $- 1$ and $4$ .

${(\frac{d^{3}}{5 c^{10}})}^{- 2}$

Step 5

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

${(\frac{5 c^{10}}{d^{3}})}^{2}$

Step 6

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

Apply the product rule to $\frac{5 c^{10}}{d^{3}}$ .

$\frac{{(5 c^{10})}^{2}}{{(d^{3})}^{2}}$

Apply the product rule to $5 c^{10}$ .

$\frac{5^{2} {(c^{10})}^{2}}{{(d^{3})}^{2}}$

Step 7

Simplify the numerator.

Raise $5$ to the power of $2$ .

$\frac{25 {(c^{10})}^{2}}{{(d^{3})}^{2}}$

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

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

$\frac{25 c^{10 \cdot 2}}{{(d^{3})}^{2}}$

Multiply $10$ by $2$ .

$\frac{25 c^{20}}{{(d^{3})}^{2}}$

Step 8

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

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

$\frac{25 c^{20}}{d^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$\frac{25 c^{20}}{d^{6}}$