Simplify ((16c^6c^-2)/((2c^2)^3))^-1

${(\frac{16 c^{6} c^{- 2}}{{(2 c^{2})}^{3}})}^{- 1}$

Step 1

Multiply $c^{6}$ by $c^{- 2}$ by adding the exponents.

Move $c^{- 2}$ .

${(\frac{16 (c^{- 2} c^{6})}{{(2 c^{2})}^{3}})}^{- 1}$

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

${(\frac{16 c^{- 2 + 6}}{{(2 c^{2})}^{3}})}^{- 1}$

Add $- 2$ and $6$ .

${(\frac{16 c^{4}}{{(2 c^{2})}^{3}})}^{- 1}$

Step 2

Simplify the denominator.

Apply the product rule to $2 c^{2}$ .

${(\frac{16 c^{4}}{2^{3} {(c^{2})}^{3}})}^{- 1}$

Raise $2$ to the power of $3$ .

${(\frac{16 c^{4}}{8 {(c^{2})}^{3}})}^{- 1}$

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

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

${(\frac{16 c^{4}}{8 c^{2 \cdot 3}})}^{- 1}$

Multiply $2$ by $3$ .

${(\frac{16 c^{4}}{8 c^{6}})}^{- 1}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $16$ and $8$ .

Factor $8$ out of $16 c^{4}$ .

${(\frac{8 (2 c^{4})}{8 c^{6}})}^{- 1}$

Cancel the common factors.

Factor $8$ out of $8 c^{6}$ .

${(\frac{8 (2 c^{4})}{8 (c^{6})})}^{- 1}$

Cancel the common factor.

${(\frac{8 (2 c^{4})}{8 c^{6}})}^{- 1}$

Rewrite the expression.

${(\frac{2 c^{4}}{c^{6}})}^{- 1}$

Cancel the common factor of $c^{4}$ and $c^{6}$ .

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

${(\frac{c^{4} \cdot 2}{c^{6}})}^{- 1}$

Cancel the common factors.

Factor $c^{4}$ out of $c^{6}$ .

${(\frac{c^{4} \cdot 2}{c^{4} c^{2}})}^{- 1}$

Cancel the common factor.

${(\frac{c^{4} \cdot 2}{c^{4} c^{2}})}^{- 1}$

Rewrite the expression.

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

Step 4

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

$\frac{c^{2}}{2}$