Simplify ((4w^-1c^4)^-5)/(5c^-3)

$\frac{{(4 w^{- 1} c^{4})}^{- 5}}{5 c^{- 3}}$

Step 1

Simplify the expression.

Move ${(4 w^{- 1} c^{4})}^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{5 c^{- 3} {(4 w^{- 1} c^{4})}^{5}}$

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

$\frac{c^{3}}{5 {(4 w^{- 1} c^{4})}^{5}}$

Step 2

Simplify the denominator.

Apply the product rule to $4 w^{- 1} c^{4}$ .

$\frac{c^{3}}{5 {(4 w^{- 1})}^{5} {(c^{4})}^{5}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{c^{3}}{5 {(4 \frac{1}{w})}^{5} {(c^{4})}^{5}}$

Combine $4$ and $\frac{1}{w}$ .

$\frac{c^{3}}{5 {(\frac{4}{w})}^{5} {(c^{4})}^{5}}$

Apply the product rule to $\frac{4}{w}$ .

$\frac{c^{3}}{5 \frac{4^{5}}{w^{5}} {(c^{4})}^{5}}$

Raise $4$ to the power of $5$ .

$\frac{c^{3}}{5 \frac{1024}{w^{5}} {(c^{4})}^{5}}$

Multiply the exponents in ${(c^{4})}^{5}$ .

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

$\frac{c^{3}}{5 \frac{1024}{w^{5}} c^{4 \cdot 5}}$

Multiply $4$ by $5$ .

$\frac{c^{3}}{5 \frac{1024}{w^{5}} c^{20}}$

Combine exponents.

Combine $5$ and $\frac{1024}{w^{5}}$ .

$\frac{c^{3}}{\frac{5 \cdot 1024}{w^{5}} c^{20}}$

Multiply $5$ by $1024$ .

$\frac{c^{3}}{\frac{5120}{w^{5}} c^{20}}$

Combine $\frac{5120}{w^{5}}$ and $c^{20}$ .

$\frac{c^{3}}{\frac{5120 c^{20}}{w^{5}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$c^{3} \frac{w^{5}}{5120 c^{20}}$

Step 4

Cancel the common factor of $c^{3}$ .

Factor $c^{3}$ out of $5120 c^{20}$ .

$c^{3} \frac{w^{5}}{c^{3} (5120 c^{17})}$

Cancel the common factor.

$c^{3} \frac{w^{5}}{c^{3} (5120 c^{17})}$

Rewrite the expression.

$\frac{w^{5}}{5120 c^{17}}$