Simplify ((4c^-5)/(8d^-2))^3

${(\frac{4 c^{- 5}}{8 d^{- 2}})}^{3}$

Step 1

Simplify.

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

${(\frac{4}{8 d^{- 2} c^{5}})}^{3}$

Move $d^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{4 d^{2}}{8 c^{5}})}^{3}$

Step 2

Cancel the common factor of $4$ and $8$ .

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

${(\frac{4 (d^{2})}{8 c^{5}})}^{3}$

Cancel the common factors.

Factor $4$ out of $8 c^{5}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

Apply the product rule to $\frac{d^{2}}{2 c^{5}}$ .

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

Apply the product rule to $2 c^{5}$ .

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

Step 4

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

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

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

Multiply $2$ by $3$ .

$\frac{d^{6}}{2^{3} {(c^{5})}^{3}}$

Step 5

Simplify the denominator.

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

$\frac{d^{6}}{8 {(c^{5})}^{3}}$

Multiply the exponents in ${(c^{5})}^{3}$ .

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

$\frac{d^{6}}{8 c^{5 \cdot 3}}$

Multiply $5$ by $3$ .

$\frac{d^{6}}{8 c^{15}}$