Simplify the Radical Expression ((w^-8)/(w^-4))^(5/12)

${(\frac{w^{- 8}}{w^{- 4}})}^{\frac{5}{12}}$

Move $w^{- 8}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{w^{- 4} w^{8}})}^{\frac{5}{12}}$

Multiply $w^{- 4}$ by $w^{8}$ by adding the exponents.

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

${(\frac{1}{w^{- 4 + 8}})}^{\frac{5}{12}}$

Add $- 4$ and $8$ .

${(\frac{1}{w^{4}})}^{\frac{5}{12}}$

Apply the product rule to $\frac{1}{w^{4}}$ .

$\frac{1^{\frac{5}{12}}}{{(w^{4})}^{\frac{5}{12}}}$

One to any power is one.

$\frac{1}{{(w^{4})}^{\frac{5}{12}}}$

Multiply the exponents in ${(w^{4})}^{\frac{5}{12}}$ .

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

$\frac{1}{w^{4 (\frac{5}{12})}}$

Cancel the common factor of $4$ .

Factor $4$ out of $12$ .

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

Cancel the common factor.

$\frac{1}{w^{4 \frac{5}{4 \cdot 3}}}$

Rewrite the expression.

$\frac{1}{w^{\frac{5}{3}}}$