Simplify the Radical Expression (1/(q^-8*q^2))^(1/8)

${(\frac{1}{q^{- 8} \cdot q^{2}})}^{\frac{1}{8}}$

Multiply $q^{- 8}$ by $q^{2}$ by adding the exponents.

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

${(\frac{1}{q^{- 8 + 2}})}^{\frac{1}{8}}$

Add $- 8$ and $2$ .

${(\frac{1}{q^{- 6}})}^{\frac{1}{8}}$

Move $q^{- 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(q^{6})}^{\frac{1}{8}}$

Multiply the exponents in ${(q^{6})}^{\frac{1}{8}}$ .

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

$q^{6 (\frac{1}{8})}$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$q^{2 (3) \frac{1}{8}}$

Factor $2$ out of $8$ .

$q^{2 \cdot 3 \frac{1}{2 \cdot 4}}$

Cancel the common factor.

$q^{2 \cdot 3 \frac{1}{2 \cdot 4}}$

Rewrite the expression.

$q^{3 (\frac{1}{4})}$

Combine $3$ and $\frac{1}{4}$ .

$q^{\frac{3}{4}}$