Simplify (((x^-8)^(1/4))^-1 cube root of x^2)/( sixth root of x^4)

$\frac{{({(x^{- 8})}^{\frac{1}{4}})}^{- 1} \sqrt[3]{x^{2}}}{\sqrt[6]{x^{4}}}$

Move ${({(x^{- 8})}^{\frac{1}{4}})}^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\sqrt[3]{x^{2}}}{\sqrt[6]{x^{4}} {(x^{- 8})}^{\frac{1}{4}}}$

Simplify the denominator.

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{\sqrt[3]{x^{2}}}{\sqrt[6]{{(x^{2})}^{2}} {(x^{- 8})}^{\frac{1}{4}}}$

Rewrite $\sqrt[6]{{(x^{2})}^{2}}$ as $\sqrt[3]{\sqrt{{(x^{2})}^{2}}}$ .

$\frac{\sqrt[3]{x^{2}}}{\sqrt[3]{\sqrt{{(x^{2})}^{2}}} {(x^{- 8})}^{\frac{1}{4}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt[3]{x^{2}}}{\sqrt[3]{x^{2}} {(x^{- 8})}^{\frac{1}{4}}}$

Multiply the exponents in ${(x^{- 8})}^{\frac{1}{4}}$ .

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

$\frac{\sqrt[3]{x^{2}}}{\sqrt[3]{x^{2}} x^{- 8 (\frac{1}{4})}}$

Cancel the common factor of $4$ .

Factor $4$ out of $- 8$ .

$\frac{\sqrt[3]{x^{2}}}{\sqrt[3]{x^{2}} x^{4 (- 2) \frac{1}{4}}}$

Cancel the common factor.

$\frac{\sqrt[3]{x^{2}}}{\sqrt[3]{x^{2}} x^{4 \cdot - 2 \frac{1}{4}}}$

Rewrite the expression.

$\frac{\sqrt[3]{x^{2}}}{\sqrt[3]{x^{2}} x^{- 2}}$

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

$\frac{\sqrt[3]{x^{2}}}{\sqrt[3]{x^{2}} \frac{1}{x^{2}}}$

Combine $\sqrt[3]{x^{2}}$ and $\frac{1}{x^{2}}$ .

$\frac{\sqrt[3]{x^{2}}}{\frac{\sqrt[3]{x^{2}}}{x^{2}}}$

Multiply the numerator by the reciprocal of the denominator.

$\sqrt[3]{x^{2}} \frac{x^{2}}{\sqrt[3]{x^{2}}}$

Cancel the common factor of $\sqrt[3]{x^{2}}$ .

Cancel the common factor.

$\sqrt[3]{x^{2}} \frac{x^{2}}{\sqrt[3]{x^{2}}}$

Rewrite the expression.

$x^{2}$