Simplify ( cube root of x^8)/((x^4)^(1/3))

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

Simplify the numerator.

Rewrite $x^{8}$ as ${(x^{2})}^{3} x^{2}$ .

Factor out $x^{6}$ .

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

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

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

Pull terms out from under the radical.

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

Multiply the exponents in ${(x^{4})}^{\frac{1}{3}}$ .

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

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

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

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

Move $x^{\frac{4}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Multiply $x^{2}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

Move $x^{- \frac{4}{3}}$ .

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

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

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

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Combine $2$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$x^{\frac{- 4 + 6}{3}} \sqrt[3]{x^{2}}$

Add $- 4$ and $6$ .

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$x^{\frac{2}{3}} x^{\frac{2}{3}}$

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

$x^{\frac{2}{3} + \frac{2}{3}}$

Combine the numerators over the common denominator.

$x^{\frac{2 + 2}{3}}$

Add $2$ and $2$ .

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