Simplify fourth root of (x^2)/(x^(2/3))

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

Step 1

Reduce the expression by cancelling the common factors.

Move $x^{\frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

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

Subtract $2$ from $6$ .

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

Step 2

Rewrite $x^{\frac{4}{3}}$ as ${(x^{\frac{1}{3}})}^{4}$ .

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

Step 3

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

$x^{\frac{1}{3}}$