Simplify square root of z/(z^(1/3))

$\sqrt{\frac{z}{z^{\frac{1}{3}}}}$

Step 1

Reduce the expression by cancelling the common factors.

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

$\sqrt{z \cdot z^{- \frac{1}{3}}}$

Multiply $z$ by $z^{- \frac{1}{3}}$ by adding the exponents.

Multiply $z$ by $z^{- \frac{1}{3}}$ .

Raise $z$ to the power of $1$ .

$\sqrt{z^{1} z^{- \frac{1}{3}}}$

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

$\sqrt{z^{1 - \frac{1}{3}}}$

Write $1$ as a fraction with a common denominator.

$\sqrt{z^{\frac{3}{3} - \frac{1}{3}}}$

Combine the numerators over the common denominator.

$\sqrt{z^{\frac{3 - 1}{3}}}$

Subtract $1$ from $3$ .

$\sqrt{z^{\frac{2}{3}}}$

Step 2

Rewrite $z^{\frac{2}{3}}$ as ${(z^{\frac{1}{3}})}^{2}$ .

$\sqrt{{(z^{\frac{1}{3}})}^{2}}$

Step 3

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

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