Simplify fifth root of z^4z^(-3/2)

$\sqrt[5]{z^{4} z^{- \frac{3}{2}}}$

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

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

$\sqrt[5]{z^{4 - \frac{3}{2}}}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt[5]{z^{4 \cdot \frac{2}{2} - \frac{3}{2}}}$

Combine $4$ and $\frac{2}{2}$ .

$\sqrt[5]{z^{\frac{4 \cdot 2}{2} - \frac{3}{2}}}$

Combine the numerators over the common denominator.

$\sqrt[5]{z^{\frac{4 \cdot 2 - 3}{2}}}$

Simplify the numerator.

Multiply $4$ by $2$ .

$\sqrt[5]{z^{\frac{8 - 3}{2}}}$

Subtract $3$ from $8$ .

$\sqrt[5]{z^{\frac{5}{2}}}$

Rewrite $z^{\frac{5}{2}}$ as ${(z^{\frac{1}{2}})}^{5}$ .

$\sqrt[5]{{(z^{\frac{1}{2}})}^{5}}$

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

$z^{\frac{1}{2}}$