Simplify (z^(1/3))/(z^(-3/2)z^(1/2))

$\frac{z^{\frac{1}{3}}}{z^{- \frac{3}{2}} z^{\frac{1}{2}}}$

Multiply $z^{- \frac{3}{2}}$ by $z^{\frac{1}{2}}$ by adding the exponents.

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

$\frac{z^{\frac{1}{3}}}{z^{- \frac{3}{2} + \frac{1}{2}}}$

Combine the numerators over the common denominator.

$\frac{z^{\frac{1}{3}}}{z^{\frac{- 3 + 1}{2}}}$

Add $- 3$ and $1$ .

$\frac{z^{\frac{1}{3}}}{z^{\frac{- 2}{2}}}$

Divide $- 2$ by $2$ .

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $3$ .

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