Simplify (z^(1/3))/(z^(-1/2)z^(1/4))

$\frac{z^{\frac{1}{3}}}{z^{- \frac{1}{2}} z^{\frac{1}{4}}}$

Step 1

Multiply $z^{- \frac{1}{2}}$ by $z^{\frac{1}{4}}$ by adding the exponents.

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

$\frac{z^{\frac{1}{3}}}{z^{- \frac{1}{2} + \frac{1}{4}}}$

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{z^{\frac{1}{3}}}{z^{- \frac{1}{2} \cdot \frac{2}{2} + \frac{1}{4}}}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$\frac{z^{\frac{1}{3}}}{z^{- \frac{2}{2 \cdot 2} + \frac{1}{4}}}$

Multiply $2$ by $2$ .

$\frac{z^{\frac{1}{3}}}{z^{- \frac{2}{4} + \frac{1}{4}}}$

Combine the numerators over the common denominator.

$\frac{z^{\frac{1}{3}}}{z^{\frac{- 2 + 1}{4}}}$

Add $- 2$ and $1$ .

$\frac{z^{\frac{1}{3}}}{z^{\frac{- 1}{4}}}$

Move the negative in front of the fraction.

$\frac{z^{\frac{1}{3}}}{z^{- \frac{1}{4}}}$

Step 2

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

$z^{\frac{1}{3}} z^{\frac{1}{4}}$

Step 3

Multiply $z^{\frac{1}{3}}$ by $z^{\frac{1}{4}}$ by adding the exponents.

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

$z^{\frac{1}{3} + \frac{1}{4}}$

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$z^{\frac{1}{3} \cdot \frac{4}{4} + \frac{1}{4}}$

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$z^{\frac{1}{3} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{3}{3}}$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$z^{\frac{4}{3 \cdot 4} + \frac{1}{4} \cdot \frac{3}{3}}$

Multiply $3$ by $4$ .

$z^{\frac{4}{12} + \frac{1}{4} \cdot \frac{3}{3}}$

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$z^{\frac{4}{12} + \frac{3}{4 \cdot 3}}$

Multiply $4$ by $3$ .

$z^{\frac{4}{12} + \frac{3}{12}}$

Combine the numerators over the common denominator.

$z^{\frac{4 + 3}{12}}$

Add $4$ and $3$ .

$z^{\frac{7}{12}}$