Expand the Logarithmic Expression natural log of cube root of z square root of e

$\ln (\sqrt[3]{z \sqrt{e}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{e}$ as $e^{\frac{1}{2}}$ .

$\ln (\sqrt[3]{z e^{\frac{1}{2}}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{z e^{\frac{1}{2}}}$ as ${(z e^{\frac{1}{2}})}^{\frac{1}{3}}$ .

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

Expand $\ln ({(z e^{\frac{1}{2}})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

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

Rewrite $\ln (z e^{\frac{1}{2}})$ as $\ln (z) + \ln (e^{\frac{1}{2}})$ .

$\frac{1}{3} (\ln (z) + \ln (e^{\frac{1}{2}}))$

Expand $\ln (e^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{3} (\ln (z) + \frac{1}{2} \ln (e))$

The natural logarithm of $e$ is $1$ .

$\frac{1}{3} (\ln (z) + \frac{1}{2} \cdot 1)$

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

$\frac{1}{3} (\ln (z) + \frac{1}{2})$

Apply the distributive property.

$\frac{1}{3} \ln (z) + \frac{1}{3} \cdot \frac{1}{2}$

Combine $\frac{1}{3}$ and $\ln (z)$ .

$\frac{\ln (z)}{3} + \frac{1}{3} \cdot \frac{1}{2}$

Multiply $\frac{1}{3} \cdot \frac{1}{2}$ .

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

$\frac{\ln (z)}{3} + \frac{1}{3 \cdot 2}$

Multiply $3$ by $2$ .

$\frac{\ln (z)}{3} + \frac{1}{6}$