Precalculus Examples

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

Step 1

Multiply $\frac{1}{\sqrt[3]{e^{2}}}$ by $\frac{{\sqrt[3]{e^{2}}}^{2}}{{\sqrt[3]{e^{2}}}^{2}}$ .

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

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{1}{\sqrt[3]{e^{2}}}$ by $\frac{{\sqrt[3]{e^{2}}}^{2}}{{\sqrt[3]{e^{2}}}^{2}}$ .

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

Step 2.2

Raise $\sqrt[3]{e^{2}}$ to the power of $1$ .

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

Step 2.3

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

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

Step 2.4

Add $1$ and $2$ .

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

Step 2.5

Rewrite ${\sqrt[3]{e^{2}}}^{3}$ as $e^{2}$ .

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Step 2.5.1

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

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

Step 2.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.5.3

Combine $\frac{2}{3}$ and $3$ .

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

Step 2.5.4

Multiply $2$ by $3$ .

$\ln (\frac{{\sqrt[3]{e^{2}}}^{2}}{e^{\frac{6}{3}}})$

Step 2.5.5

Cancel the common factor of $6$ and $3$ .

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Step 2.5.5.1

Factor $3$ out of $6$ .

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

Step 2.5.5.2

Cancel the common factors.

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Step 2.5.5.2.1

Factor $3$ out of $3$ .

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

Step 2.5.5.2.2

Cancel the common factor.

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

Step 2.5.5.2.3

Rewrite the expression.

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

Step 2.5.5.2.4

Divide $2$ by $1$ .

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

Step 3

Move $e^{2}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\ln ({\sqrt[3]{e^{2}}}^{2} e^{- 2})$

Step 4

Rewrite $\ln ({\sqrt[3]{e^{2}}}^{2} e^{- 2})$ as $\ln ({\sqrt[3]{e^{2}}}^{2}) + \ln (e^{- 2})$ .

$\ln ({\sqrt[3]{e^{2}}}^{2}) + \ln (e^{- 2})$

Step 5

Use logarithm rules to move $- 2$ out of the exponent.

$\ln ({\sqrt[3]{e^{2}}}^{2}) - 2 \ln (e)$

Step 6

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

$\ln ({\sqrt[3]{e^{2}}}^{2}) - 2 \cdot 1$

Step 7

Multiply $- 2$ by $1$ .

$\ln ({\sqrt[3]{e^{2}}}^{2}) - 2$

Step 8

Expand $\ln ({\sqrt[3]{e^{2}}}^{2})$ by moving $2$ outside the logarithm.

$2 \ln (\sqrt[3]{e^{2}}) - 2$

Step 9

The result can be shown in multiple forms.

Exact Form:

$2 \ln (\sqrt[3]{e^{2}}) - 2$

Decimal Form:

$- 0.\overline{6}$