Precalculus Examples

$\ln (\frac{3 \sqrt{5}}{\sqrt[3]{6}})$

Step 1

Multiply $\frac{3 \sqrt{5}}{\sqrt[3]{6}}$ by $\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}}$ .

$\ln (\frac{3 \sqrt{5}}{\sqrt[3]{6}} \cdot \frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}})$

Step 2

Simplify terms.

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Step 2.1

Combine and simplify the denominator.

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Step 2.1.1

Multiply $\frac{3 \sqrt{5}}{\sqrt[3]{6}}$ by $\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}}$ .

$\ln (\frac{3 \sqrt{5} {\sqrt[3]{6}}^{2}}{\sqrt[3]{6} {\sqrt[3]{6}}^{2}})$

Step 2.1.2

Raise $\sqrt[3]{6}$ to the power of $1$ .

$\ln (\frac{3 \sqrt{5} {\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{1} {\sqrt[3]{6}}^{2}})$

Step 2.1.3

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

$\ln (\frac{3 \sqrt{5} {\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{1 + 2}})$

Step 2.1.4

Add $1$ and $2$ .

$\ln (\frac{3 \sqrt{5} {\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{3}})$

Step 2.1.5

Rewrite ${\sqrt[3]{6}}^{3}$ as $6$ .

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Step 2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6}$ as $6^{\frac{1}{3}}$ .

$\ln (\frac{3 \sqrt{5} {\sqrt[3]{6}}^{2}}{{(6^{\frac{1}{3}})}^{3}})$

Step 2.1.5.2

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

$\ln (\frac{3 \sqrt{5} {\sqrt[3]{6}}^{2}}{6^{\frac{1}{3} \cdot 3}})$

Step 2.1.5.3

Combine $\frac{1}{3}$ and $3$ .

$\ln (\frac{3 \sqrt{5} {\sqrt[3]{6}}^{2}}{6^{\frac{3}{3}}})$

Step 2.1.5.4

Cancel the common factor of $3$ .

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Step 2.1.5.4.1

Cancel the common factor.

$\ln (\frac{3 \sqrt{5} {\sqrt[3]{6}}^{2}}{6^{\frac{3}{3}}})$

Step 2.1.5.4.2

Rewrite the expression.

$\ln (\frac{3 \sqrt{5} {\sqrt[3]{6}}^{2}}{6^{1}})$

Step 2.1.5.5

Evaluate the exponent.

$\ln (\frac{3 \sqrt{5} {\sqrt[3]{6}}^{2}}{6})$

Step 2.2

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

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Step 2.2.1

Factor $3$ out of $3 \sqrt{5} {\sqrt[3]{6}}^{2}$ .

$\ln (\frac{3 (\sqrt{5} {\sqrt[3]{6}}^{2})}{6})$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $3$ out of $6$ .

$\ln (\frac{3 (\sqrt{5} {\sqrt[3]{6}}^{2})}{3 \cdot 2})$

Step 2.2.2.2

Cancel the common factor.

$\ln (\frac{3 (\sqrt{5} {\sqrt[3]{6}}^{2})}{3 \cdot 2})$

Step 2.2.2.3

Rewrite the expression.

$\ln (\frac{\sqrt{5} {\sqrt[3]{6}}^{2}}{2})$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[3]{6}}^{2}$ as $\sqrt[3]{6^{2}}$ .

$\ln (\frac{\sqrt{5} \sqrt[3]{6^{2}}}{2})$

Step 3.2

Raise $6$ to the power of $2$ .

$\ln (\frac{\sqrt{5} \sqrt[3]{36}}{2})$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of $6$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\ln (\frac{5^{\frac{1}{2}} \sqrt[3]{36}}{2})$

Step 4.1.2

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{3}{6}}$ .

$\ln (\frac{5^{\frac{3}{6}} \sqrt[3]{36}}{2})$

Step 4.1.3

Rewrite $5^{\frac{3}{6}}$ as $\sqrt[6]{5^{3}}$ .

$\ln (\frac{\sqrt[6]{5^{3}} \sqrt[3]{36}}{2})$

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{36}$ as $36^{\frac{1}{3}}$ .

$\ln (\frac{\sqrt[6]{5^{3}} \cdot 36^{\frac{1}{3}}}{2})$

Step 4.1.5

Rewrite $36^{\frac{1}{3}}$ as $36^{\frac{2}{6}}$ .

$\ln (\frac{\sqrt[6]{5^{3}} \cdot 36^{\frac{2}{6}}}{2})$

Step 4.1.6

Rewrite $36^{\frac{2}{6}}$ as $\sqrt[6]{36^{2}}$ .

$\ln (\frac{\sqrt[6]{5^{3}} \sqrt[6]{36^{2}}}{2})$

Step 4.2

Combine using the product rule for radicals.

$\ln (\frac{\sqrt[6]{5^{3} \cdot 36^{2}}}{2})$

Step 4.3

Raise $5$ to the power of $3$ .

$\ln (\frac{\sqrt[6]{125 \cdot 36^{2}}}{2})$

Step 4.4

Raise $36$ to the power of $2$ .

$\ln (\frac{\sqrt[6]{125 \cdot 1296}}{2})$

Step 5

Multiply $125$ by $1296$ .

$\ln (\frac{\sqrt[6]{162000}}{2})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{\sqrt[6]{162000}}{2})$

Decimal Form:

$1.30607808 \dots$