Algebra Examples

$\frac{4 + \sqrt{6}}{3 \sqrt{6}}$

Step 1

Multiply $\frac{4 + \sqrt{6}}{3 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{4 + \sqrt{6}}{3 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

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 $\frac{4 + \sqrt{6}}{3 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{(4 + \sqrt{6}) \sqrt{6}}{3 \sqrt{6} \sqrt{6}}$

Step 2.1.2

Move $\sqrt{6}$ .

$\frac{(4 + \sqrt{6}) \sqrt{6}}{3 (\sqrt{6} \sqrt{6})}$

Step 2.1.3

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

$\frac{(4 + \sqrt{6}) \sqrt{6}}{3 ({\sqrt{6}}^{1} \sqrt{6})}$

Step 2.1.4

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

$\frac{(4 + \sqrt{6}) \sqrt{6}}{3 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}$

Step 2.1.5

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

$\frac{(4 + \sqrt{6}) \sqrt{6}}{3 {\sqrt{6}}^{1 + 1}}$

Step 2.1.6

Add $1$ and $1$ .

$\frac{(4 + \sqrt{6}) \sqrt{6}}{3 {\sqrt{6}}^{2}}$

Step 2.1.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{(4 + \sqrt{6}) \sqrt{6}}{3 {(6^{\frac{1}{2}})}^{2}}$

Step 2.1.7.2

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

$\frac{(4 + \sqrt{6}) \sqrt{6}}{3 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 2.1.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{(4 + \sqrt{6}) \sqrt{6}}{3 \cdot 6^{\frac{2}{2}}}$

Step 2.1.7.4

Cancel the common factor of $2$ .

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Step 2.1.7.4.1

Cancel the common factor.

$\frac{(4 + \sqrt{6}) \sqrt{6}}{3 \cdot 6^{\frac{2}{2}}}$

Step 2.1.7.4.2

Rewrite the expression.

$\frac{(4 + \sqrt{6}) \sqrt{6}}{3 \cdot 6^{1}}$

Step 2.1.7.5

Evaluate the exponent.

$\frac{(4 + \sqrt{6}) \sqrt{6}}{3 \cdot 6}$

Step 2.2

Multiply $3$ by $6$ .

$\frac{(4 + \sqrt{6}) \sqrt{6}}{18}$

Step 2.3

Apply the distributive property.

$\frac{4 \sqrt{6} + \sqrt{6} \sqrt{6}}{18}$

Step 2.4

Combine using the product rule for radicals.

$\frac{4 \sqrt{6} + \sqrt{6 \cdot 6}}{18}$

Step 3

Simplify each term.

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Step 3.1

Multiply $6$ by $6$ .

$\frac{4 \sqrt{6} + \sqrt{36}}{18}$

Step 3.2

Rewrite $36$ as $6^{2}$ .

$\frac{4 \sqrt{6} + \sqrt{6^{2}}}{18}$

Step 3.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{4 \sqrt{6} + 6}{18}$

Step 4

Cancel the common factor of $4 \sqrt{6} + 6$ and $18$ .

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Step 4.1

Factor $2$ out of $4 \sqrt{6}$ .

$\frac{2 (2 \sqrt{6}) + 6}{18}$

Step 4.2

Factor $2$ out of $6$ .

$\frac{2 (2 \sqrt{6}) + 2 (3)}{18}$

Step 4.3

Factor $2$ out of $2 (2 \sqrt{6}) + 2 (3)$ .

$\frac{2 (2 \sqrt{6} + 3)}{18}$

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $2$ out of $18$ .

$\frac{2 (2 \sqrt{6} + 3)}{2 (9)}$

Step 4.4.2

Cancel the common factor.

$\frac{2 (2 \sqrt{6} + 3)}{2 \cdot 9}$

Step 4.4.3

Rewrite the expression.

$\frac{2 \sqrt{6} + 3}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{6} + 3}{9}$

Decimal Form:

$0.87766438 \dots$