Algebra Examples

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

Step 1

Multiply $\frac{2 + \sqrt{6}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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 $\frac{2 + \sqrt{6}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Add $1$ and $1$ .

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

Step 2.1.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.6.1

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

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

Step 2.1.6.2

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

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

Step 2.1.6.3

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

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

Step 2.1.6.4

Cancel the common factor of $2$ .

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Step 2.1.6.4.1

Cancel the common factor.

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

Step 2.1.6.4.2

Rewrite the expression.

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

Step 2.1.6.5

Evaluate the exponent.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Combine using the product rule for radicals.

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

Step 3

Simplify each term.

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Step 3.1

Multiply $6$ by $3$ .

$\frac{2 \sqrt{3} + \sqrt{18}}{3}$

Step 3.2

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 3.2.1

Factor $9$ out of $18$ .

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

Step 3.2.2

Rewrite $9$ as $3^{2}$ .

$\frac{2 \sqrt{3} + \sqrt{3^{2} \cdot 2}}{3}$

Step 3.3

Pull terms out from under the radical.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$2.56891410 \dots$