Algebra Examples

$\frac{\sqrt{3}}{3 - \sqrt{6}}$

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Move $3$ to the left of $\sqrt{3}$ .

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

Step 2.6

Combine using the product rule for radicals.

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

Step 3

Simplify each term.

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Step 3.1

Multiply $3$ by $6$ .

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

Step 3.2

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

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Step 3.2.1

Factor $9$ out of $18$ .

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

Step 3.2.2

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

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

Step 3.3

Pull terms out from under the radical.

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

Step 4

Cancel the common factor of $3 \sqrt{3} + 3 \sqrt{2}$ and $3$ .

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Step 4.1

Factor $3$ out of $3 \sqrt{3}$ .

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

Step 4.2

Factor $3$ out of $3 \sqrt{2}$ .

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

Step 4.3

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

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

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $3$ out of $3$ .

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Rewrite the expression.

$\frac{\sqrt{3} + \sqrt{2}}{1}$

Step 4.4.4

Divide $\sqrt{3} + \sqrt{2}$ by $1$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$3.14626436 \dots$