Finite Math Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{3} (3 \sqrt{6} + \sqrt{3})}{9 {\sqrt{6}}^{2} + 3 \sqrt{18} - 3 \sqrt{18} - {\sqrt{3}}^{2}}$

Step 4

Simplify.

$\frac{\sqrt{3} (3 \sqrt{6} + \sqrt{3})}{51}$

Step 5

Apply the distributive property.

$\frac{\sqrt{3} (3 \sqrt{6}) + \sqrt{3} \sqrt{3}}{51}$

Step 6

Multiply $\sqrt{3} (3 \sqrt{6})$ .

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Step 6.1

Combine using the product rule for radicals.

$\frac{3 \sqrt{3 \cdot 6} + \sqrt{3} \sqrt{3}}{51}$

Step 6.2

Multiply $3$ by $6$ .

$\frac{3 \sqrt{18} + \sqrt{3} \sqrt{3}}{51}$

Step 7

Combine using the product rule for radicals.

$\frac{3 \sqrt{18} + \sqrt{3 \cdot 3}}{51}$

Step 8

Simplify each term.

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Step 8.1

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

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Step 8.1.1

Factor $9$ out of $18$ .

$\frac{3 \sqrt{9 (2)} + \sqrt{3 \cdot 3}}{51}$

Step 8.1.2

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

$\frac{3 \sqrt{3^{2} \cdot 2} + \sqrt{3 \cdot 3}}{51}$

Step 8.2

Pull terms out from under the radical.

$\frac{3 (3 \sqrt{2}) + \sqrt{3 \cdot 3}}{51}$

Step 8.3

Multiply $3$ by $3$ .

$\frac{9 \sqrt{2} + \sqrt{3 \cdot 3}}{51}$

Step 8.4

Multiply $3$ by $3$ .

$\frac{9 \sqrt{2} + \sqrt{9}}{51}$

Step 8.5

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

$\frac{9 \sqrt{2} + \sqrt{3^{2}}}{51}$

Step 8.6

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

$\frac{9 \sqrt{2} + 3}{51}$

Step 9

Cancel the common factor of $9 \sqrt{2} + 3$ and $51$ .

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Step 9.1

Factor $3$ out of $9 \sqrt{2}$ .

$\frac{3 (3 \sqrt{2}) + 3}{51}$

Step 9.2

Factor $3$ out of $3$ .

$\frac{3 (3 \sqrt{2}) + 3 (1)}{51}$

Step 9.3

Factor $3$ out of $3 (3 \sqrt{2}) + 3 (1)$ .

$\frac{3 (3 \sqrt{2} + 1)}{51}$

Step 9.4

Cancel the common factors.

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Step 9.4.1

Factor $3$ out of $51$ .

$\frac{3 (3 \sqrt{2} + 1)}{3 (17)}$

Step 9.4.2

Cancel the common factor.

$\frac{3 (3 \sqrt{2} + 1)}{3 \cdot 17}$

Step 9.4.3

Rewrite the expression.

$\frac{3 \sqrt{2} + 1}{17}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{2} + 1}{17}$

Decimal Form:

$0.30839062 \dots$