Finite Math Examples

$\frac{4 \sqrt{3}}{2 - \sqrt{2}} \cdot \frac{2 + \sqrt{2}}{2 + \sqrt{2}}$

Step 1

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

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Step 1.1

Cancel the common factor.

$\frac{4 \sqrt{3}}{2 - \sqrt{2}} \cdot \frac{2 + \sqrt{2}}{2 + \sqrt{2}}$

Step 1.2

Rewrite the expression.

$\frac{4 \sqrt{3}}{2 - \sqrt{2}} \cdot 1$

Step 2

Multiply $\frac{4 \sqrt{3}}{2 - \sqrt{2}}$ by $1$ .

$\frac{4 \sqrt{3}}{2 - \sqrt{2}}$

Step 3

Multiply $\frac{4 \sqrt{3}}{2 - \sqrt{2}}$ by $\frac{2 + \sqrt{2}}{2 + \sqrt{2}}$ .

$\frac{4 \sqrt{3}}{2 - \sqrt{2}} \cdot \frac{2 + \sqrt{2}}{2 + \sqrt{2}}$

Step 4

Multiply $\frac{4 \sqrt{3}}{2 - \sqrt{2}}$ by $\frac{2 + \sqrt{2}}{2 + \sqrt{2}}$ .

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

Step 5

Expand the denominator using the FOIL method.

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

Step 6

Simplify.

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

Step 7

Cancel the common factor of $4$ and $2$ .

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Step 7.1

Factor $2$ out of $4 \sqrt{3} (2 + \sqrt{2})$ .

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $2$ out of $2$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 7.2.4

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

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

Step 8

Apply the distributive property.

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

Step 9

Multiply $2$ by $2$ .

$4 \sqrt{3} + 2 \sqrt{3} \sqrt{2}$

Step 10

Multiply $2 \sqrt{3} \sqrt{2}$ .

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Step 10.1

Combine using the product rule for radicals.

$4 \sqrt{3} + 2 \sqrt{2 \cdot 3}$

Step 10.2

Multiply $2$ by $3$ .

$4 \sqrt{3} + 2 \sqrt{6}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{3} + 2 \sqrt{6}$

Decimal Form:

$11.82718271 \dots$