Basic Math Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(3 + \sqrt{2}) (6 + 3 \sqrt{2})}{36 + 18 \sqrt{2} - 18 \sqrt{2} - 9 {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

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

Step 2.4

Cancel the common factor of $6 + 3 \sqrt{2}$ and $18$ .

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Step 2.4.1

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

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

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $3$ out of $18$ .

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

Step 2.4.2.2

Cancel the common factor.

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

Step 2.4.2.3

Rewrite the expression.

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

Step 3

Expand $(3 + \sqrt{2}) (2 + \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $3$ by $2$ .

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

Step 4.1.2

Move $2$ to the left of $\sqrt{2}$ .

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

Step 4.1.3

Combine using the product rule for radicals.

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

Step 4.1.4

Multiply $2$ by $2$ .

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

Step 4.1.5

Rewrite $4$ as $2^{2}$ .

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

Step 4.1.6

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

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

Step 4.2

Add $6$ and $2$ .

$\frac{8 + 3 \sqrt{2} + 2 \sqrt{2}}{6}$

Step 4.3

Add $3 \sqrt{2}$ and $2 \sqrt{2}$ .

$\frac{8 + 5 \sqrt{2}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{8 + 5 \sqrt{2}}{6}$

Decimal Form:

$2.51184463 \dots$