Algebra Examples

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

Step 1

Multiply $\frac{4 + \sqrt{6}}{4 + \sqrt{2}}$ by $\frac{4 - \sqrt{2}}{4 - \sqrt{2}}$ .

$\frac{4 + \sqrt{6}}{4 + \sqrt{2}} \cdot \frac{4 - \sqrt{2}}{4 - \sqrt{2}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{4 + \sqrt{6}}{4 + \sqrt{2}}$ by $\frac{4 - \sqrt{2}}{4 - \sqrt{2}}$ .

$\frac{(4 + \sqrt{6}) (4 - \sqrt{2})}{(4 + \sqrt{2}) (4 - \sqrt{2})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(4 + \sqrt{6}) (4 - \sqrt{2})}{16 - 4 \sqrt{2} + \sqrt{2} \cdot 4 - {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

$\frac{(4 + \sqrt{6}) (4 - \sqrt{2})}{14}$

Step 3

Expand $(4 + \sqrt{6}) (4 - \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{4 (4 - \sqrt{2}) + \sqrt{6} (4 - \sqrt{2})}{14}$

Step 3.2

Apply the distributive property.

$\frac{4 \cdot 4 + 4 (- \sqrt{2}) + \sqrt{6} (4 - \sqrt{2})}{14}$

Step 3.3

Apply the distributive property.

$\frac{4 \cdot 4 + 4 (- \sqrt{2}) + \sqrt{6} \cdot 4 + \sqrt{6} (- \sqrt{2})}{14}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $4$ by $4$ .

$\frac{16 + 4 (- \sqrt{2}) + \sqrt{6} \cdot 4 + \sqrt{6} (- \sqrt{2})}{14}$

Step 4.1.2

Multiply $- 1$ by $4$ .

$\frac{16 - 4 \sqrt{2} + \sqrt{6} \cdot 4 + \sqrt{6} (- \sqrt{2})}{14}$

Step 4.1.3

Move $4$ to the left of $\sqrt{6}$ .

$\frac{16 - 4 \sqrt{2} + 4 \cdot \sqrt{6} + \sqrt{6} (- \sqrt{2})}{14}$

Step 4.1.4

Multiply $\sqrt{6} (- \sqrt{2})$ .

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Step 4.1.4.1

Combine using the product rule for radicals.

$\frac{16 - 4 \sqrt{2} + 4 \sqrt{6} - \sqrt{6 \cdot 2}}{14}$

Step 4.1.4.2

Multiply $6$ by $2$ .

$\frac{16 - 4 \sqrt{2} + 4 \sqrt{6} - \sqrt{12}}{14}$

Step 4.1.5

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.1.5.1

Factor $4$ out of $12$ .

$\frac{16 - 4 \sqrt{2} + 4 \sqrt{6} - \sqrt{4 (3)}}{14}$

Step 4.1.5.2

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

$\frac{16 - 4 \sqrt{2} + 4 \sqrt{6} - \sqrt{2^{2} \cdot 3}}{14}$

Step 4.1.6

Pull terms out from under the radical.

$\frac{16 - 4 \sqrt{2} + 4 \sqrt{6} - (2 \sqrt{3})}{14}$

Step 4.1.7

Multiply $2$ by $- 1$ .

$\frac{16 - 4 \sqrt{2} + 4 \sqrt{6} - 2 \sqrt{3}}{14}$

Step 4.2

Cancel the common factor of $16 - 4 \sqrt{2} + 4 \sqrt{6} - 2 \sqrt{3}$ and $14$ .

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Step 4.2.1

Factor $2$ out of $16$ .

$\frac{2 (8) - 4 \sqrt{2} + 4 \sqrt{6} - 2 \sqrt{3}}{14}$

Step 4.2.2

Factor $2$ out of $- 4 \sqrt{2}$ .

$\frac{2 (8) + 2 (- 2 \sqrt{2}) + 4 \sqrt{6} - 2 \sqrt{3}}{14}$

Step 4.2.3

Factor $2$ out of $2 (8) + 2 (- 2 \sqrt{2})$ .

$\frac{2 (8 - 2 \sqrt{2}) + 4 \sqrt{6} - 2 \sqrt{3}}{14}$

Step 4.2.4

Factor $2$ out of $4 \sqrt{6}$ .

$\frac{2 (8 - 2 \sqrt{2}) + 2 (2 \sqrt{6}) - 2 \sqrt{3}}{14}$

Step 4.2.5

Factor $2$ out of $2 (8 - 2 \sqrt{2}) + 2 (2 \sqrt{6})$ .

$\frac{2 (8 - 2 \sqrt{2} + 2 \sqrt{6}) - 2 \sqrt{3}}{14}$

Step 4.2.6

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

$\frac{2 (8 - 2 \sqrt{2} + 2 \sqrt{6}) + 2 (- \sqrt{3})}{14}$

Step 4.2.7

Factor $2$ out of $2 (8 - 2 \sqrt{2} + 2 \sqrt{6}) + 2 (- \sqrt{3})$ .

$\frac{2 (8 - 2 \sqrt{2} + 2 \sqrt{6} - \sqrt{3})}{14}$

Step 4.2.8

Cancel the common factors.

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Step 4.2.8.1

Factor $2$ out of $14$ .

$\frac{2 (8 - 2 \sqrt{2} + 2 \sqrt{6} - \sqrt{3})}{2 \cdot 7}$

Step 4.2.8.2

Cancel the common factor.

$\frac{2 (8 - 2 \sqrt{2} + 2 \sqrt{6} - \sqrt{3})}{2 \cdot 7}$

Step 4.2.8.3

Rewrite the expression.

$\frac{8 - 2 \sqrt{2} + 2 \sqrt{6} - \sqrt{3}}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{8 - 2 \sqrt{2} + 2 \sqrt{6} - \sqrt{3}}{7}$

Decimal Form:

$1.19121450 \dots$