Trigonometry Examples

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

Step 1

Multiply $\frac{6 + \sqrt{7}}{2 - \sqrt{2}}$ by $\frac{2 + \sqrt{2}}{2 + \sqrt{2}}$ .

$\frac{6 + \sqrt{7}}{2 - \sqrt{2}} \cdot \frac{2 + \sqrt{2}}{2 + \sqrt{2}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{6 + \sqrt{7}}{2 - \sqrt{2}}$ by $\frac{2 + \sqrt{2}}{2 + \sqrt{2}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(6 + \sqrt{7}) (2 + \sqrt{2})}{4 + 2 \sqrt{2} - 2 \sqrt{2} - {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

$\frac{(6 + \sqrt{7}) (2 + \sqrt{2})}{2}$

Step 3

Expand $(6 + \sqrt{7}) (2 + \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{6 (2 + \sqrt{2}) + \sqrt{7} (2 + \sqrt{2})}{2}$

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

$\frac{6 \cdot 2 + 6 \sqrt{2} + \sqrt{7} \cdot 2 + \sqrt{7} \sqrt{2}}{2}$

Step 4

Simplify each term.

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Step 4.1

Multiply $6$ by $2$ .

$\frac{12 + 6 \sqrt{2} + \sqrt{7} \cdot 2 + \sqrt{7} \sqrt{2}}{2}$

Step 4.2

Move $2$ to the left of $\sqrt{7}$ .

$\frac{12 + 6 \sqrt{2} + 2 \cdot \sqrt{7} + \sqrt{7} \sqrt{2}}{2}$

Step 4.3

Combine using the product rule for radicals.

$\frac{12 + 6 \sqrt{2} + 2 \sqrt{7} + \sqrt{7 \cdot 2}}{2}$

Step 4.4

Multiply $7$ by $2$ .

$\frac{12 + 6 \sqrt{2} + 2 \sqrt{7} + \sqrt{14}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{12 + 6 \sqrt{2} + 2 \sqrt{7} + \sqrt{14}}{2}$

Decimal Form:

$14.75922069 \dots$