Trigonometry Examples

$\frac{5 - \sqrt{3}}{8 - \sqrt{2}}$

Step 1

Multiply $\frac{5 - \sqrt{3}}{8 - \sqrt{2}}$ by $\frac{8 + \sqrt{2}}{8 + \sqrt{2}}$ .

$\frac{5 - \sqrt{3}}{8 - \sqrt{2}} \cdot \frac{8 + \sqrt{2}}{8 + \sqrt{2}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{5 - \sqrt{3}}{8 - \sqrt{2}}$ by $\frac{8 + \sqrt{2}}{8 + \sqrt{2}}$ .

$\frac{(5 - \sqrt{3}) (8 + \sqrt{2})}{(8 - \sqrt{2}) (8 + \sqrt{2})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(5 - \sqrt{3}) (8 + \sqrt{2})}{64 + 8 \sqrt{2} - 8 \sqrt{2} - {\sqrt{2}}^{2}}$

Step 2.3

Simplify.

$\frac{(5 - \sqrt{3}) (8 + \sqrt{2})}{62}$

Step 3

Expand $(5 - \sqrt{3}) (8 + \sqrt{2})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{5 (8 + \sqrt{2}) - \sqrt{3} (8 + \sqrt{2})}{62}$

Step 3.2

Apply the distributive property.

$\frac{5 \cdot 8 + 5 \sqrt{2} - \sqrt{3} (8 + \sqrt{2})}{62}$

Step 3.3

Apply the distributive property.

$\frac{5 \cdot 8 + 5 \sqrt{2} - \sqrt{3} \cdot 8 - \sqrt{3} \sqrt{2}}{62}$

Step 4

Simplify each term.

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Step 4.1

Multiply $5$ by $8$ .

$\frac{40 + 5 \sqrt{2} - \sqrt{3} \cdot 8 - \sqrt{3} \sqrt{2}}{62}$

Step 4.2

Multiply $8$ by $- 1$ .

$\frac{40 + 5 \sqrt{2} - 8 \sqrt{3} - \sqrt{3} \sqrt{2}}{62}$

Step 4.3

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

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Step 4.3.1

Combine using the product rule for radicals.

$\frac{40 + 5 \sqrt{2} - 8 \sqrt{3} - \sqrt{2 \cdot 3}}{62}$

Step 4.3.2

Multiply $2$ by $3$ .

$\frac{40 + 5 \sqrt{2} - 8 \sqrt{3} - \sqrt{6}}{62}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{40 + 5 \sqrt{2} - 8 \sqrt{3} - \sqrt{6}}{62}$

Decimal Form:

$0.49621244 \dots$