Trigonometry Examples

$\frac{\sqrt{3} - \sqrt{5}}{\sqrt{15} - \sqrt{14}}$

Step 1

Multiply $\frac{\sqrt{3} - \sqrt{5}}{\sqrt{15} - \sqrt{14}}$ by $\frac{\sqrt{15} + \sqrt{14}}{\sqrt{15} + \sqrt{14}}$ .

$\frac{\sqrt{3} - \sqrt{5}}{\sqrt{15} - \sqrt{14}} \cdot \frac{\sqrt{15} + \sqrt{14}}{\sqrt{15} + \sqrt{14}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{3} - \sqrt{5}}{\sqrt{15} - \sqrt{14}}$ by $\frac{\sqrt{15} + \sqrt{14}}{\sqrt{15} + \sqrt{14}}$ .

$\frac{(\sqrt{3} - \sqrt{5}) (\sqrt{15} + \sqrt{14})}{(\sqrt{15} - \sqrt{14}) (\sqrt{15} + \sqrt{14})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{3} - \sqrt{5}) (\sqrt{15} + \sqrt{14})}{{\sqrt{15}}^{2} + \sqrt{210} - \sqrt{210} - {\sqrt{14}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{3} - \sqrt{5}) (\sqrt{15} + \sqrt{14})}{1}$

Step 2.4

Divide $(\sqrt{3} - \sqrt{5}) (\sqrt{15} + \sqrt{14})$ by $1$ .

$(\sqrt{3} - \sqrt{5}) (\sqrt{15} + \sqrt{14})$

Step 3

Expand $(\sqrt{3} - \sqrt{5}) (\sqrt{15} + \sqrt{14})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\sqrt{3} (\sqrt{15} + \sqrt{14}) - \sqrt{5} (\sqrt{15} + \sqrt{14})$

Step 3.2

Apply the distributive property.

$\sqrt{3} \sqrt{15} + \sqrt{3} \sqrt{14} - \sqrt{5} (\sqrt{15} + \sqrt{14})$

Step 3.3

Apply the distributive property.

$\sqrt{3} \sqrt{15} + \sqrt{3} \sqrt{14} - \sqrt{5} \sqrt{15} - \sqrt{5} \sqrt{14}$

Step 4

Simplify each term.

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Step 4.1

Combine using the product rule for radicals.

$\sqrt{3 \cdot 15} + \sqrt{3} \sqrt{14} - \sqrt{5} \sqrt{15} - \sqrt{5} \sqrt{14}$

Step 4.2

Multiply $3$ by $15$ .

$\sqrt{45} + \sqrt{3} \sqrt{14} - \sqrt{5} \sqrt{15} - \sqrt{5} \sqrt{14}$

Step 4.3

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 4.3.1

Factor $9$ out of $45$ .

$\sqrt{9 (5)} + \sqrt{3} \sqrt{14} - \sqrt{5} \sqrt{15} - \sqrt{5} \sqrt{14}$

Step 4.3.2

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} \cdot 5} + \sqrt{3} \sqrt{14} - \sqrt{5} \sqrt{15} - \sqrt{5} \sqrt{14}$

Step 4.4

Pull terms out from under the radical.

$3 \sqrt{5} + \sqrt{3} \sqrt{14} - \sqrt{5} \sqrt{15} - \sqrt{5} \sqrt{14}$

Step 4.5

Combine using the product rule for radicals.

$3 \sqrt{5} + \sqrt{3 \cdot 14} - \sqrt{5} \sqrt{15} - \sqrt{5} \sqrt{14}$

Step 4.6

Multiply $3$ by $14$ .

$3 \sqrt{5} + \sqrt{42} - \sqrt{5} \sqrt{15} - \sqrt{5} \sqrt{14}$

Step 4.7

Multiply $- \sqrt{5} \sqrt{15}$ .

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Step 4.7.1

Combine using the product rule for radicals.

$3 \sqrt{5} + \sqrt{42} - \sqrt{15 \cdot 5} - \sqrt{5} \sqrt{14}$

Step 4.7.2

Multiply $15$ by $5$ .

$3 \sqrt{5} + \sqrt{42} - \sqrt{75} - \sqrt{5} \sqrt{14}$

Step 4.8

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 4.8.1

Factor $25$ out of $75$ .

$3 \sqrt{5} + \sqrt{42} - \sqrt{25 (3)} - \sqrt{5} \sqrt{14}$

Step 4.8.2

Rewrite $25$ as $5^{2}$ .

$3 \sqrt{5} + \sqrt{42} - \sqrt{5^{2} \cdot 3} - \sqrt{5} \sqrt{14}$

Step 4.9

Pull terms out from under the radical.

$3 \sqrt{5} + \sqrt{42} - (5 \sqrt{3}) - \sqrt{5} \sqrt{14}$

Step 4.10

Multiply $5$ by $- 1$ .

$3 \sqrt{5} + \sqrt{42} - 5 \sqrt{3} - \sqrt{5} \sqrt{14}$

Step 4.11

Multiply $- \sqrt{5} \sqrt{14}$ .

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Step 4.11.1

Combine using the product rule for radicals.

$3 \sqrt{5} + \sqrt{42} - 5 \sqrt{3} - \sqrt{14 \cdot 5}$

Step 4.11.2

Multiply $14$ by $5$ .

$3 \sqrt{5} + \sqrt{42} - 5 \sqrt{3} - \sqrt{70}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{5} + \sqrt{42} - 5 \sqrt{3} - \sqrt{70}$

Decimal Form:

$- 3.83790967 \dots$