Trigonometry Examples

$\frac{3 (\sqrt{5})}{9 - \sqrt{5}}$

Step 1

Multiply $\frac{3 \sqrt{5}}{9 - \sqrt{5}}$ by $\frac{9 + \sqrt{5}}{9 + \sqrt{5}}$ .

$\frac{3 \sqrt{5}}{9 - \sqrt{5}} \cdot \frac{9 + \sqrt{5}}{9 + \sqrt{5}}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{3 \sqrt{5}}{9 - \sqrt{5}}$ by $\frac{9 + \sqrt{5}}{9 + \sqrt{5}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{3 \sqrt{5} (9 + \sqrt{5})}{81 + 9 \sqrt{5} - 9 \sqrt{5} - {\sqrt{5}}^{2}}$

Step 2.3

Simplify.

$\frac{3 \sqrt{5} (9 + \sqrt{5})}{76}$

Step 2.4

Group $9 + \sqrt{5}$ and $\sqrt{5}$ together.

$\frac{3 ((9 + \sqrt{5}) \sqrt{5})}{76}$

Step 2.5

Apply the distributive property.

$\frac{3 (9 \sqrt{5} + \sqrt{5} \sqrt{5})}{76}$

Step 2.6

Combine using the product rule for radicals.

$\frac{3 (9 \sqrt{5} + \sqrt{5 \cdot 5})}{76}$

Step 3

Simplify each term.

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Step 3.1

Multiply $5$ by $5$ .

$\frac{3 (9 \sqrt{5} + \sqrt{25})}{76}$

Step 3.2

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

$\frac{3 (9 \sqrt{5} + \sqrt{5^{2}})}{76}$

Step 3.3

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

$\frac{3 (9 \sqrt{5} + 5)}{76}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{3 (9 \sqrt{5} + 5)}{76}$

Decimal Form:

$0.99176099 \dots$