Trigonometry Examples

$\frac{4 \sqrt{3}}{3 \sqrt{2} - 5 \sqrt{5}}$

Step 1

Multiply $\frac{4 \sqrt{3}}{3 \sqrt{2} - 5 \sqrt{5}}$ by $\frac{3 \sqrt{2} + 5 \sqrt{5}}{3 \sqrt{2} + 5 \sqrt{5}}$ .

$\frac{4 \sqrt{3}}{3 \sqrt{2} - 5 \sqrt{5}} \cdot \frac{3 \sqrt{2} + 5 \sqrt{5}}{3 \sqrt{2} + 5 \sqrt{5}}$

Step 2

Multiply $\frac{4 \sqrt{3}}{3 \sqrt{2} - 5 \sqrt{5}}$ by $\frac{3 \sqrt{2} + 5 \sqrt{5}}{3 \sqrt{2} + 5 \sqrt{5}}$ .

$\frac{4 \sqrt{3} (3 \sqrt{2} + 5 \sqrt{5})}{(3 \sqrt{2} - 5 \sqrt{5}) (3 \sqrt{2} + 5 \sqrt{5})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{4 \sqrt{3} (3 \sqrt{2} + 5 \sqrt{5})}{9 {\sqrt{2}}^{2} + 15 \sqrt{10} - 15 \sqrt{10} - 25 {\sqrt{5}}^{2}}$

Step 4

Simplify.

$\frac{4 \sqrt{3} (3 \sqrt{2} + 5 \sqrt{5})}{- 107}$

Step 5

Group $3 \sqrt{2} + 5 \sqrt{5}$ and $\sqrt{3}$ together.

$\frac{4 ((3 \sqrt{2} + 5 \sqrt{5}) \sqrt{3})}{- 107}$

Step 6

Apply the distributive property.

$\frac{4 (3 \sqrt{2} \sqrt{3} + 5 \sqrt{5} \sqrt{3})}{- 107}$

Step 7

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

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Step 7.1

Combine using the product rule for radicals.

$\frac{4 (3 \sqrt{3 \cdot 2} + 5 \sqrt{5} \sqrt{3})}{- 107}$

Step 7.2

Multiply $3$ by $2$ .

$\frac{4 (3 \sqrt{6} + 5 \sqrt{5} \sqrt{3})}{- 107}$

Step 8

Multiply $5 \sqrt{5} \sqrt{3}$ .

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Step 8.1

Combine using the product rule for radicals.

$\frac{4 (3 \sqrt{6} + 5 \sqrt{3 \cdot 5})}{- 107}$

Step 8.2

Multiply $3$ by $5$ .

$\frac{4 (3 \sqrt{6} + 5 \sqrt{15})}{- 107}$

Step 9

Move the negative in front of the fraction.

$- \frac{4 (3 \sqrt{6} + 5 \sqrt{15})}{107}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{4 (3 \sqrt{6} + 5 \sqrt{15})}{107}$

Decimal Form:

$- 0.99863125 \dots$