Trigonometry Examples

$\frac{\sqrt{125} - \sqrt{6}}{\sqrt{30} - 5}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 1.1.1

Factor $25$ out of $125$ .

$\frac{\sqrt{25 (5)} - \sqrt{6}}{\sqrt{30} - 5}$

Step 1.1.2

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

$\frac{\sqrt{5^{2} \cdot 5} - \sqrt{6}}{\sqrt{30} - 5}$

Step 1.2

Pull terms out from under the radical.

$\frac{5 \sqrt{5} - \sqrt{6}}{\sqrt{30} - 5}$

Step 2

Multiply $\frac{5 \sqrt{5} - \sqrt{6}}{\sqrt{30} - 5}$ by $\frac{\sqrt{30} + 5}{\sqrt{30} + 5}$ .

$\frac{5 \sqrt{5} - \sqrt{6}}{\sqrt{30} - 5} \cdot \frac{\sqrt{30} + 5}{\sqrt{30} + 5}$

Step 3

Combine fractions.

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Step 3.1

Multiply $\frac{5 \sqrt{5} - \sqrt{6}}{\sqrt{30} - 5}$ by $\frac{\sqrt{30} + 5}{\sqrt{30} + 5}$ .

$\frac{(5 \sqrt{5} - \sqrt{6}) (\sqrt{30} + 5)}{(\sqrt{30} - 5) (\sqrt{30} + 5)}$

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(5 \sqrt{5} - \sqrt{6}) (\sqrt{30} + 5)}{{\sqrt{30}}^{2} + \sqrt{30} \cdot 5 - 5 \sqrt{30} - 25}$

Step 3.3

Simplify.

$\frac{(5 \sqrt{5} - \sqrt{6}) (\sqrt{30} + 5)}{5}$

Step 4

Expand $(5 \sqrt{5} - \sqrt{6}) (\sqrt{30} + 5)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$\frac{5 \sqrt{5} (\sqrt{30} + 5) - \sqrt{6} (\sqrt{30} + 5)}{5}$

Step 4.2

Apply the distributive property.

$\frac{5 \sqrt{5} \sqrt{30} + 5 \sqrt{5} \cdot 5 - \sqrt{6} (\sqrt{30} + 5)}{5}$

Step 4.3

Apply the distributive property.

$\frac{5 \sqrt{5} \sqrt{30} + 5 \sqrt{5} \cdot 5 - \sqrt{6} \sqrt{30} - \sqrt{6} \cdot 5}{5}$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $5 \sqrt{5} \sqrt{30}$ .

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Step 5.1.1.1

Combine using the product rule for radicals.

$\frac{5 \sqrt{30 \cdot 5} + 5 \sqrt{5} \cdot 5 - \sqrt{6} \sqrt{30} - \sqrt{6} \cdot 5}{5}$

Step 5.1.1.2

Multiply $30$ by $5$ .

$\frac{5 \sqrt{150} + 5 \sqrt{5} \cdot 5 - \sqrt{6} \sqrt{30} - \sqrt{6} \cdot 5}{5}$

Step 5.1.2

Rewrite $150$ as $5^{2} \cdot 6$ .

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Step 5.1.2.1

Factor $25$ out of $150$ .

$\frac{5 \sqrt{25 (6)} + 5 \sqrt{5} \cdot 5 - \sqrt{6} \sqrt{30} - \sqrt{6} \cdot 5}{5}$

Step 5.1.2.2

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

$\frac{5 \sqrt{5^{2} \cdot 6} + 5 \sqrt{5} \cdot 5 - \sqrt{6} \sqrt{30} - \sqrt{6} \cdot 5}{5}$

Step 5.1.3

Pull terms out from under the radical.

$\frac{5 (5 \sqrt{6}) + 5 \sqrt{5} \cdot 5 - \sqrt{6} \sqrt{30} - \sqrt{6} \cdot 5}{5}$

Step 5.1.4

Multiply $5$ by $5$ .

$\frac{25 \sqrt{6} + 5 \sqrt{5} \cdot 5 - \sqrt{6} \sqrt{30} - \sqrt{6} \cdot 5}{5}$

Step 5.1.5

Multiply $5$ by $5$ .

$\frac{25 \sqrt{6} + 25 \sqrt{5} - \sqrt{6} \sqrt{30} - \sqrt{6} \cdot 5}{5}$

Step 5.1.6

Multiply $- \sqrt{6} \sqrt{30}$ .

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Step 5.1.6.1

Combine using the product rule for radicals.

$\frac{25 \sqrt{6} + 25 \sqrt{5} - \sqrt{30 \cdot 6} - \sqrt{6} \cdot 5}{5}$

Step 5.1.6.2

Multiply $30$ by $6$ .

$\frac{25 \sqrt{6} + 25 \sqrt{5} - \sqrt{180} - \sqrt{6} \cdot 5}{5}$

Step 5.1.7

Rewrite $180$ as $6^{2} \cdot 5$ .

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Step 5.1.7.1

Factor $36$ out of $180$ .

$\frac{25 \sqrt{6} + 25 \sqrt{5} - \sqrt{36 (5)} - \sqrt{6} \cdot 5}{5}$

Step 5.1.7.2

Rewrite $36$ as $6^{2}$ .

$\frac{25 \sqrt{6} + 25 \sqrt{5} - \sqrt{6^{2} \cdot 5} - \sqrt{6} \cdot 5}{5}$

Step 5.1.8

Pull terms out from under the radical.

$\frac{25 \sqrt{6} + 25 \sqrt{5} - (6 \sqrt{5}) - \sqrt{6} \cdot 5}{5}$

Step 5.1.9

Multiply $6$ by $- 1$ .

$\frac{25 \sqrt{6} + 25 \sqrt{5} - 6 \sqrt{5} - \sqrt{6} \cdot 5}{5}$

Step 5.1.10

Multiply $5$ by $- 1$ .

$\frac{25 \sqrt{6} + 25 \sqrt{5} - 6 \sqrt{5} - 5 \sqrt{6}}{5}$

Step 5.2

Subtract $5 \sqrt{6}$ from $25 \sqrt{6}$ .

$\frac{20 \sqrt{6} + 25 \sqrt{5} - 6 \sqrt{5}}{5}$

Step 5.3

Subtract $6 \sqrt{5}$ from $25 \sqrt{5}$ .

$\frac{20 \sqrt{6} + 19 \sqrt{5}}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{20 \sqrt{6} + 19 \sqrt{5}}{5}$

Decimal Form:

$18.29501728 \dots$