Trigonometry Examples

$\frac{5 + \sqrt{13}}{4 - \sqrt{13}}$

Step 1

Multiply $\frac{5 + \sqrt{13}}{4 - \sqrt{13}}$ by $\frac{4 + \sqrt{13}}{4 + \sqrt{13}}$ .

$\frac{5 + \sqrt{13}}{4 - \sqrt{13}} \cdot \frac{4 + \sqrt{13}}{4 + \sqrt{13}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{5 + \sqrt{13}}{4 - \sqrt{13}}$ by $\frac{4 + \sqrt{13}}{4 + \sqrt{13}}$ .

$\frac{(5 + \sqrt{13}) (4 + \sqrt{13})}{(4 - \sqrt{13}) (4 + \sqrt{13})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(5 + \sqrt{13}) (4 + \sqrt{13})}{16 + 4 \sqrt{13} - 4 \sqrt{13} - {\sqrt{13}}^{2}}$

Step 2.3

Simplify.

$\frac{(5 + \sqrt{13}) (4 + \sqrt{13})}{3}$

Step 3

Expand $(5 + \sqrt{13}) (4 + \sqrt{13})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{5 (4 + \sqrt{13}) + \sqrt{13} (4 + \sqrt{13})}{3}$

Step 3.2

Apply the distributive property.

$\frac{5 \cdot 4 + 5 \sqrt{13} + \sqrt{13} (4 + \sqrt{13})}{3}$

Step 3.3

Apply the distributive property.

$\frac{5 \cdot 4 + 5 \sqrt{13} + \sqrt{13} \cdot 4 + \sqrt{13} \sqrt{13}}{3}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $5$ by $4$ .

$\frac{20 + 5 \sqrt{13} + \sqrt{13} \cdot 4 + \sqrt{13} \sqrt{13}}{3}$

Step 4.1.2

Move $4$ to the left of $\sqrt{13}$ .

$\frac{20 + 5 \sqrt{13} + 4 \cdot \sqrt{13} + \sqrt{13} \sqrt{13}}{3}$

Step 4.1.3

Combine using the product rule for radicals.

$\frac{20 + 5 \sqrt{13} + 4 \sqrt{13} + \sqrt{13 \cdot 13}}{3}$

Step 4.1.4

Multiply $13$ by $13$ .

$\frac{20 + 5 \sqrt{13} + 4 \sqrt{13} + \sqrt{169}}{3}$

Step 4.1.5

Rewrite $169$ as $13^{2}$ .

$\frac{20 + 5 \sqrt{13} + 4 \sqrt{13} + \sqrt{13^{2}}}{3}$

Step 4.1.6

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

$\frac{20 + 5 \sqrt{13} + 4 \sqrt{13} + 13}{3}$

Step 4.2

Add $20$ and $13$ .

$\frac{33 + 5 \sqrt{13} + 4 \sqrt{13}}{3}$

Step 4.3

Add $5 \sqrt{13}$ and $4 \sqrt{13}$ .

$\frac{33 + 9 \sqrt{13}}{3}$

Step 5

Cancel the common factor of $33 + 9 \sqrt{13}$ and $3$ .

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Step 5.1

Factor $3$ out of $33$ .

$\frac{3 \cdot 11 + 9 \sqrt{13}}{3}$

Step 5.2

Factor $3$ out of $9 \sqrt{13}$ .

$\frac{3 \cdot 11 + 3 (3 \sqrt{13})}{3}$

Step 5.3

Factor $3$ out of $3 (11) + 3 (3 \sqrt{13})$ .

$\frac{3 (11 + 3 \sqrt{13})}{3}$

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $3$ out of $3$ .

$\frac{3 (11 + 3 \sqrt{13})}{3 (1)}$

Step 5.4.2

Cancel the common factor.

$\frac{3 (11 + 3 \sqrt{13})}{3 \cdot 1}$

Step 5.4.3

Rewrite the expression.

$\frac{11 + 3 \sqrt{13}}{1}$

Step 5.4.4

Divide $11 + 3 \sqrt{13}$ by $1$ .

$11 + 3 \sqrt{13}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$11 + 3 \sqrt{13}$

Decimal Form:

$21.81665382 \dots$