Trigonometry Examples

$\frac{4 + \sqrt{2}}{4 - \sqrt{2}}$

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

$\frac{(4 + \sqrt{2}) (4 + \sqrt{2})}{14}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $4 + \sqrt{2}$ to the power of $1$ .

$\frac{{(4 + \sqrt{2})}^{1} (4 + \sqrt{2})}{14}$

Step 3.2

Raise $4 + \sqrt{2}$ to the power of $1$ .

$\frac{{(4 + \sqrt{2})}^{1} {(4 + \sqrt{2})}^{1}}{14}$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(4 + \sqrt{2})}^{1 + 1}}{14}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(4 + \sqrt{2})}^{2}}{14}$

Step 4

Rewrite ${(4 + \sqrt{2})}^{2}$ as $(4 + \sqrt{2}) (4 + \sqrt{2})$ .

$\frac{(4 + \sqrt{2}) (4 + \sqrt{2})}{14}$

Step 5

Expand $(4 + \sqrt{2}) (4 + \sqrt{2})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{4 (4 + \sqrt{2}) + \sqrt{2} (4 + \sqrt{2})}{14}$

Step 5.2

Apply the distributive property.

$\frac{4 \cdot 4 + 4 \sqrt{2} + \sqrt{2} (4 + \sqrt{2})}{14}$

Step 5.3

Apply the distributive property.

$\frac{4 \cdot 4 + 4 \sqrt{2} + \sqrt{2} \cdot 4 + \sqrt{2} \sqrt{2}}{14}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $4$ by $4$ .

$\frac{16 + 4 \sqrt{2} + \sqrt{2} \cdot 4 + \sqrt{2} \sqrt{2}}{14}$

Step 6.1.2

Move $4$ to the left of $\sqrt{2}$ .

$\frac{16 + 4 \sqrt{2} + 4 \cdot \sqrt{2} + \sqrt{2} \sqrt{2}}{14}$

Step 6.1.3

Combine using the product rule for radicals.

$\frac{16 + 4 \sqrt{2} + 4 \sqrt{2} + \sqrt{2 \cdot 2}}{14}$

Step 6.1.4

Multiply $2$ by $2$ .

$\frac{16 + 4 \sqrt{2} + 4 \sqrt{2} + \sqrt{4}}{14}$

Step 6.1.5

Rewrite $4$ as $2^{2}$ .

$\frac{16 + 4 \sqrt{2} + 4 \sqrt{2} + \sqrt{2^{2}}}{14}$

Step 6.1.6

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

$\frac{16 + 4 \sqrt{2} + 4 \sqrt{2} + 2}{14}$

Step 6.2

Add $16$ and $2$ .

$\frac{18 + 4 \sqrt{2} + 4 \sqrt{2}}{14}$

Step 6.3

Add $4 \sqrt{2}$ and $4 \sqrt{2}$ .

$\frac{18 + 8 \sqrt{2}}{14}$

Step 7

Cancel the common factor of $18 + 8 \sqrt{2}$ and $14$ .

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Step 7.1

Factor $2$ out of $18$ .

$\frac{2 (9) + 8 \sqrt{2}}{14}$

Step 7.2

Factor $2$ out of $8 \sqrt{2}$ .

$\frac{2 (9) + 2 (4 \sqrt{2})}{14}$

Step 7.3

Factor $2$ out of $2 (9) + 2 (4 \sqrt{2})$ .

$\frac{2 (9 + 4 \sqrt{2})}{14}$

Step 7.4

Cancel the common factors.

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Step 7.4.1

Factor $2$ out of $14$ .

$\frac{2 (9 + 4 \sqrt{2})}{2 \cdot 7}$

Step 7.4.2

Cancel the common factor.

$\frac{2 (9 + 4 \sqrt{2})}{2 \cdot 7}$

Step 7.4.3

Rewrite the expression.

$\frac{9 + 4 \sqrt{2}}{7}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{9 + 4 \sqrt{2}}{7}$

Decimal Form:

$2.09383632 \dots$