Trigonometry Examples

$\frac{7 + \sqrt{5}}{3 + \sqrt{5}}$

Step 1

Multiply $\frac{7 + \sqrt{5}}{3 + \sqrt{5}}$ by $\frac{3 - \sqrt{5}}{3 - \sqrt{5}}$ .

$\frac{7 + \sqrt{5}}{3 + \sqrt{5}} \cdot \frac{3 - \sqrt{5}}{3 - \sqrt{5}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{7 + \sqrt{5}}{3 + \sqrt{5}}$ by $\frac{3 - \sqrt{5}}{3 - \sqrt{5}}$ .

$\frac{(7 + \sqrt{5}) (3 - \sqrt{5})}{(3 + \sqrt{5}) (3 - \sqrt{5})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(7 + \sqrt{5}) (3 - \sqrt{5})}{9 - 3 \sqrt{5} + \sqrt{5} \cdot 3 - {\sqrt{5}}^{2}}$

Step 2.3

Simplify.

$\frac{(7 + \sqrt{5}) (3 - \sqrt{5})}{4}$

Step 3

Expand $(7 + \sqrt{5}) (3 - \sqrt{5})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{7 (3 - \sqrt{5}) + \sqrt{5} (3 - \sqrt{5})}{4}$

Step 3.2

Apply the distributive property.

$\frac{7 \cdot 3 + 7 (- \sqrt{5}) + \sqrt{5} (3 - \sqrt{5})}{4}$

Step 3.3

Apply the distributive property.

$\frac{7 \cdot 3 + 7 (- \sqrt{5}) + \sqrt{5} \cdot 3 + \sqrt{5} (- \sqrt{5})}{4}$

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 $7$ by $3$ .

$\frac{21 + 7 (- \sqrt{5}) + \sqrt{5} \cdot 3 + \sqrt{5} (- \sqrt{5})}{4}$

Step 4.1.2

Multiply $- 1$ by $7$ .

$\frac{21 - 7 \sqrt{5} + \sqrt{5} \cdot 3 + \sqrt{5} (- \sqrt{5})}{4}$

Step 4.1.3

Move $3$ to the left of $\sqrt{5}$ .

$\frac{21 - 7 \sqrt{5} + 3 \cdot \sqrt{5} + \sqrt{5} (- \sqrt{5})}{4}$

Step 4.1.4

Multiply $\sqrt{5} (- \sqrt{5})$ .

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Step 4.1.4.1

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{21 - 7 \sqrt{5} + 3 \sqrt{5} - ({\sqrt{5}}^{1} \sqrt{5})}{4}$

Step 4.1.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{21 - 7 \sqrt{5} + 3 \sqrt{5} - ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}{4}$

Step 4.1.4.3

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

$\frac{21 - 7 \sqrt{5} + 3 \sqrt{5} - {\sqrt{5}}^{1 + 1}}{4}$

Step 4.1.4.4

Add $1$ and $1$ .

$\frac{21 - 7 \sqrt{5} + 3 \sqrt{5} - {\sqrt{5}}^{2}}{4}$

Step 4.1.5

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{21 - 7 \sqrt{5} + 3 \sqrt{5} - {(5^{\frac{1}{2}})}^{2}}{4}$

Step 4.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{21 - 7 \sqrt{5} + 3 \sqrt{5} - 5^{\frac{1}{2} \cdot 2}}{4}$

Step 4.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{21 - 7 \sqrt{5} + 3 \sqrt{5} - 5^{\frac{2}{2}}}{4}$

Step 4.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.5.4.1

Cancel the common factor.

$\frac{21 - 7 \sqrt{5} + 3 \sqrt{5} - 5^{\frac{2}{2}}}{4}$

Step 4.1.5.4.2

Rewrite the expression.

$\frac{21 - 7 \sqrt{5} + 3 \sqrt{5} - 5^{1}}{4}$

Step 4.1.5.5

Evaluate the exponent.

$\frac{21 - 7 \sqrt{5} + 3 \sqrt{5} - 1 \cdot 5}{4}$

Step 4.1.6

Multiply $- 1$ by $5$ .

$\frac{21 - 7 \sqrt{5} + 3 \sqrt{5} - 5}{4}$

Step 4.2

Subtract $5$ from $21$ .

$\frac{16 - 7 \sqrt{5} + 3 \sqrt{5}}{4}$

Step 4.3

Add $- 7 \sqrt{5}$ and $3 \sqrt{5}$ .

$\frac{16 - 4 \sqrt{5}}{4}$

Step 5

Cancel the common factor of $16 - 4 \sqrt{5}$ and $4$ .

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Step 5.1

Factor $4$ out of $16$ .

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

Step 5.2

Factor $4$ out of $- 4 \sqrt{5}$ .

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

Step 5.3

Factor $4$ out of $4 (4) + 4 (- \sqrt{5})$ .

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

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $4$ out of $4$ .

$\frac{4 (4 - \sqrt{5})}{4 (1)}$

Step 5.4.2

Cancel the common factor.

$\frac{4 (4 - \sqrt{5})}{4 \cdot 1}$

Step 5.4.3

Rewrite the expression.

$\frac{4 - \sqrt{5}}{1}$

Step 5.4.4

Divide $4 - \sqrt{5}$ by $1$ .

$4 - \sqrt{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$4 - \sqrt{5}$

Decimal Form:

$1.76393202 \dots$