Trigonometry Examples

$\sqrt{\frac{1 - \frac{2}{\sqrt{7}}}{1 + \frac{2}{\sqrt{7}}}}$

Step 1

Multiply $\frac{2}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sqrt{\frac{1 - (\frac{2}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})}{1 + \frac{2}{\sqrt{7}}}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{2}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{7}}{\sqrt{7} \sqrt{7}}}{1 + \frac{2}{\sqrt{7}}}}$

Step 2.2

Raise $\sqrt{7}$ to the power of $1$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}}{1 + \frac{2}{\sqrt{7}}}}$

Step 2.3

Raise $\sqrt{7}$ to the power of $1$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}}{1 + \frac{2}{\sqrt{7}}}}$

Step 2.4

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

$\sqrt{\frac{1 - \frac{2 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}}{1 + \frac{2}{\sqrt{7}}}}$

Step 2.5

Add $1$ and $1$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{7}}{{\sqrt{7}}^{2}}}{1 + \frac{2}{\sqrt{7}}}}$

Step 2.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{\frac{1 - \frac{2 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}}{1 + \frac{2}{\sqrt{7}}}}$

Step 2.6.2

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

$\sqrt{\frac{1 - \frac{2 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}}{1 + \frac{2}{\sqrt{7}}}}$

Step 2.6.3

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

$\sqrt{\frac{1 - \frac{2 \sqrt{7}}{7^{\frac{2}{2}}}}{1 + \frac{2}{\sqrt{7}}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{1 - \frac{2 \sqrt{7}}{7^{\frac{2}{2}}}}{1 + \frac{2}{\sqrt{7}}}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{1 - \frac{2 \sqrt{7}}{7^{1}}}{1 + \frac{2}{\sqrt{7}}}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{1 - \frac{2 \sqrt{7}}{7}}{1 + \frac{2}{\sqrt{7}}}}$

Step 3

Simplify the expression.

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Step 3.1

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{7}{7} - \frac{2 \sqrt{7}}{7}}{1 + \frac{2}{\sqrt{7}}}}$

Step 3.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{7 - 2 \sqrt{7}}{7}}{1 + \frac{2}{\sqrt{7}}}}$

Step 3.3

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{7 - 2 \sqrt{7}}{7}}{\frac{\sqrt{7}}{\sqrt{7}} + \frac{2}{\sqrt{7}}}}$

Step 3.4

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{7 - 2 \sqrt{7}}{7}}{\frac{\sqrt{7} + 2}{\sqrt{7}}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{7 - 2 \sqrt{7}}{7} \cdot \frac{\sqrt{7}}{\sqrt{7} + 2}}$

Step 5

Multiply $\frac{\sqrt{7}}{\sqrt{7} + 2}$ by $\frac{\sqrt{7} - 2}{\sqrt{7} - 2}$ .

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

Step 6

Simplify terms.

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Step 6.1

Multiply $\frac{\sqrt{7}}{\sqrt{7} + 2}$ by $\frac{\sqrt{7} - 2}{\sqrt{7} - 2}$ .

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

Step 6.2

Expand the denominator using the FOIL method.

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

Step 6.3

Simplify.

$\sqrt{\frac{7 - 2 \sqrt{7}}{7} \cdot \frac{\sqrt{7} (\sqrt{7} - 2)}{3}}$

Step 6.4

Apply the distributive property.

$\sqrt{\frac{7 - 2 \sqrt{7}}{7} \cdot \frac{\sqrt{7} \sqrt{7} + \sqrt{7} \cdot - 2}{3}}$

Step 6.5

Combine using the product rule for radicals.

$\sqrt{\frac{7 - 2 \sqrt{7}}{7} \cdot \frac{\sqrt{7 \cdot 7} + \sqrt{7} \cdot - 2}{3}}$

Step 6.6

Move $- 2$ to the left of $\sqrt{7}$ .

$\sqrt{\frac{7 - 2 \sqrt{7}}{7} \cdot \frac{\sqrt{7 \cdot 7} - 2 \cdot \sqrt{7}}{3}}$

Step 7

Simplify each term.

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Step 7.1

Multiply $7$ by $7$ .

$\sqrt{\frac{7 - 2 \sqrt{7}}{7} \cdot \frac{\sqrt{49} - 2 \cdot \sqrt{7}}{3}}$

Step 7.2

Rewrite $49$ as $7^{2}$ .

$\sqrt{\frac{7 - 2 \sqrt{7}}{7} \cdot \frac{\sqrt{7^{2}} - 2 \cdot \sqrt{7}}{3}}$

Step 7.3

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

$\sqrt{\frac{7 - 2 \sqrt{7}}{7} \cdot \frac{7 - 2 \sqrt{7}}{3}}$

Step 8

Multiply $\frac{7 - 2 \sqrt{7}}{7} \cdot \frac{7 - 2 \sqrt{7}}{3}$ .

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Step 8.1

Multiply $\frac{7 - 2 \sqrt{7}}{7}$ by $\frac{7 - 2 \sqrt{7}}{3}$ .

$\sqrt{\frac{(7 - 2 \sqrt{7}) (7 - 2 \sqrt{7})}{7 \cdot 3}}$

Step 8.2

Raise $7 - 2 \sqrt{7}$ to the power of $1$ .

$\sqrt{\frac{{(7 - 2 \sqrt{7})}^{1} (7 - 2 \sqrt{7})}{7 \cdot 3}}$

Step 8.3

Raise $7 - 2 \sqrt{7}$ to the power of $1$ .

$\sqrt{\frac{{(7 - 2 \sqrt{7})}^{1} {(7 - 2 \sqrt{7})}^{1}}{7 \cdot 3}}$

Step 8.4

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

$\sqrt{\frac{{(7 - 2 \sqrt{7})}^{1 + 1}}{7 \cdot 3}}$

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

Multiply $7$ by $3$ .

$\sqrt{\frac{{(7 - 2 \sqrt{7})}^{2}}{21}}$

Step 9

Rewrite ${(7 - 2 \sqrt{7})}^{2}$ as $(7 - 2 \sqrt{7}) (7 - 2 \sqrt{7})$ .

$\sqrt{\frac{(7 - 2 \sqrt{7}) (7 - 2 \sqrt{7})}{21}}$

Step 10

Expand $(7 - 2 \sqrt{7}) (7 - 2 \sqrt{7})$ using the FOIL Method.

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Step 10.1

Apply the distributive property.

$\sqrt{\frac{7 (7 - 2 \sqrt{7}) - 2 \sqrt{7} (7 - 2 \sqrt{7})}{21}}$

Step 10.2

Apply the distributive property.

$\sqrt{\frac{7 \cdot 7 + 7 (- 2 \sqrt{7}) - 2 \sqrt{7} (7 - 2 \sqrt{7})}{21}}$

Step 10.3

Apply the distributive property.

$\sqrt{\frac{7 \cdot 7 + 7 (- 2 \sqrt{7}) - 2 \sqrt{7} \cdot 7 - 2 \sqrt{7} (- 2 \sqrt{7})}{21}}$

Step 11

Simplify and combine like terms.

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Step 11.1

Simplify each term.

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Step 11.1.1

Multiply $7$ by $7$ .

$\sqrt{\frac{49 + 7 (- 2 \sqrt{7}) - 2 \sqrt{7} \cdot 7 - 2 \sqrt{7} (- 2 \sqrt{7})}{21}}$

Step 11.1.2

Multiply $- 2$ by $7$ .

$\sqrt{\frac{49 - 14 \sqrt{7} - 2 \sqrt{7} \cdot 7 - 2 \sqrt{7} (- 2 \sqrt{7})}{21}}$

Step 11.1.3

Multiply $7$ by $- 2$ .

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} - 2 \sqrt{7} (- 2 \sqrt{7})}{21}}$

Step 11.1.4

Multiply $- 2 \sqrt{7} (- 2 \sqrt{7})$ .

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Step 11.1.4.1

Multiply $- 2$ by $- 2$ .

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} + 4 \sqrt{7} \sqrt{7}}{21}}$

Step 11.1.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} + 4 ({\sqrt{7}}^{1} \sqrt{7})}{21}}$

Step 11.1.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} + 4 ({\sqrt{7}}^{1} {\sqrt{7}}^{1})}{21}}$

Step 11.1.4.4

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

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} + 4 {\sqrt{7}}^{1 + 1}}{21}}$

Step 11.1.4.5

Add $1$ and $1$ .

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} + 4 {\sqrt{7}}^{2}}{21}}$

Step 11.1.5

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 11.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} + 4 {(7^{\frac{1}{2}})}^{2}}{21}}$

Step 11.1.5.2

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

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} + 4 \cdot 7^{\frac{1}{2} \cdot 2}}{21}}$

Step 11.1.5.3

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

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} + 4 \cdot 7^{\frac{2}{2}}}{21}}$

Step 11.1.5.4

Cancel the common factor of $2$ .

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Step 11.1.5.4.1

Cancel the common factor.

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} + 4 \cdot 7^{\frac{2}{2}}}{21}}$

Step 11.1.5.4.2

Rewrite the expression.

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} + 4 \cdot 7^{1}}{21}}$

Step 11.1.5.5

Evaluate the exponent.

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} + 4 \cdot 7}{21}}$

Step 11.1.6

Multiply $4$ by $7$ .

$\sqrt{\frac{49 - 14 \sqrt{7} - 14 \sqrt{7} + 28}{21}}$

Step 11.2

Add $49$ and $28$ .

$\sqrt{\frac{77 - 14 \sqrt{7} - 14 \sqrt{7}}{21}}$

Step 11.3

Subtract $14 \sqrt{7}$ from $- 14 \sqrt{7}$ .

$\sqrt{\frac{77 - 28 \sqrt{7}}{21}}$

Step 12

Cancel the common factor of $77 - 28 \sqrt{7}$ and $21$ .

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Step 12.1

Factor $7$ out of $77$ .

$\sqrt{\frac{7 (11) - 28 \sqrt{7}}{21}}$

Step 12.2

Factor $7$ out of $- 28 \sqrt{7}$ .

$\sqrt{\frac{7 (11) + 7 (- 4 \sqrt{7})}{21}}$

Step 12.3

Factor $7$ out of $7 (11) + 7 (- 4 \sqrt{7})$ .

$\sqrt{\frac{7 (11 - 4 \sqrt{7})}{21}}$

Step 12.4

Cancel the common factors.

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Step 12.4.1

Factor $7$ out of $21$ .

$\sqrt{\frac{7 (11 - 4 \sqrt{7})}{7 \cdot 3}}$

Step 12.4.2

Cancel the common factor.

$\sqrt{\frac{7 (11 - 4 \sqrt{7})}{7 \cdot 3}}$

Step 12.4.3

Rewrite the expression.

$\sqrt{\frac{11 - 4 \sqrt{7}}{3}}$

Step 13

Rewrite $\sqrt{\frac{11 - 4 \sqrt{7}}{3}}$ as $\frac{\sqrt{11 - 4 \sqrt{7}}}{\sqrt{3}}$ .

$\frac{\sqrt{11 - 4 \sqrt{7}}}{\sqrt{3}}$

Step 14

Multiply $\frac{\sqrt{11 - 4 \sqrt{7}}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{11 - 4 \sqrt{7}}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 15

Combine and simplify the denominator.

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Step 15.1

Multiply $\frac{\sqrt{11 - 4 \sqrt{7}}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{11 - 4 \sqrt{7}} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 15.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{11 - 4 \sqrt{7}} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 15.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{11 - 4 \sqrt{7}} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 15.4

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

$\frac{\sqrt{11 - 4 \sqrt{7}} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 15.5

Add $1$ and $1$ .

$\frac{\sqrt{11 - 4 \sqrt{7}} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 15.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 15.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{11 - 4 \sqrt{7}} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 15.6.2

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

$\frac{\sqrt{11 - 4 \sqrt{7}} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 15.6.3

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

$\frac{\sqrt{11 - 4 \sqrt{7}} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 15.6.4

Cancel the common factor of $2$ .

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Step 15.6.4.1

Cancel the common factor.

$\frac{\sqrt{11 - 4 \sqrt{7}} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 15.6.4.2

Rewrite the expression.

$\frac{\sqrt{11 - 4 \sqrt{7}} \sqrt{3}}{3^{1}}$

Step 15.6.5

Evaluate the exponent.

$\frac{\sqrt{11 - 4 \sqrt{7}} \sqrt{3}}{3}$

Step 16

Combine using the product rule for radicals.

$\frac{\sqrt{(11 - 4 \sqrt{7}) \cdot 3}}{3}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(11 - 4 \sqrt{7}) \cdot 3}}{3}$

Decimal Form:

$0.37282469 \dots$