Trigonometry Examples

$\frac{2 \tan (\frac{5 π}{12})}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1

The exact value of $\tan (\frac{5 π}{12})$ is $2 + \sqrt{3}$ .

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Step 1.1

Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\frac{2 \tan (\frac{π}{6} + \frac{π}{4})}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.2

Apply the sum of angles identity.

$\frac{2 \frac{\tan (\frac{π}{6}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.3

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{2 \frac{\frac{\sqrt{3}}{3} + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.4

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{2 \frac{\frac{\sqrt{3}}{3} + 1}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.5

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{2 \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \tan (\frac{π}{4})}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.6

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{2 \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7

Simplify $\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ .

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Step 1.7.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 1.7.1.1

Multiply $\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ by $\frac{3}{3}$ .

$\frac{2 (\frac{3}{3} \cdot \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1})}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.1.2

Combine.

$\frac{2 \frac{3 (\frac{\sqrt{3}}{3} + 1)}{3 (1 - \frac{\sqrt{3}}{3} \cdot 1)}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.2

Apply the distributive property.

$\frac{2 \frac{3 \frac{\sqrt{3}}{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.3

Cancel the common factor of $3$ .

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Step 1.7.3.1

Cancel the common factor.

$\frac{2 \frac{3 \frac{\sqrt{3}}{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.3.2

Rewrite the expression.

$\frac{2 \frac{\sqrt{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.4

Multiply $3$ by $1$ .

$\frac{2 \frac{\sqrt{3} + 3}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.5

Simplify the denominator.

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Step 1.7.5.1

Multiply $3$ by $1$ .

$\frac{2 \frac{\sqrt{3} + 3}{3 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.5.2

Multiply $- 1$ by $1$ .

$\frac{2 \frac{\sqrt{3} + 3}{3 + 3 (- \frac{\sqrt{3}}{3})}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.5.3

Cancel the common factor of $3$ .

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Step 1.7.5.3.1

Move the leading negative in $- \frac{\sqrt{3}}{3}$ into the numerator.

$\frac{2 \frac{\sqrt{3} + 3}{3 + 3 \frac{- \sqrt{3}}{3}}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.5.3.2

Cancel the common factor.

$\frac{2 \frac{\sqrt{3} + 3}{3 + 3 \frac{- \sqrt{3}}{3}}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.5.3.3

Rewrite the expression.

$\frac{2 \frac{\sqrt{3} + 3}{3 - \sqrt{3}}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.6

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

$\frac{2 (\frac{\sqrt{3} + 3}{3 - \sqrt{3}} \cdot \frac{3 + \sqrt{3}}{3 + \sqrt{3}})}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.7

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

$\frac{2 \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{(3 - \sqrt{3}) (3 + \sqrt{3})}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.8

Expand the denominator using the FOIL method.

$\frac{2 \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{9 + 3 \sqrt{3} - 3 \sqrt{3} - {\sqrt{3}}^{2}}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.9

Simplify.

$\frac{2 \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.10

Simplify the numerator.

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Step 1.7.10.1

Reorder terms.

$\frac{2 \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.10.2

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

$\frac{2 \frac{{(3 + \sqrt{3})}^{1} (3 + \sqrt{3})}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.10.3

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

$\frac{2 \frac{{(3 + \sqrt{3})}^{1} {(3 + \sqrt{3})}^{1}}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.10.4

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

$\frac{2 \frac{{(3 + \sqrt{3})}^{1 + 1}}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.10.5

Add $1$ and $1$ .

$\frac{2 \frac{{(3 + \sqrt{3})}^{2}}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.11

Rewrite ${(3 + \sqrt{3})}^{2}$ as $(3 + \sqrt{3}) (3 + \sqrt{3})$ .

$\frac{2 \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.12

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

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Step 1.7.12.1

Apply the distributive property.

$\frac{2 \frac{3 (3 + \sqrt{3}) + \sqrt{3} (3 + \sqrt{3})}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.12.2

Apply the distributive property.

$\frac{2 \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} (3 + \sqrt{3})}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.12.3

Apply the distributive property.

$\frac{2 \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.13

Simplify and combine like terms.

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Step 1.7.13.1

Simplify each term.

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Step 1.7.13.1.1

Multiply $3$ by $3$ .

$\frac{2 \frac{9 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.13.1.2

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

$\frac{2 \frac{9 + 3 \sqrt{3} + 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.13.1.3

Combine using the product rule for radicals.

$\frac{2 \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3 \cdot 3}}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.13.1.4

Multiply $3$ by $3$ .

$\frac{2 \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{9}}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.13.1.5

Rewrite $9$ as $3^{2}$ .

$\frac{2 \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3^{2}}}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.13.1.6

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

$\frac{2 \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + 3}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.13.2

Add $9$ and $3$ .

$\frac{2 \frac{12 + 3 \sqrt{3} + 3 \sqrt{3}}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.13.3

Add $3 \sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{2 \frac{12 + 6 \sqrt{3}}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.14

Cancel the common factor of $12 + 6 \sqrt{3}$ and $6$ .

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Step 1.7.14.1

Factor $6$ out of $12$ .

$\frac{2 \frac{6 \cdot 2 + 6 \sqrt{3}}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.14.2

Factor $6$ out of $6 \sqrt{3}$ .

$\frac{2 \frac{6 \cdot 2 + 6 (\sqrt{3})}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.14.3

Factor $6$ out of $6 (2) + 6 (\sqrt{3})$ .

$\frac{2 \frac{6 (2 + \sqrt{3})}{6}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.14.4

Cancel the common factors.

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Step 1.7.14.4.1

Factor $6$ out of $6$ .

$\frac{2 \frac{6 (2 + \sqrt{3})}{6 (1)}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.14.4.2

Cancel the common factor.

$\frac{2 \frac{6 (2 + \sqrt{3})}{6 \cdot 1}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.14.4.3

Rewrite the expression.

$\frac{2 \frac{2 + \sqrt{3}}{1}}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 1.7.14.4.4

Divide $2 + \sqrt{3}$ by $1$ .

$\frac{2 (2 + \sqrt{3})}{1 - \tan^{2} (\frac{5 π}{12})}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

$\frac{2 (2 + \sqrt{3})}{1^{2} - \tan^{2} (\frac{5 π}{12})}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \tan (\frac{5 π}{12})$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \tan (\frac{5 π}{12})) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3

Simplify.

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Step 2.3.1

The exact value of $\tan (\frac{5 π}{12})$ is $2 + \sqrt{3}$ .

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Step 2.3.1.1

Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\frac{2 (2 + \sqrt{3})}{(1 + \tan (\frac{π}{6} + \frac{π}{4})) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.2

Apply the sum of angles identity.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\tan (\frac{π}{6}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.3

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\frac{\sqrt{3}}{3} + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.4

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\frac{\sqrt{3}}{3} + 1}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.5

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \tan (\frac{π}{4})}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.6

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7

Simplify $\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ .

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Step 2.3.1.7.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 2.3.1.7.1.1

Multiply $\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ by $\frac{3}{3}$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{3}{3} \cdot \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.1.2

Combine.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{3 (\frac{\sqrt{3}}{3} + 1)}{3 (1 - \frac{\sqrt{3}}{3} \cdot 1)}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.2

Apply the distributive property.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{3 \frac{\sqrt{3}}{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.3

Cancel the common factor of $3$ .

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Step 2.3.1.7.3.1

Cancel the common factor.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{3 \frac{\sqrt{3}}{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.3.2

Rewrite the expression.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\sqrt{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.4

Multiply $3$ by $1$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\sqrt{3} + 3}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.5

Simplify the denominator.

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Step 2.3.1.7.5.1

Multiply $3$ by $1$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\sqrt{3} + 3}{3 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.5.2

Multiply $- 1$ by $1$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\sqrt{3} + 3}{3 + 3 (- \frac{\sqrt{3}}{3})}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.5.3

Cancel the common factor of $3$ .

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Step 2.3.1.7.5.3.1

Move the leading negative in $- \frac{\sqrt{3}}{3}$ into the numerator.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\sqrt{3} + 3}{3 + 3 \frac{- \sqrt{3}}{3}}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.5.3.2

Cancel the common factor.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\sqrt{3} + 3}{3 + 3 \frac{- \sqrt{3}}{3}}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.5.3.3

Rewrite the expression.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\sqrt{3} + 3}{3 - \sqrt{3}}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.6

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

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{\sqrt{3} + 3}{3 - \sqrt{3}} \cdot \frac{3 + \sqrt{3}}{3 + \sqrt{3}}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.7

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

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{(3 - \sqrt{3}) (3 + \sqrt{3})}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.8

Expand the denominator using the FOIL method.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{9 + 3 \sqrt{3} - 3 \sqrt{3} - {\sqrt{3}}^{2}}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.9

Simplify.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.10

Simplify the numerator.

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Step 2.3.1.7.10.1

Reorder terms.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.10.2

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

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{{(3 + \sqrt{3})}^{1} (3 + \sqrt{3})}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.10.3

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

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{{(3 + \sqrt{3})}^{1} {(3 + \sqrt{3})}^{1}}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.10.4

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

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{{(3 + \sqrt{3})}^{1 + 1}}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.10.5

Add $1$ and $1$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{{(3 + \sqrt{3})}^{2}}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.11

Rewrite ${(3 + \sqrt{3})}^{2}$ as $(3 + \sqrt{3}) (3 + \sqrt{3})$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.12

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

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Step 2.3.1.7.12.1

Apply the distributive property.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{3 (3 + \sqrt{3}) + \sqrt{3} (3 + \sqrt{3})}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.12.2

Apply the distributive property.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} (3 + \sqrt{3})}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.12.3

Apply the distributive property.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.13

Simplify and combine like terms.

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Step 2.3.1.7.13.1

Simplify each term.

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Step 2.3.1.7.13.1.1

Multiply $3$ by $3$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{9 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.13.1.2

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

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{9 + 3 \sqrt{3} + 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.13.1.3

Combine using the product rule for radicals.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3 \cdot 3}}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.13.1.4

Multiply $3$ by $3$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{9}}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.13.1.5

Rewrite $9$ as $3^{2}$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3^{2}}}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.13.1.6

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

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + 3}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.13.2

Add $9$ and $3$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{12 + 3 \sqrt{3} + 3 \sqrt{3}}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.13.3

Add $3 \sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{12 + 6 \sqrt{3}}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.14

Cancel the common factor of $12 + 6 \sqrt{3}$ and $6$ .

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Step 2.3.1.7.14.1

Factor $6$ out of $12$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{6 \cdot 2 + 6 \sqrt{3}}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.14.2

Factor $6$ out of $6 \sqrt{3}$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{6 \cdot 2 + 6 (\sqrt{3})}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.14.3

Factor $6$ out of $6 (2) + 6 (\sqrt{3})$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{6 (2 + \sqrt{3})}{6}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.14.4

Cancel the common factors.

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Step 2.3.1.7.14.4.1

Factor $6$ out of $6$ .

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{6 (2 + \sqrt{3})}{6 (1)}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.14.4.2

Cancel the common factor.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{6 (2 + \sqrt{3})}{6 \cdot 1}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.14.4.3

Rewrite the expression.

$\frac{2 (2 + \sqrt{3})}{(1 + \frac{2 + \sqrt{3}}{1}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.1.7.14.4.4

Divide $2 + \sqrt{3}$ by $1$ .

$\frac{2 (2 + \sqrt{3})}{(1 + 2 + \sqrt{3}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.2

Add $1$ and $2$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \tan (\frac{5 π}{12}))}$

Step 2.3.3

The exact value of $\tan (\frac{5 π}{12})$ is $2 + \sqrt{3}$ .

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Step 2.3.3.1

Split $\frac{5 π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \tan (\frac{π}{6} + \frac{π}{4}))}$

Step 2.3.3.2

Apply the sum of angles identity.

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{\tan (\frac{π}{6}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})})}$

Step 2.3.3.3

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{\frac{\sqrt{3}}{3} + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})})}$

Step 2.3.3.4

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{\frac{\sqrt{3}}{3} + 1}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{4})})}$

Step 2.3.3.5

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \tan (\frac{π}{4})})}$

Step 2.3.3.6

The exact value of $\tan (\frac{π}{4})$ is $1$ .

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

Step 2.3.3.7

Simplify $\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ .

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Step 2.3.3.7.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 2.3.3.7.1.1

Multiply $\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ by $\frac{3}{3}$ .

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

Step 2.3.3.7.1.2

Combine.

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

Step 2.3.3.7.2

Apply the distributive property.

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

Step 2.3.3.7.3

Cancel the common factor of $3$ .

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Step 2.3.3.7.3.1

Cancel the common factor.

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

Step 2.3.3.7.3.2

Rewrite the expression.

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

Step 2.3.3.7.4

Multiply $3$ by $1$ .

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

Step 2.3.3.7.5

Simplify the denominator.

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Step 2.3.3.7.5.1

Multiply $3$ by $1$ .

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

Step 2.3.3.7.5.2

Multiply $- 1$ by $1$ .

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

Step 2.3.3.7.5.3

Cancel the common factor of $3$ .

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Step 2.3.3.7.5.3.1

Move the leading negative in $- \frac{\sqrt{3}}{3}$ into the numerator.

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

Step 2.3.3.7.5.3.2

Cancel the common factor.

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

Step 2.3.3.7.5.3.3

Rewrite the expression.

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

Step 2.3.3.7.6

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

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

Step 2.3.3.7.7

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

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

Step 2.3.3.7.8

Expand the denominator using the FOIL method.

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{9 + 3 \sqrt{3} - 3 \sqrt{3} - {\sqrt{3}}^{2}})}$

Step 2.3.3.7.9

Simplify.

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{6})}$

Step 2.3.3.7.10

Simplify the numerator.

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Step 2.3.3.7.10.1

Reorder terms.

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6})}$

Step 2.3.3.7.10.2

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

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{{(3 + \sqrt{3})}^{1} (3 + \sqrt{3})}{6})}$

Step 2.3.3.7.10.3

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

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{{(3 + \sqrt{3})}^{1} {(3 + \sqrt{3})}^{1}}{6})}$

Step 2.3.3.7.10.4

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

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{{(3 + \sqrt{3})}^{1 + 1}}{6})}$

Step 2.3.3.7.10.5

Add $1$ and $1$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{{(3 + \sqrt{3})}^{2}}{6})}$

Step 2.3.3.7.11

Rewrite ${(3 + \sqrt{3})}^{2}$ as $(3 + \sqrt{3}) (3 + \sqrt{3})$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6})}$

Step 2.3.3.7.12

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

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Step 2.3.3.7.12.1

Apply the distributive property.

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{3 (3 + \sqrt{3}) + \sqrt{3} (3 + \sqrt{3})}{6})}$

Step 2.3.3.7.12.2

Apply the distributive property.

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} (3 + \sqrt{3})}{6})}$

Step 2.3.3.7.12.3

Apply the distributive property.

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6})}$

Step 2.3.3.7.13

Simplify and combine like terms.

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Step 2.3.3.7.13.1

Simplify each term.

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Step 2.3.3.7.13.1.1

Multiply $3$ by $3$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{9 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6})}$

Step 2.3.3.7.13.1.2

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

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{9 + 3 \sqrt{3} + 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{6})}$

Step 2.3.3.7.13.1.3

Combine using the product rule for radicals.

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3 \cdot 3}}{6})}$

Step 2.3.3.7.13.1.4

Multiply $3$ by $3$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{9}}{6})}$

Step 2.3.3.7.13.1.5

Rewrite $9$ as $3^{2}$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3^{2}}}{6})}$

Step 2.3.3.7.13.1.6

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

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + 3}{6})}$

Step 2.3.3.7.13.2

Add $9$ and $3$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{12 + 3 \sqrt{3} + 3 \sqrt{3}}{6})}$

Step 2.3.3.7.13.3

Add $3 \sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{12 + 6 \sqrt{3}}{6})}$

Step 2.3.3.7.14

Cancel the common factor of $12 + 6 \sqrt{3}$ and $6$ .

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Step 2.3.3.7.14.1

Factor $6$ out of $12$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{6 \cdot 2 + 6 \sqrt{3}}{6})}$

Step 2.3.3.7.14.2

Factor $6$ out of $6 \sqrt{3}$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{6 \cdot 2 + 6 (\sqrt{3})}{6})}$

Step 2.3.3.7.14.3

Factor $6$ out of $6 (2) + 6 (\sqrt{3})$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{6 (2 + \sqrt{3})}{6})}$

Step 2.3.3.7.14.4

Cancel the common factors.

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Step 2.3.3.7.14.4.1

Factor $6$ out of $6$ .

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{6 (2 + \sqrt{3})}{6 (1)})}$

Step 2.3.3.7.14.4.2

Cancel the common factor.

$\frac{2 (2 + \sqrt{3})}{(3 + \sqrt{3}) (1 - \frac{6 (2 + \sqrt{3})}{6 \cdot 1})}$

Step 2.3.3.7.14.4.3

Rewrite the expression.

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

Step 2.3.3.7.14.4.4

Divide $2 + \sqrt{3}$ by $1$ .

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

Step 2.3.4

Apply the distributive property.

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

Step 2.3.5

Multiply $- 1$ by $2$ .

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

Step 2.3.6

Subtract $2$ from $1$ .

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

Step 3

Expand $(3 + \sqrt{3}) (- 1 - \sqrt{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

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 $3$ by $- 1$ .

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

Step 4.1.2

Multiply $- 1$ by $3$ .

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

Step 4.1.3

Move $- 1$ to the left of $\sqrt{3}$ .

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

Step 4.1.4

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$\frac{2 (2 + \sqrt{3})}{- 3 - 3 \sqrt{3} - \sqrt{3} + \sqrt{3} (- \sqrt{3})}$

Step 4.1.5

Multiply $\sqrt{3} (- \sqrt{3})$ .

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Step 4.1.5.1

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

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

Step 4.1.5.2

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

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

Step 4.1.5.3

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

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

Step 4.1.5.4

Add $1$ and $1$ .

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

Step 4.1.6

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

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Step 4.1.6.1

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

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

Step 4.1.6.2

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

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

Step 4.1.6.3

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

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

Step 4.1.6.4

Cancel the common factor of $2$ .

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Step 4.1.6.4.1

Cancel the common factor.

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

Step 4.1.6.4.2

Rewrite the expression.

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

Step 4.1.6.5

Evaluate the exponent.

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

Step 4.1.7

Multiply $- 1$ by $3$ .

$\frac{2 (2 + \sqrt{3})}{- 3 - 3 \sqrt{3} - \sqrt{3} - 3}$

Step 4.2

Subtract $3$ from $- 3$ .

$\frac{2 (2 + \sqrt{3})}{- 6 - 3 \sqrt{3} - \sqrt{3}}$

Step 4.3

Subtract $\sqrt{3}$ from $- 3 \sqrt{3}$ .

$\frac{2 (2 + \sqrt{3})}{- 6 - 4 \sqrt{3}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $2$ out of $- 6$ .

$\frac{2 (2 + \sqrt{3})}{2 (- 3) - 4 \sqrt{3}}$

Step 5.2

Factor $2$ out of $- 4 \sqrt{3}$ .

$\frac{2 (2 + \sqrt{3})}{2 (- 3) + 2 (- 2 \sqrt{3})}$

Step 5.3

Factor $2$ out of $2 (- 3) + 2 (- 2 \sqrt{3})$ .

$\frac{2 (2 + \sqrt{3})}{2 (- 3 - 2 \sqrt{3})}$

Step 5.4

Cancel the common factor.

$\frac{2 (2 + \sqrt{3})}{2 (- 3 - 2 \sqrt{3})}$

Step 5.5

Rewrite the expression.

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

Step 6

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

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

Step 7

Combine fractions.

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Step 7.1

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

$\frac{(2 + \sqrt{3}) (- 3 + 2 \sqrt{3})}{(- 3 - 2 \sqrt{3}) (- 3 + 2 \sqrt{3})}$

Step 7.2

Expand the denominator using the FOIL method.

$\frac{(2 + \sqrt{3}) (- 3 + 2 \sqrt{3})}{9 - 6 \sqrt{3} + 6 \sqrt{3} - 4 {\sqrt{3}}^{2}}$

Step 7.3

Simplify.

$\frac{(2 + \sqrt{3}) (- 3 + 2 \sqrt{3})}{- 3}$

Step 8

Simplify the numerator.

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Step 8.1

Expand $(2 + \sqrt{3}) (- 3 + 2 \sqrt{3})$ using the FOIL Method.

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Step 8.1.1

Apply the distributive property.

$\frac{2 (- 3 + 2 \sqrt{3}) + \sqrt{3} (- 3 + 2 \sqrt{3})}{- 3}$

Step 8.1.2

Apply the distributive property.

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

Step 8.1.3

Apply the distributive property.

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

Step 8.2

Simplify and combine like terms.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Multiply $2$ by $- 3$ .

$\frac{- 6 + 2 (2 \sqrt{3}) + \sqrt{3} \cdot - 3 + \sqrt{3} (2 \sqrt{3})}{- 3}$

Step 8.2.1.2

Multiply $2$ by $2$ .

$\frac{- 6 + 4 \sqrt{3} + \sqrt{3} \cdot - 3 + \sqrt{3} (2 \sqrt{3})}{- 3}$

Step 8.2.1.3

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

$\frac{- 6 + 4 \sqrt{3} - 3 \cdot \sqrt{3} + \sqrt{3} (2 \sqrt{3})}{- 3}$

Step 8.2.1.4

Multiply $\sqrt{3} (2 \sqrt{3})$ .

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Step 8.2.1.4.1

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

$\frac{- 6 + 4 \sqrt{3} - 3 \sqrt{3} + 2 ({\sqrt{3}}^{1} \sqrt{3})}{- 3}$

Step 8.2.1.4.2

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

$\frac{- 6 + 4 \sqrt{3} - 3 \sqrt{3} + 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{- 3}$

Step 8.2.1.4.3

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

$\frac{- 6 + 4 \sqrt{3} - 3 \sqrt{3} + 2 {\sqrt{3}}^{1 + 1}}{- 3}$

Step 8.2.1.4.4

Add $1$ and $1$ .

$\frac{- 6 + 4 \sqrt{3} - 3 \sqrt{3} + 2 {\sqrt{3}}^{2}}{- 3}$

Step 8.2.1.5

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

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Step 8.2.1.5.1

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

$\frac{- 6 + 4 \sqrt{3} - 3 \sqrt{3} + 2 {(3^{\frac{1}{2}})}^{2}}{- 3}$

Step 8.2.1.5.2

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

$\frac{- 6 + 4 \sqrt{3} - 3 \sqrt{3} + 2 \cdot 3^{\frac{1}{2} \cdot 2}}{- 3}$

Step 8.2.1.5.3

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

$\frac{- 6 + 4 \sqrt{3} - 3 \sqrt{3} + 2 \cdot 3^{\frac{2}{2}}}{- 3}$

Step 8.2.1.5.4

Cancel the common factor of $2$ .

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Step 8.2.1.5.4.1

Cancel the common factor.

$\frac{- 6 + 4 \sqrt{3} - 3 \sqrt{3} + 2 \cdot 3^{\frac{2}{2}}}{- 3}$

Step 8.2.1.5.4.2

Rewrite the expression.

$\frac{- 6 + 4 \sqrt{3} - 3 \sqrt{3} + 2 \cdot 3^{1}}{- 3}$

Step 8.2.1.5.5

Evaluate the exponent.

$\frac{- 6 + 4 \sqrt{3} - 3 \sqrt{3} + 2 \cdot 3}{- 3}$

Step 8.2.1.6

Multiply $2$ by $3$ .

$\frac{- 6 + 4 \sqrt{3} - 3 \sqrt{3} + 6}{- 3}$

Step 8.2.2

Add $- 6$ and $6$ .

$\frac{0 + 4 \sqrt{3} - 3 \sqrt{3}}{- 3}$

Step 8.2.3

Add $0$ and $4 \sqrt{3}$ .

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

Step 8.2.4

Subtract $3 \sqrt{3}$ from $4 \sqrt{3}$ .

$\frac{\sqrt{3}}{- 3}$

Step 9

Move the negative in front of the fraction.

$- \frac{\sqrt{3}}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{3}}{3}$

Decimal Form:

$- 0.57735026 \dots$