Algebra Examples

$\frac{\tan (\frac{5 π}{4}) + \tan (\frac{7 π}{12})}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\tan (\frac{π}{4}) + \tan (\frac{7 π}{12})}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.2

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

$\frac{1 + \tan (\frac{7 π}{12})}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3

The exact value of $\tan (\frac{7 π}{12})$ is $- \sqrt{7 + 4 \sqrt{3}}$ .

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Step 1.3.1

Rewrite $\frac{7 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\frac{1 + \tan (\frac{\frac{7 π}{6}}{2})}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.2

Apply the tangent half-angle identity.

$\frac{1 \pm \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{7 π}{6})}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.3

Change the $\pm$ to $-$ because tangent is negative in the second quadrant.

$\frac{1 - \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{7 π}{6})}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4

Simplify $- \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{7 π}{6})}}$ .

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Step 1.3.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$\frac{1 - \sqrt{\frac{1 - - \cos (\frac{π}{6})}{1 + \cos (\frac{7 π}{6})}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1 - \sqrt{\frac{1 - - \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.3

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

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Step 1.3.4.3.1

Multiply $- 1$ by $- 1$ .

$\frac{1 - \sqrt{\frac{1 + 1 \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.3.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$\frac{1 - \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.4

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

$\frac{1 - \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.5

Combine the numerators over the common denominator.

$\frac{1 - \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$\frac{1 - \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{1 - \cos (\frac{π}{6})}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.7

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1 - \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{1 - \frac{\sqrt{3}}{2}}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.8

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

$\frac{1 - \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{\frac{2}{2} - \frac{\sqrt{3}}{2}}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.9

Combine the numerators over the common denominator.

$\frac{1 - \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{\frac{2 - \sqrt{3}}{2}}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.10

Multiply the numerator by the reciprocal of the denominator.

$\frac{1 - \sqrt{\frac{2 + \sqrt{3}}{2} \cdot \frac{2}{2 - \sqrt{3}}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.11

Cancel the common factor of $2$ .

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Step 1.3.4.11.1

Cancel the common factor.

$\frac{1 - \sqrt{\frac{2 + \sqrt{3}}{2} \cdot \frac{2}{2 - \sqrt{3}}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.11.2

Rewrite the expression.

$\frac{1 - \sqrt{(2 + \sqrt{3}) \frac{1}{2 - \sqrt{3}}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.12

Multiply $\frac{1}{2 - \sqrt{3}}$ by $\frac{2 + \sqrt{3}}{2 + \sqrt{3}}$ .

$\frac{1 - \sqrt{(2 + \sqrt{3}) (\frac{1}{2 - \sqrt{3}} \cdot \frac{2 + \sqrt{3}}{2 + \sqrt{3}})}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.13

Multiply $\frac{1}{2 - \sqrt{3}}$ by $\frac{2 + \sqrt{3}}{2 + \sqrt{3}}$ .

$\frac{1 - \sqrt{(2 + \sqrt{3}) \frac{2 + \sqrt{3}}{(2 - \sqrt{3}) (2 + \sqrt{3})}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.14

Expand the denominator using the FOIL method.

$\frac{1 - \sqrt{(2 + \sqrt{3}) \frac{2 + \sqrt{3}}{4 + 2 \sqrt{3} - 2 \sqrt{3} - {\sqrt{3}}^{2}}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.15

Simplify.

$\frac{1 - \sqrt{(2 + \sqrt{3}) \frac{2 + \sqrt{3}}{1}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.16

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

$\frac{1 - \sqrt{(2 + \sqrt{3}) (2 + \sqrt{3})}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.17

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

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Step 1.3.4.17.1

Apply the distributive property.

$\frac{1 - \sqrt{2 (2 + \sqrt{3}) + \sqrt{3} (2 + \sqrt{3})}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.17.2

Apply the distributive property.

$\frac{1 - \sqrt{2 \cdot 2 + 2 \sqrt{3} + \sqrt{3} (2 + \sqrt{3})}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.17.3

Apply the distributive property.

$\frac{1 - \sqrt{2 \cdot 2 + 2 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.18

Simplify and combine like terms.

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Step 1.3.4.18.1

Simplify each term.

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Step 1.3.4.18.1.1

Multiply $2$ by $2$ .

$\frac{1 - \sqrt{4 + 2 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.18.1.2

Move $2$ to the left of $\sqrt{3}$ .

$\frac{1 - \sqrt{4 + 2 \sqrt{3} + 2 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.18.1.3

Combine using the product rule for radicals.

$\frac{1 - \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3 \cdot 3}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.18.1.4

Multiply $3$ by $3$ .

$\frac{1 - \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{9}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.18.1.5

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

$\frac{1 - \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3^{2}}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.18.1.6

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

$\frac{1 - \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + 3}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.18.2

Add $4$ and $3$ .

$\frac{1 - \sqrt{7 + 2 \sqrt{3} + 2 \sqrt{3}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 1.3.4.18.3

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - \tan (\frac{5 π}{4}) \tan (\frac{7 π}{12})}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - \tan (\frac{π}{4}) \tan (\frac{7 π}{12})}$

Step 2.2

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

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 \cdot 1 \tan (\frac{7 π}{12})}$

Step 2.3

Multiply $- 1$ by $1$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 \tan (\frac{7 π}{12})}$

Step 2.4

The exact value of $\tan (\frac{7 π}{12})$ is $- \sqrt{7 + 4 \sqrt{3}}$ .

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Step 2.4.1

Rewrite $\frac{7 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 \tan (\frac{\frac{7 π}{6}}{2})}$

Step 2.4.2

Apply the tangent half-angle identity.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (\pm \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{7 π}{6})}})}$

Step 2.4.3

Change the $\pm$ to $-$ because tangent is negative in the second quadrant.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{7 π}{6})}})}$

Step 2.4.4

Simplify $- \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{1 + \cos (\frac{7 π}{6})}}$ .

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Step 2.4.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{\frac{1 - - \cos (\frac{π}{6})}{1 + \cos (\frac{7 π}{6})}})}$

Step 2.4.4.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{\frac{1 - - \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}})}$

Step 2.4.4.3

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

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Step 2.4.4.3.1

Multiply $- 1$ by $- 1$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{\frac{1 + 1 \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}})}$

Step 2.4.4.3.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}})}$

Step 2.4.4.4

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

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}})}$

Step 2.4.4.5

Combine the numerators over the common denominator.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{1 + \cos (\frac{7 π}{6})}})}$

Step 2.4.4.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{1 - \cos (\frac{π}{6})}})}$

Step 2.4.4.7

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 2.4.4.8

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

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

Step 2.4.4.9

Combine the numerators over the common denominator.

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

Step 2.4.4.10

Multiply the numerator by the reciprocal of the denominator.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{\frac{2 + \sqrt{3}}{2} \cdot \frac{2}{2 - \sqrt{3}}})}$

Step 2.4.4.11

Cancel the common factor of $2$ .

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Step 2.4.4.11.1

Cancel the common factor.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{\frac{2 + \sqrt{3}}{2} \cdot \frac{2}{2 - \sqrt{3}}})}$

Step 2.4.4.11.2

Rewrite the expression.

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

Step 2.4.4.12

Multiply $\frac{1}{2 - \sqrt{3}}$ by $\frac{2 + \sqrt{3}}{2 + \sqrt{3}}$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{(2 + \sqrt{3}) (\frac{1}{2 - \sqrt{3}} \cdot \frac{2 + \sqrt{3}}{2 + \sqrt{3}})})}$

Step 2.4.4.13

Multiply $\frac{1}{2 - \sqrt{3}}$ by $\frac{2 + \sqrt{3}}{2 + \sqrt{3}}$ .

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

Step 2.4.4.14

Expand the denominator using the FOIL method.

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

Step 2.4.4.15

Simplify.

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

Step 2.4.4.16

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

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

Step 2.4.4.17

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

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Step 2.4.4.17.1

Apply the distributive property.

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

Step 2.4.4.17.2

Apply the distributive property.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{2 \cdot 2 + 2 \sqrt{3} + \sqrt{3} (2 + \sqrt{3})})}$

Step 2.4.4.17.3

Apply the distributive property.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{2 \cdot 2 + 2 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}})}$

Step 2.4.4.18

Simplify and combine like terms.

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Step 2.4.4.18.1

Simplify each term.

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Step 2.4.4.18.1.1

Multiply $2$ by $2$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{4 + 2 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}})}$

Step 2.4.4.18.1.2

Move $2$ to the left of $\sqrt{3}$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{4 + 2 \sqrt{3} + 2 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}})}$

Step 2.4.4.18.1.3

Combine using the product rule for radicals.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3 \cdot 3}})}$

Step 2.4.4.18.1.4

Multiply $3$ by $3$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - 1 (- \sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{9}})}$

Step 2.4.4.18.1.5

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

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

Step 2.4.4.18.1.6

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

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

Step 2.4.4.18.2

Add $4$ and $3$ .

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

Step 2.4.4.18.3

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

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

Step 2.5

Multiply $- 1 (- \sqrt{7 + 4 \sqrt{3}})$ .

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Step 2.5.1

Multiply $- 1$ by $- 1$ .

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

Step 2.5.2

Multiply $\sqrt{7 + 4 \sqrt{3}}$ by $1$ .

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

Step 3

Multiply $\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 + \sqrt{7 + 4 \sqrt{3}}}$ by $\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - \sqrt{7 + 4 \sqrt{3}}}$ .

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

Step 4

Combine fractions.

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Step 4.1

Multiply $\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 + \sqrt{7 + 4 \sqrt{3}}}$ by $\frac{1 - \sqrt{7 + 4 \sqrt{3}}}{1 - \sqrt{7 + 4 \sqrt{3}}}$ .

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

Step 4.2

Expand the denominator using the FOIL method.

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

Step 4.3

Simplify.

$\frac{(1 - \sqrt{7 + 4 \sqrt{3}}) (1 - \sqrt{7 + 4 \sqrt{3}})}{- 6 - 4 \sqrt{3}}$

Step 5

Simplify the numerator.

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Step 5.1

Raise $1 - \sqrt{7 + 4 \sqrt{3}}$ to the power of $1$ .

$\frac{{(1 - \sqrt{7 + 4 \sqrt{3}})}^{1} (1 - \sqrt{7 + 4 \sqrt{3}})}{- 6 - 4 \sqrt{3}}$

Step 5.2

Raise $1 - \sqrt{7 + 4 \sqrt{3}}$ to the power of $1$ .

$\frac{{(1 - \sqrt{7 + 4 \sqrt{3}})}^{1} {(1 - \sqrt{7 + 4 \sqrt{3}})}^{1}}{- 6 - 4 \sqrt{3}}$

Step 5.3

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

$\frac{{(1 - \sqrt{7 + 4 \sqrt{3}})}^{1 + 1}}{- 6 - 4 \sqrt{3}}$

Step 5.4

Add $1$ and $1$ .

$\frac{{(1 - \sqrt{7 + 4 \sqrt{3}})}^{2}}{- 6 - 4 \sqrt{3}}$

Step 6

Rewrite ${(1 - \sqrt{7 + 4 \sqrt{3}})}^{2}$ as $(1 - \sqrt{7 + 4 \sqrt{3}}) (1 - \sqrt{7 + 4 \sqrt{3}})$ .

$\frac{(1 - \sqrt{7 + 4 \sqrt{3}}) (1 - \sqrt{7 + 4 \sqrt{3}})}{- 6 - 4 \sqrt{3}}$

Step 7

Expand $(1 - \sqrt{7 + 4 \sqrt{3}}) (1 - \sqrt{7 + 4 \sqrt{3}})$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$\frac{1 (1 - \sqrt{7 + 4 \sqrt{3}}) - \sqrt{7 + 4 \sqrt{3}} (1 - \sqrt{7 + 4 \sqrt{3}})}{- 6 - 4 \sqrt{3}}$

Step 7.2

Apply the distributive property.

$\frac{1 \cdot 1 + 1 (- \sqrt{7 + 4 \sqrt{3}}) - \sqrt{7 + 4 \sqrt{3}} (1 - \sqrt{7 + 4 \sqrt{3}})}{- 6 - 4 \sqrt{3}}$

Step 7.3

Apply the distributive property.

$\frac{1 \cdot 1 + 1 (- \sqrt{7 + 4 \sqrt{3}}) - \sqrt{7 + 4 \sqrt{3}} \cdot 1 - \sqrt{7 + 4 \sqrt{3}} (- \sqrt{7 + 4 \sqrt{3}})}{- 6 - 4 \sqrt{3}}$

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $1$ by $1$ .

$\frac{1 + 1 (- \sqrt{7 + 4 \sqrt{3}}) - \sqrt{7 + 4 \sqrt{3}} \cdot 1 - \sqrt{7 + 4 \sqrt{3}} (- \sqrt{7 + 4 \sqrt{3}})}{- 6 - 4 \sqrt{3}}$

Step 8.1.2

Multiply $- \sqrt{7 + 4 \sqrt{3}}$ by $1$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} \cdot 1 - \sqrt{7 + 4 \sqrt{3}} (- \sqrt{7 + 4 \sqrt{3}})}{- 6 - 4 \sqrt{3}}$

Step 8.1.3

Multiply $- 1$ by $1$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} (- \sqrt{7 + 4 \sqrt{3}})}{- 6 - 4 \sqrt{3}}$

Step 8.1.4

Multiply $- \sqrt{7 + 4 \sqrt{3}} (- \sqrt{7 + 4 \sqrt{3}})$ .

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Step 8.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + 1 \sqrt{7 + 4 \sqrt{3}} \sqrt{7 + 4 \sqrt{3}}}{- 6 - 4 \sqrt{3}}$

Step 8.1.4.2

Multiply $\sqrt{7 + 4 \sqrt{3}}$ by $1$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + \sqrt{7 + 4 \sqrt{3}} \sqrt{7 + 4 \sqrt{3}}}{- 6 - 4 \sqrt{3}}$

Step 8.1.4.3

Raise $\sqrt{7 + 4 \sqrt{3}}$ to the power of $1$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + {\sqrt{7 + 4 \sqrt{3}}}^{1} \sqrt{7 + 4 \sqrt{3}}}{- 6 - 4 \sqrt{3}}$

Step 8.1.4.4

Raise $\sqrt{7 + 4 \sqrt{3}}$ to the power of $1$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + {\sqrt{7 + 4 \sqrt{3}}}^{1} {\sqrt{7 + 4 \sqrt{3}}}^{1}}{- 6 - 4 \sqrt{3}}$

Step 8.1.4.5

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

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + {\sqrt{7 + 4 \sqrt{3}}}^{1 + 1}}{- 6 - 4 \sqrt{3}}$

Step 8.1.4.6

Add $1$ and $1$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + {\sqrt{7 + 4 \sqrt{3}}}^{2}}{- 6 - 4 \sqrt{3}}$

Step 8.1.5

Rewrite ${\sqrt{7 + 4 \sqrt{3}}}^{2}$ as $7 + 4 \sqrt{3}$ .

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Step 8.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 + 4 \sqrt{3}}$ as ${(7 + 4 \sqrt{3})}^{\frac{1}{2}}$ .

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + {({(7 + 4 \sqrt{3})}^{\frac{1}{2}})}^{2}}{- 6 - 4 \sqrt{3}}$

Step 8.1.5.2

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

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + {(7 + 4 \sqrt{3})}^{\frac{1}{2} \cdot 2}}{- 6 - 4 \sqrt{3}}$

Step 8.1.5.3

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

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + {(7 + 4 \sqrt{3})}^{\frac{2}{2}}}{- 6 - 4 \sqrt{3}}$

Step 8.1.5.4

Cancel the common factor of $2$ .

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Step 8.1.5.4.1

Cancel the common factor.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + {(7 + 4 \sqrt{3})}^{\frac{2}{2}}}{- 6 - 4 \sqrt{3}}$

Step 8.1.5.4.2

Rewrite the expression.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + {(7 + 4 \sqrt{3})}^{1}}{- 6 - 4 \sqrt{3}}$

Step 8.1.5.5

Simplify.

$\frac{1 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + 7 + 4 \sqrt{3}}{- 6 - 4 \sqrt{3}}$

Step 8.2

Add $1$ and $7$ .

$\frac{8 - \sqrt{7 + 4 \sqrt{3}} - \sqrt{7 + 4 \sqrt{3}} + 4 \sqrt{3}}{- 6 - 4 \sqrt{3}}$

Step 8.3

Subtract $\sqrt{7 + 4 \sqrt{3}}$ from $- \sqrt{7 + 4 \sqrt{3}}$ .

$\frac{8 - 2 \sqrt{7 + 4 \sqrt{3}} + 4 \sqrt{3}}{- 6 - 4 \sqrt{3}}$

Step 9

Cancel the common factor of $8 - 2 \sqrt{7 + 4 \sqrt{3}} + 4 \sqrt{3}$ and $- 6 - 4 \sqrt{3}$ .

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Step 9.1

Factor $2$ out of $8$ .

$\frac{2 (4) - 2 \sqrt{7 + 4 \sqrt{3}} + 4 \sqrt{3}}{- 6 - 4 \sqrt{3}}$

Step 9.2

Factor $2$ out of $- 2 \sqrt{7 + 4 \sqrt{3}}$ .

$\frac{2 (4) + 2 (- \sqrt{7 + 4 \sqrt{3}}) + 4 \sqrt{3}}{- 6 - 4 \sqrt{3}}$

Step 9.3

Factor $2$ out of $2 (4) + 2 (- \sqrt{7 + 4 \sqrt{3}})$ .

$\frac{2 (4 - \sqrt{7 + 4 \sqrt{3}}) + 4 \sqrt{3}}{- 6 - 4 \sqrt{3}}$

Step 9.4

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

$\frac{2 (4 - \sqrt{7 + 4 \sqrt{3}}) + 2 (2 \sqrt{3})}{- 6 - 4 \sqrt{3}}$

Step 9.5

Factor $2$ out of $2 (4 - \sqrt{7 + 4 \sqrt{3}}) + 2 (2 \sqrt{3})$ .

$\frac{2 (4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3})}{- 6 - 4 \sqrt{3}}$

Step 9.6

Cancel the common factors.

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Step 9.6.1

Factor $2$ out of $- 6$ .

$\frac{2 (4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3})}{2 \cdot - 3 - 4 \sqrt{3}}$

Step 9.6.2

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

$\frac{2 (4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3})}{2 \cdot - 3 + 2 (- 2 \sqrt{3})}$

Step 9.6.3

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

$\frac{2 (4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3})}{2 (- 3 - 2 \sqrt{3})}$

Step 9.6.4

Cancel the common factor.

$\frac{2 (4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3})}{2 (- 3 - 2 \sqrt{3})}$

Step 9.6.5

Rewrite the expression.

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

Step 10

Multiply $\frac{4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3}}{- 3 - 2 \sqrt{3}}$ by $\frac{- 3 + 2 \sqrt{3}}{- 3 + 2 \sqrt{3}}$ .

$\frac{4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3}}{- 3 - 2 \sqrt{3}} \cdot \frac{- 3 + 2 \sqrt{3}}{- 3 + 2 \sqrt{3}}$

Step 11

Multiply $\frac{4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3}}{- 3 - 2 \sqrt{3}}$ by $\frac{- 3 + 2 \sqrt{3}}{- 3 + 2 \sqrt{3}}$ .

$\frac{(4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3}) (- 3 + 2 \sqrt{3})}{(- 3 - 2 \sqrt{3}) (- 3 + 2 \sqrt{3})}$

Step 12

Expand the denominator using the FOIL method.

$\frac{(4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3}) (- 3 + 2 \sqrt{3})}{9 - 6 \sqrt{3} + 6 \sqrt{3} - 4 {\sqrt{3}}^{2}}$

Step 13

Simplify.

$\frac{(4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3}) (- 3 + 2 \sqrt{3})}{- 3}$

Step 14

Move the negative in front of the fraction.

$- \frac{(4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3}) (- 3 + 2 \sqrt{3})}{3}$

Step 15

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 16

Factor $- 1$ out of $2 \sqrt{3}$ .

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

Step 17

Factor $- 1$ out of $- 1 (3) - (- 2 \sqrt{3})$ .

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

Step 18

Simplify the expression.

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Step 18.1

Move the negative in front of the fraction.

$- - \frac{(4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3}) (3 - 2 \sqrt{3})}{3}$

Step 18.2

Multiply $- 1$ by $- 1$ .

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

Step 18.3

Multiply $\frac{(4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3}) (3 - 2 \sqrt{3})}{3}$ by $1$ .

$\frac{(4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3}) (3 - 2 \sqrt{3})}{3}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$\frac{(4 - \sqrt{7 + 4 \sqrt{3}} + 2 \sqrt{3}) (3 - 2 \sqrt{3})}{3}$

Decimal Form:

$- 0.57735026 \dots$