Trigonometry Examples

$\sqrt{\frac{1 - \cos (\frac{5 π}{8})}{1 + \cos (\frac{5 π}{8})}}$

Step 1

The exact value of $\cos (\frac{5 π}{8})$ is $- \frac{\sqrt{2 - \sqrt{2}}}{2}$ .

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Step 1.1

Rewrite $\frac{5 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\sqrt{\frac{1 - \cos (\frac{\frac{5 π}{4}}{2})}{1 + \cos (\frac{5 π}{8})}}$

Step 1.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$\sqrt{\frac{1 - (\pm \sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}})}{1 + \cos (\frac{5 π}{8})}}$

Step 1.3

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

$\sqrt{\frac{1 - - \sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}}}{1 + \cos (\frac{5 π}{8})}}$

Step 1.4

Simplify $- \sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}}$ .

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Step 1.4.1

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.

$\sqrt{\frac{1 - - \sqrt{\frac{1 - \cos (\frac{π}{4})}{2}}}{1 + \cos (\frac{5 π}{8})}}$

Step 1.4.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\sqrt{\frac{1 - - \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}}{1 + \cos (\frac{5 π}{8})}}$

Step 1.4.3

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

$\sqrt{\frac{1 - - \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{2}}}{1 + \cos (\frac{5 π}{8})}}$

Step 1.4.4

Combine the numerators over the common denominator.

$\sqrt{\frac{1 - - \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}}}{1 + \cos (\frac{5 π}{8})}}$

Step 1.4.5

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{1 - - \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}}}{1 + \cos (\frac{5 π}{8})}}$

Step 1.4.6

Multiply $\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 1.4.6.1

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

$\sqrt{\frac{1 - - \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}}}{1 + \cos (\frac{5 π}{8})}}$

Step 1.4.6.2

Multiply $2$ by $2$ .

$\sqrt{\frac{1 - - \sqrt{\frac{2 - \sqrt{2}}{4}}}{1 + \cos (\frac{5 π}{8})}}$

Step 1.4.7

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$\sqrt{\frac{1 - - \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}}{1 + \cos (\frac{5 π}{8})}}$

Step 1.4.8

Simplify the denominator.

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Step 1.4.8.1

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

$\sqrt{\frac{1 - - \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{2^{2}}}}{1 + \cos (\frac{5 π}{8})}}$

Step 1.4.8.2

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

$\sqrt{\frac{1 - - \frac{\sqrt{2 - \sqrt{2}}}{2}}{1 + \cos (\frac{5 π}{8})}}$

Step 2

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

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Step 2.1

Multiply $- 1$ by $- 1$ .

$\sqrt{\frac{1 + 1 \frac{\sqrt{2 - \sqrt{2}}}{2}}{1 + \cos (\frac{5 π}{8})}}$

Step 2.2

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

$\sqrt{\frac{1 + \frac{\sqrt{2 - \sqrt{2}}}{2}}{1 + \cos (\frac{5 π}{8})}}$

Step 3

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

$\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2 - \sqrt{2}}}{2}}{1 + \cos (\frac{5 π}{8})}}$

Step 4

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{2 + \sqrt{2 - \sqrt{2}}}{2}}{1 + \cos (\frac{5 π}{8})}}$

Step 5

The exact value of $\cos (\frac{5 π}{8})$ is $- \frac{\sqrt{2 - \sqrt{2}}}{2}$ .

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Step 5.1

Rewrite $\frac{5 π}{8}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\sqrt{\frac{\frac{2 + \sqrt{2 - \sqrt{2}}}{2}}{1 + \cos (\frac{\frac{5 π}{4}}{2})}}$

Step 5.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$\sqrt{\frac{\frac{2 + \sqrt{2 - \sqrt{2}}}{2}}{1 \pm \sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}}}}$

Step 5.3

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

$\sqrt{\frac{\frac{2 + \sqrt{2 - \sqrt{2}}}{2}}{1 - \sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}}}}$

Step 5.4

Simplify $- \sqrt{\frac{1 + \cos (\frac{5 π}{4})}{2}}$ .

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Step 5.4.1

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.

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

Step 5.4.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 5.4.3

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

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

Step 5.4.4

Combine the numerators over the common denominator.

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

Step 5.4.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.4.6

Multiply $\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 5.4.6.1

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

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

Step 5.4.6.2

Multiply $2$ by $2$ .

$\sqrt{\frac{\frac{2 + \sqrt{2 - \sqrt{2}}}{2}}{1 - \sqrt{\frac{2 - \sqrt{2}}{4}}}}$

Step 5.4.7

Rewrite $\sqrt{\frac{2 - \sqrt{2}}{4}}$ as $\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}$ .

$\sqrt{\frac{\frac{2 + \sqrt{2 - \sqrt{2}}}{2}}{1 - \frac{\sqrt{2 - \sqrt{2}}}{\sqrt{4}}}}$

Step 5.4.8

Simplify the denominator.

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Step 5.4.8.1

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

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

Step 5.4.8.2

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

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Multiply the numerator by the reciprocal of the denominator.

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

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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

Step 10

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

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

Step 11

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

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

Step 12

Expand the denominator using the FOIL method.

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

Step 13

Simplify.

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

Step 14

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

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

Step 15

Combine fractions.

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Step 15.1

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

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

Step 15.2

Expand the denominator using the FOIL method.

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

Step 15.3

Simplify.

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

Step 16

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

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Step 16.1

Apply the distributive property.

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

Step 16.2

Apply the distributive property.

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

Step 16.3

Apply the distributive property.

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

Step 17

Simplify terms.

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Step 17.1

Simplify each term.

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Step 17.1.1

Multiply $2$ by $2$ .

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

Step 17.1.2

Multiply $- 1$ by $2$ .

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

Step 17.1.3

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

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

Step 17.1.4

Combine using the product rule for radicals.

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

Step 17.2

Simplify terms.

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Step 17.2.1

Apply the distributive property.

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

Step 17.2.2

Cancel the common factor of $2$ .

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Step 17.2.2.1

Cancel the common factor.

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

Step 17.2.2.2

Rewrite the expression.

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

Step 17.2.3

Combine $\sqrt{2 - \sqrt{2}}$ and $\frac{4 - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}{2}$ .

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

Step 17.3

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

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

Step 18

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 19

Combine fractions.

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Step 19.1

Combine $4$ and $\frac{2}{2}$ .

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

Step 19.2

Simplify the expression.

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Step 19.2.1

Combine the numerators over the common denominator.

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

Step 19.2.2

Multiply $4$ by $2$ .

$\sqrt{\frac{8 + \sqrt{2 - \sqrt{2}} (4 - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{2 (2 - \sqrt{2})})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20

Simplify each term.

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Step 20.1

Simplify the numerator.

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Step 20.1.1

Apply the distributive property.

$\sqrt{\frac{8 + \sqrt{2 - \sqrt{2}} \cdot 4 + \sqrt{2 - \sqrt{2}} (- 2 \sqrt{2}) + \sqrt{2 - \sqrt{2}} (2 \sqrt{2 - \sqrt{2}}) + \sqrt{2 - \sqrt{2}} (- \sqrt{2 (2 - \sqrt{2})})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.2

Simplify.

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Step 20.1.2.1

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

$\sqrt{\frac{8 + 4 \cdot \sqrt{2 - \sqrt{2}} + \sqrt{2 - \sqrt{2}} (- 2 \sqrt{2}) + \sqrt{2 - \sqrt{2}} (2 \sqrt{2 - \sqrt{2}}) + \sqrt{2 - \sqrt{2}} (- \sqrt{2 (2 - \sqrt{2})})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.2.2

Combine using the product rule for radicals.

$\sqrt{\frac{8 + 4 \cdot \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + \sqrt{2 - \sqrt{2}} (2 \sqrt{2 - \sqrt{2}}) + \sqrt{2 - \sqrt{2}} (- \sqrt{2 (2 - \sqrt{2})})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.2.3

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

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Step 20.1.2.3.1

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

$\sqrt{\frac{8 + 4 \cdot \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 ({\sqrt{2 - \sqrt{2}}}^{1} \sqrt{2 - \sqrt{2}}) + \sqrt{2 - \sqrt{2}} (- \sqrt{2 (2 - \sqrt{2})})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.2.3.2

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

$\sqrt{\frac{8 + 4 \cdot \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 ({\sqrt{2 - \sqrt{2}}}^{1} {\sqrt{2 - \sqrt{2}}}^{1}) + \sqrt{2 - \sqrt{2}} (- \sqrt{2 (2 - \sqrt{2})})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.2.3.3

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

$\sqrt{\frac{8 + 4 \cdot \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 {\sqrt{2 - \sqrt{2}}}^{1 + 1} + \sqrt{2 - \sqrt{2}} (- \sqrt{2 (2 - \sqrt{2})})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.2.3.4

Add $1$ and $1$ .

$\sqrt{\frac{8 + 4 \cdot \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 {\sqrt{2 - \sqrt{2}}}^{2} + \sqrt{2 - \sqrt{2}} (- \sqrt{2 (2 - \sqrt{2})})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.2.4

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

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Step 20.1.2.4.1

Combine using the product rule for radicals.

$\sqrt{\frac{8 + 4 \cdot \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 {\sqrt{2 - \sqrt{2}}}^{2} - \sqrt{(2 - \sqrt{2}) (2 (2 - \sqrt{2}))}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.2.4.2

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

$\sqrt{\frac{8 + 4 \cdot \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 {\sqrt{2 - \sqrt{2}}}^{2} - \sqrt{2 ({(2 - \sqrt{2})}^{1} (2 - \sqrt{2}))}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.2.4.3

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

$\sqrt{\frac{8 + 4 \cdot \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 {\sqrt{2 - \sqrt{2}}}^{2} - \sqrt{2 ({(2 - \sqrt{2})}^{1} {(2 - \sqrt{2})}^{1})}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.2.4.4

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

$\sqrt{\frac{8 + 4 \cdot \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 {\sqrt{2 - \sqrt{2}}}^{2} - \sqrt{2 {(2 - \sqrt{2})}^{1 + 1}}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.2.4.5

Add $1$ and $1$ .

$\sqrt{\frac{8 + 4 \cdot \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 {\sqrt{2 - \sqrt{2}}}^{2} - \sqrt{2 {(2 - \sqrt{2})}^{2}}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3

Simplify each term.

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Step 20.1.3.1

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

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Step 20.1.3.1.1

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

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 {({(2 - \sqrt{2})}^{\frac{1}{2}})}^{2} - \sqrt{2 {(2 - \sqrt{2})}^{2}}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.1.2

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

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 {(2 - \sqrt{2})}^{\frac{1}{2} \cdot 2} - \sqrt{2 {(2 - \sqrt{2})}^{2}}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.1.3

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

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 {(2 - \sqrt{2})}^{\frac{2}{2}} - \sqrt{2 {(2 - \sqrt{2})}^{2}}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.1.4

Cancel the common factor of $2$ .

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Step 20.1.3.1.4.1

Cancel the common factor.

Step 20.1.3.1.4.2

Rewrite the expression.

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 {(2 - \sqrt{2})}^{1} - \sqrt{2 {(2 - \sqrt{2})}^{2}}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.1.5

Simplify.

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 (2 - \sqrt{2}) - \sqrt{2 {(2 - \sqrt{2})}^{2}}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.2

Apply the distributive property.

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 2 \cdot 2 + 2 (- \sqrt{2}) - \sqrt{2 {(2 - \sqrt{2})}^{2}}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.3

Multiply $2$ by $2$ .

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 + 2 (- \sqrt{2}) - \sqrt{2 {(2 - \sqrt{2})}^{2}}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.4

Multiply $- 1$ by $2$ .

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - \sqrt{2 {(2 - \sqrt{2})}^{2}}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.5

Rewrite $2 {(2 - \sqrt{2})}^{2}$ as ${(2 + i^{2} \sqrt{2})}^{2} \cdot 2$ .

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Step 20.1.3.5.1

Reorder $2$ and ${(2 - \sqrt{2})}^{2}$ .

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - \sqrt{{(2 - \sqrt{2})}^{2} \cdot 2}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.5.2

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

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - \sqrt{{(2 + i^{2} \sqrt{2})}^{2} \cdot 2}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.6

Pull terms out from under the radical.

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - ((2 + i^{2} \sqrt{2}) \sqrt{2})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.7

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

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - ((2 - \sqrt{2}) \sqrt{2})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.8

Apply the distributive property.

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2} - \sqrt{2} \sqrt{2})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.9

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

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Step 20.1.3.9.1

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

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2} - ({\sqrt{2}}^{1} \sqrt{2}))}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.9.2

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

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1}))}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.9.3

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

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2} - {\sqrt{2}}^{1 + 1})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.9.4

Add $1$ and $1$ .

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2} - {\sqrt{2}}^{2})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.10

Simplify each term.

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Step 20.1.3.10.1

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

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Step 20.1.3.10.1.1

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

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2} - {(2^{\frac{1}{2}})}^{2})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.10.1.2

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

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2} - 2^{\frac{1}{2} \cdot 2})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.10.1.3

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

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2} - 2^{\frac{2}{2}})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.10.1.4

Cancel the common factor of $2$ .

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Step 20.1.3.10.1.4.1

Cancel the common factor.

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2} - 2^{\frac{2}{2}})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.10.1.4.2

Rewrite the expression.

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2} - 2^{1})}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.10.1.5

Evaluate the exponent.

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2} - 1 \cdot 2)}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.10.2

Multiply $- 1$ by $2$ .

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2} - 2)}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.11

Apply the distributive property.

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - (2 \sqrt{2}) - - 2}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.12

Multiply $2$ by $- 1$ .

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - 2 \sqrt{2} - - 2}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.3.13

Multiply $- 1$ by $- 2$ .

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 4 - 2 \sqrt{2} - 2 \sqrt{2} + 2}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.4

Add $4$ and $2$ .

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 6 - 2 \sqrt{2} - 2 \sqrt{2}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.5

Subtract $2 \sqrt{2}$ from $- 2 \sqrt{2}$ .

$\sqrt{\frac{8 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} + 6 - 4 \sqrt{2}}{2} - 2 \sqrt{2} + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2}}$

Step 20.1.6

Add $8$ and $6$ .

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

Step 20.2

Cancel the common factor of $14 + 4 \sqrt{2 - \sqrt{2}} - 2 \sqrt{(2 - \sqrt{2}) \cdot 2} - 4 \sqrt{2}$ and $2$ .

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Step 20.2.1

Factor $2$ out of $14$ .

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

Step 20.2.2

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

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

Step 20.2.3

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

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

Step 20.2.4

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

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

Step 20.2.5

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

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

Step 20.2.6

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

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

Step 20.2.7

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

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

Step 20.2.8

Cancel the common factors.

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Step 20.2.8.1

Factor $2$ out of $2$ .

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

Step 20.2.8.2

Cancel the common factor.

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

Step 20.2.8.3

Rewrite the expression.

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

Step 20.2.8.4

Divide $7 + 2 \sqrt{2 - \sqrt{2}} - \sqrt{(2 - \sqrt{2}) \cdot 2} - 2 \sqrt{2}$ by $1$ .

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

Step 21

Simplify by adding terms.

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Step 21.1

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

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

Step 21.2

Subtract $\sqrt{(2 - \sqrt{2}) \cdot 2}$ from $- \sqrt{(2 - \sqrt{2}) \cdot 2}$ .

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

Step 21.3

Subtract $2 \sqrt{2}$ from $- 2 \sqrt{2}$ .

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

Step 22

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.49660576 \dots$