Trigonometry Examples

$\tan (37.5)$

Step 1

Rewrite $37.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\tan (\frac{75}{2})$

Step 2

Apply the tangent half-angle identity.

$\pm \sqrt{\frac{1 - \cos (75)}{1 + \cos (75)}}$

Step 3

Change the $\pm$ to $+$ because tangent is positive in the first quadrant.

$\sqrt{\frac{1 - \cos (75)}{1 + \cos (75)}}$

Step 4

Simplify $\sqrt{\frac{1 - \cos (75)}{1 + \cos (75)}}$ .

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Step 4.1

The exact value of $\cos (75)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 4.1.1

Split $75$ into two angles where the values of the six trigonometric functions are known.

$\sqrt{\frac{1 - \cos (30 + 45)}{1 + \cos (75)}}$

Step 4.1.2

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$\sqrt{\frac{1 - (\cos (30) \cos (45) - \sin (30) \sin (45))}{1 + \cos (75)}}$

Step 4.1.3

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\sqrt{\frac{1 - (\frac{\sqrt{3}}{2} \cos (45) - \sin (30) \sin (45))}{1 + \cos (75)}}$

Step 4.1.4

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\sqrt{\frac{1 - (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45))}{1 + \cos (75)}}$

Step 4.1.5

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\sqrt{\frac{1 - (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45))}{1 + \cos (75)}}$

Step 4.1.6

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\sqrt{\frac{1 - (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})}{1 + \cos (75)}}$

Step 4.1.7

Simplify $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 4.1.7.1

Simplify each term.

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Step 4.1.7.1.1

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

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Step 4.1.7.1.1.1

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

$\sqrt{\frac{1 - (\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})}{1 + \cos (75)}}$

Step 4.1.7.1.1.2

Combine using the product rule for radicals.

$\sqrt{\frac{1 - (\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})}{1 + \cos (75)}}$

Step 4.1.7.1.1.3

Multiply $3$ by $2$ .

$\sqrt{\frac{1 - (\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})}{1 + \cos (75)}}$

Step 4.1.7.1.1.4

Multiply $2$ by $2$ .

$\sqrt{\frac{1 - (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})}{1 + \cos (75)}}$

Step 4.1.7.1.2

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

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Step 4.1.7.1.2.1

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

$\sqrt{\frac{1 - (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})}{1 + \cos (75)}}$

Step 4.1.7.1.2.2

Multiply $2$ by $2$ .

$\sqrt{\frac{1 - (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})}{1 + \cos (75)}}$

Step 4.1.7.2

Combine the numerators over the common denominator.

$\sqrt{\frac{1 - \frac{\sqrt{6} - \sqrt{2}}{4}}{1 + \cos (75)}}$

Step 4.2

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

$\sqrt{\frac{\frac{4}{4} - \frac{\sqrt{6} - \sqrt{2}}{4}}{1 + \cos (75)}}$

Step 4.3

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{4 - (\sqrt{6} - \sqrt{2})}{4}}{1 + \cos (75)}}$

Step 4.4

Rewrite $\frac{4 - (\sqrt{6} - \sqrt{2})}{4}$ in a factored form.

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Step 4.4.1

Apply the distributive property.

$\sqrt{\frac{\frac{4 - \sqrt{6} - - \sqrt{2}}{4}}{1 + \cos (75)}}$

Step 4.4.2

Multiply $- - \sqrt{2}$ .

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Step 4.4.2.1

Multiply $- 1$ by $- 1$ .

$\sqrt{\frac{\frac{4 - \sqrt{6} + 1 \sqrt{2}}{4}}{1 + \cos (75)}}$

Step 4.4.2.2

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

$\sqrt{\frac{\frac{4 - \sqrt{6} + \sqrt{2}}{4}}{1 + \cos (75)}}$

Step 4.5

The exact value of $\cos (75)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 4.5.1

Split $75$ into two angles where the values of the six trigonometric functions are known.

$\sqrt{\frac{\frac{4 - \sqrt{6} + \sqrt{2}}{4}}{1 + \cos (30 + 45)}}$

Step 4.5.2

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$\sqrt{\frac{\frac{4 - \sqrt{6} + \sqrt{2}}{4}}{1 + \cos (30) \cos (45) - \sin (30) \sin (45)}}$

Step 4.5.3

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\sqrt{\frac{\frac{4 - \sqrt{6} + \sqrt{2}}{4}}{1 + \frac{\sqrt{3}}{2} \cos (45) - \sin (30) \sin (45)}}$

Step 4.5.4

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\sqrt{\frac{\frac{4 - \sqrt{6} + \sqrt{2}}{4}}{1 + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45)}}$

Step 4.5.5

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\sqrt{\frac{\frac{4 - \sqrt{6} + \sqrt{2}}{4}}{1 + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45)}}$

Step 4.5.6

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

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

Step 4.5.7

Simplify $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 4.5.7.1

Simplify each term.

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Step 4.5.7.1.1

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

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Step 4.5.7.1.1.1

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

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

Step 4.5.7.1.1.2

Combine using the product rule for radicals.

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

Step 4.5.7.1.1.3

Multiply $3$ by $2$ .

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

Step 4.5.7.1.1.4

Multiply $2$ by $2$ .

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

Step 4.5.7.1.2

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

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Step 4.5.7.1.2.1

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

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

Step 4.5.7.1.2.2

Multiply $2$ by $2$ .

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

Step 4.5.7.2

Combine the numerators over the common denominator.

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

Step 4.6

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

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

Step 4.7

Combine the numerators over the common denominator.

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

Step 4.8

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.9

Cancel the common factor of $4$ .

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Step 4.9.1

Cancel the common factor.

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

Step 4.9.2

Rewrite the expression.

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

Step 4.10

Multiply $\frac{1}{4 + \sqrt{6} - \sqrt{2}}$ by $\frac{4 + \sqrt{6} + \sqrt{2}}{4 + \sqrt{6} + \sqrt{2}}$ .

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

Step 4.11

Multiply $\frac{1}{4 + \sqrt{6} - \sqrt{2}}$ by $\frac{4 + \sqrt{6} + \sqrt{2}}{4 + \sqrt{6} + \sqrt{2}}$ .

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

Step 4.12

Expand the denominator using the FOIL method.

$\sqrt{(4 - \sqrt{6} + \sqrt{2}) \frac{4 + \sqrt{6} + \sqrt{2}}{16 + 4 \sqrt{6} + 4 \sqrt{2} + \sqrt{6} \cdot 4 + {\sqrt{6}}^{2} + \sqrt{12} - 4 \sqrt{2} - \sqrt{12} - {\sqrt{2}}^{2}}}$

Step 4.13

Simplify.

$\sqrt{(4 - \sqrt{6} + \sqrt{2}) \frac{4 + \sqrt{6} + \sqrt{2}}{20 + 8 \sqrt{6}}}$

Step 4.14

Multiply $\frac{4 + \sqrt{6} + \sqrt{2}}{20 + 8 \sqrt{6}}$ by $\frac{20 - 8 \sqrt{6}}{20 - 8 \sqrt{6}}$ .

$\sqrt{(4 - \sqrt{6} + \sqrt{2}) (\frac{4 + \sqrt{6} + \sqrt{2}}{20 + 8 \sqrt{6}} \cdot \frac{20 - 8 \sqrt{6}}{20 - 8 \sqrt{6}})}$

Step 4.15

Multiply $\frac{4 + \sqrt{6} + \sqrt{2}}{20 + 8 \sqrt{6}}$ by $\frac{20 - 8 \sqrt{6}}{20 - 8 \sqrt{6}}$ .

$\sqrt{(4 - \sqrt{6} + \sqrt{2}) \frac{(4 + \sqrt{6} + \sqrt{2}) (20 - 8 \sqrt{6})}{(20 + 8 \sqrt{6}) (20 - 8 \sqrt{6})}}$

Step 4.16

Expand the denominator using the FOIL method.

$\sqrt{(4 - \sqrt{6} + \sqrt{2}) \frac{(4 + \sqrt{6} + \sqrt{2}) (20 - 8 \sqrt{6})}{400 - 160 \sqrt{6} + 160 \sqrt{6} - 64 {\sqrt{6}}^{2}}}$

Step 4.17

Simplify.

$\sqrt{(4 - \sqrt{6} + \sqrt{2}) \frac{(4 + \sqrt{6} + \sqrt{2}) (20 - 8 \sqrt{6})}{16}}$

Step 4.18

Cancel the common factor of $20 - 8 \sqrt{6}$ and $16$ .

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Step 4.18.1

Factor $4$ out of $(4 + \sqrt{6} + \sqrt{2}) (20 - 8 \sqrt{6})$ .

$\sqrt{(4 - \sqrt{6} + \sqrt{2}) \frac{4 ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{16}}$

Step 4.18.2

Cancel the common factors.

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Step 4.18.2.1

Factor $4$ out of $16$ .

$\sqrt{(4 - \sqrt{6} + \sqrt{2}) \frac{4 ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4 (4)}}$

Step 4.18.2.2

Cancel the common factor.

$\sqrt{(4 - \sqrt{6} + \sqrt{2}) \frac{4 ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4 \cdot 4}}$

Step 4.18.2.3

Rewrite the expression.

$\sqrt{(4 - \sqrt{6} + \sqrt{2}) \frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})}{4}}$

Step 4.19

Apply the distributive property.

$\sqrt{4 \frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})}{4} - \sqrt{6} \frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})}{4} + \sqrt{2} \frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})}{4}}$

Step 4.20

Simplify.

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Step 4.20.1

Cancel the common factor of $4$ .

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Step 4.20.1.1

Cancel the common factor.

$\sqrt{4 \frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})}{4} - \sqrt{6} \frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})}{4} + \sqrt{2} \frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})}{4}}$

Step 4.20.1.2

Rewrite the expression.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) - \sqrt{6} \frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})}{4} + \sqrt{2} \frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})}{4}}$

Step 4.20.2

Combine $\frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})}{4}$ and $\sqrt{6}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) - \frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) \sqrt{6}}{4} + \sqrt{2} \frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})}{4}}$

Step 4.20.3

Combine $\sqrt{2}$ and $\frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})}{4}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) - \frac{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) \sqrt{6}}{4} + \frac{\sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.21

Combine the numerators over the common denominator.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- (4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22

Simplify each term.

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Step 4.22.1

Apply the distributive property.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 1 \cdot 4 - \sqrt{6} - \sqrt{2}) (5 - 2 \sqrt{6}) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.2

Multiply $- 1$ by $4$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 4 - \sqrt{6} - \sqrt{2}) (5 - 2 \sqrt{6}) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.3

Expand $(- 4 - \sqrt{6} - \sqrt{2}) (5 - 2 \sqrt{6})$ by multiplying each term in the first expression by each term in the second expression.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 4 \cdot 5 - 4 (- 2 \sqrt{6}) - \sqrt{6} \cdot 5 - \sqrt{6} (- 2 \sqrt{6}) - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4

Simplify each term.

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Step 4.22.4.1

Multiply $- 4$ by $5$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 - 4 (- 2 \sqrt{6}) - \sqrt{6} \cdot 5 - \sqrt{6} (- 2 \sqrt{6}) - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.2

Multiply $- 2$ by $- 4$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - \sqrt{6} \cdot 5 - \sqrt{6} (- 2 \sqrt{6}) - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.3

Multiply $5$ by $- 1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} - \sqrt{6} (- 2 \sqrt{6}) - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.4

Multiply $- \sqrt{6} (- 2 \sqrt{6})$ .

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Step 4.22.4.4.1

Multiply $- 2$ by $- 1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 2 \sqrt{6} \sqrt{6} - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.4.2

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 2 ({\sqrt{6}}^{1} \sqrt{6}) - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.4.3

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 2 ({\sqrt{6}}^{1} {\sqrt{6}}^{1}) - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.4.4

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 2 {\sqrt{6}}^{1 + 1} - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.4.5

Add $1$ and $1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 2 {\sqrt{6}}^{2} - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.5

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.22.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 2 {(6^{\frac{1}{2}})}^{2} - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.5.2

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 2 \cdot 6^{\frac{1}{2} \cdot 2} - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.5.3

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 2 \cdot 6^{\frac{2}{2}} - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.5.4

Cancel the common factor of $2$ .

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Step 4.22.4.5.4.1

Cancel the common factor.

Step 4.22.4.5.4.2

Rewrite the expression.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 2 \cdot 6^{1} - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.5.5

Evaluate the exponent.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 2 \cdot 6 - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.6

Multiply $2$ by $6$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 12 - \sqrt{2} \cdot 5 - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.7

Multiply $5$ by $- 1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 12 - 5 \sqrt{2} - \sqrt{2} (- 2 \sqrt{6})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.8

Multiply $- \sqrt{2} (- 2 \sqrt{6})$ .

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Step 4.22.4.8.1

Multiply $- 2$ by $- 1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 12 - 5 \sqrt{2} + 2 \sqrt{2} \sqrt{6}) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.8.2

Combine using the product rule for radicals.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 12 - 5 \sqrt{2} + 2 \sqrt{6 \cdot 2}) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.8.3

Multiply $6$ by $2$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 12 - 5 \sqrt{2} + 2 \sqrt{12}) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.9

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.22.4.9.1

Factor $4$ out of $12$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 12 - 5 \sqrt{2} + 2 \sqrt{4 (3)}) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.9.2

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 12 - 5 \sqrt{2} + 2 \sqrt{2^{2} \cdot 3}) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.10

Pull terms out from under the radical.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 12 - 5 \sqrt{2} + 2 (2 \sqrt{3})) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.4.11

Multiply $2$ by $2$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 20 + 8 \sqrt{6} - 5 \sqrt{6} + 12 - 5 \sqrt{2} + 4 \sqrt{3}) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.5

Add $- 20$ and $12$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 8 + 8 \sqrt{6} - 5 \sqrt{6} - 5 \sqrt{2} + 4 \sqrt{3}) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.6

Subtract $5 \sqrt{6}$ from $8 \sqrt{6}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{(- 8 + 3 \sqrt{6} - 5 \sqrt{2} + 4 \sqrt{3}) \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.7

Apply the distributive property.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 \sqrt{6} \sqrt{6} - 5 \sqrt{2} \sqrt{6} + 4 \sqrt{3} \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.8

Simplify.

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Step 4.22.8.1

Multiply $3 \sqrt{6} \sqrt{6}$ .

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Step 4.22.8.1.1

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 ({\sqrt{6}}^{1} \sqrt{6}) - 5 \sqrt{2} \sqrt{6} + 4 \sqrt{3} \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.8.1.2

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 ({\sqrt{6}}^{1} {\sqrt{6}}^{1}) - 5 \sqrt{2} \sqrt{6} + 4 \sqrt{3} \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.8.1.3

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 {\sqrt{6}}^{1 + 1} - 5 \sqrt{2} \sqrt{6} + 4 \sqrt{3} \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.8.1.4

Add $1$ and $1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 {\sqrt{6}}^{2} - 5 \sqrt{2} \sqrt{6} + 4 \sqrt{3} \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.8.2

Multiply $- 5 \sqrt{2} \sqrt{6}$ .

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Step 4.22.8.2.1

Combine using the product rule for radicals.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 {\sqrt{6}}^{2} - 5 \sqrt{6 \cdot 2} + 4 \sqrt{3} \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.8.2.2

Multiply $6$ by $2$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 {\sqrt{6}}^{2} - 5 \sqrt{12} + 4 \sqrt{3} \sqrt{6} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.8.3

Multiply $4 \sqrt{3} \sqrt{6}$ .

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Step 4.22.8.3.1

Combine using the product rule for radicals.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 {\sqrt{6}}^{2} - 5 \sqrt{12} + 4 \sqrt{6 \cdot 3} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.8.3.2

Multiply $6$ by $3$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 {\sqrt{6}}^{2} - 5 \sqrt{12} + 4 \sqrt{18} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9

Simplify each term.

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Step 4.22.9.1

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.22.9.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 {(6^{\frac{1}{2}})}^{2} - 5 \sqrt{12} + 4 \sqrt{18} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.1.2

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 \cdot 6^{\frac{1}{2} \cdot 2} - 5 \sqrt{12} + 4 \sqrt{18} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.1.3

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 \cdot 6^{\frac{2}{2}} - 5 \sqrt{12} + 4 \sqrt{18} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.1.4

Cancel the common factor of $2$ .

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Step 4.22.9.1.4.1

Cancel the common factor.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 \cdot 6^{\frac{2}{2}} - 5 \sqrt{12} + 4 \sqrt{18} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.1.4.2

Rewrite the expression.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 \cdot 6^{1} - 5 \sqrt{12} + 4 \sqrt{18} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.1.5

Evaluate the exponent.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 3 \cdot 6 - 5 \sqrt{12} + 4 \sqrt{18} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.2

Multiply $3$ by $6$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 5 \sqrt{12} + 4 \sqrt{18} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.3

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.22.9.3.1

Factor $4$ out of $12$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 5 \sqrt{4 (3)} + 4 \sqrt{18} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.3.2

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 5 \sqrt{2^{2} \cdot 3} + 4 \sqrt{18} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.4

Pull terms out from under the radical.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 5 (2 \sqrt{3}) + 4 \sqrt{18} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.5

Multiply $2$ by $- 5$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 4 \sqrt{18} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.6

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.22.9.6.1

Factor $9$ out of $18$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 4 \sqrt{9 (2)} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.6.2

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 4 \sqrt{3^{2} \cdot 2} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.7

Pull terms out from under the radical.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 4 (3 \sqrt{2}) + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.9.8

Multiply $3$ by $4$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} ((4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}))}{4}}$

Step 4.22.10

Expand $(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})$ by multiplying each term in the first expression by each term in the second expression.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (4 \cdot 5 + 4 (- 2 \sqrt{6}) + \sqrt{6} \cdot 5 + \sqrt{6} (- 2 \sqrt{6}) + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11

Simplify each term.

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Step 4.22.11.1

Multiply $4$ by $5$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 + 4 (- 2 \sqrt{6}) + \sqrt{6} \cdot 5 + \sqrt{6} (- 2 \sqrt{6}) + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.2

Multiply $- 2$ by $4$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + \sqrt{6} \cdot 5 + \sqrt{6} (- 2 \sqrt{6}) + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.3

Move $5$ to the left of $\sqrt{6}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \cdot \sqrt{6} + \sqrt{6} (- 2 \sqrt{6}) + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.4

Multiply $\sqrt{6} (- 2 \sqrt{6})$ .

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Step 4.22.11.4.1

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 ({\sqrt{6}}^{1} \sqrt{6}) + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.4.2

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 ({\sqrt{6}}^{1} {\sqrt{6}}^{1}) + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.4.3

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 {\sqrt{6}}^{1 + 1} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.4.4

Add $1$ and $1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 {\sqrt{6}}^{2} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.5

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.22.11.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 {(6^{\frac{1}{2}})}^{2} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.5.2

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 \cdot 6^{\frac{1}{2} \cdot 2} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.5.3

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 \cdot 6^{\frac{2}{2}} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.5.4

Cancel the common factor of $2$ .

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Step 4.22.11.5.4.1

Cancel the common factor.

Step 4.22.11.5.4.2

Rewrite the expression.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 \cdot 6^{1} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.5.5

Evaluate the exponent.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 \cdot 6 + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.6

Multiply $- 2$ by $6$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.7

Move $5$ to the left of $\sqrt{2}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \cdot \sqrt{2} + \sqrt{2} (- 2 \sqrt{6}))}{4}}$

Step 4.22.11.8

Multiply $\sqrt{2} (- 2 \sqrt{6})$ .

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Step 4.22.11.8.1

Combine using the product rule for radicals.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \sqrt{2} - 2 \sqrt{2 \cdot 6})}{4}}$

Step 4.22.11.8.2

Multiply $2$ by $6$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \sqrt{2} - 2 \sqrt{12})}{4}}$

Step 4.22.11.9

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.22.11.9.1

Factor $4$ out of $12$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \sqrt{2} - 2 \sqrt{4 (3)})}{4}}$

Step 4.22.11.9.2

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \sqrt{2} - 2 \sqrt{2^{2} \cdot 3})}{4}}$

Step 4.22.11.10

Pull terms out from under the radical.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \sqrt{2} - 2 (2 \sqrt{3}))}{4}}$

Step 4.22.11.11

Multiply $2$ by $- 2$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \sqrt{2} - 4 \sqrt{3})}{4}}$

Step 4.22.12

Subtract $12$ from $20$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (8 - 8 \sqrt{6} + 5 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3})}{4}}$

Step 4.22.13

Add $- 8 \sqrt{6}$ and $5 \sqrt{6}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} (8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3})}{4}}$

Step 4.22.14

Apply the distributive property.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + \sqrt{2} \cdot 8 + \sqrt{2} (- 3 \sqrt{6}) + \sqrt{2} (5 \sqrt{2}) + \sqrt{2} (- 4 \sqrt{3})}{4}}$

Step 4.22.15

Simplify.

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Step 4.22.15.1

Move $8$ to the left of $\sqrt{2}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \cdot \sqrt{2} + \sqrt{2} (- 3 \sqrt{6}) + \sqrt{2} (5 \sqrt{2}) + \sqrt{2} (- 4 \sqrt{3})}{4}}$

Step 4.22.15.2

Multiply $\sqrt{2} (- 3 \sqrt{6})$ .

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Step 4.22.15.2.1

Combine using the product rule for radicals.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \cdot \sqrt{2} - 3 \sqrt{2 \cdot 6} + \sqrt{2} (5 \sqrt{2}) + \sqrt{2} (- 4 \sqrt{3})}{4}}$

Step 4.22.15.2.2

Multiply $2$ by $6$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \cdot \sqrt{2} - 3 \sqrt{12} + \sqrt{2} (5 \sqrt{2}) + \sqrt{2} (- 4 \sqrt{3})}{4}}$

Step 4.22.15.3

Multiply $\sqrt{2} (5 \sqrt{2})$ .

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Step 4.22.15.3.1

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \cdot \sqrt{2} - 3 \sqrt{12} + 5 ({\sqrt{2}}^{1} \sqrt{2}) + \sqrt{2} (- 4 \sqrt{3})}{4}}$

Step 4.22.15.3.2

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \cdot \sqrt{2} - 3 \sqrt{12} + 5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + \sqrt{2} (- 4 \sqrt{3})}{4}}$

Step 4.22.15.3.3

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \cdot \sqrt{2} - 3 \sqrt{12} + 5 {\sqrt{2}}^{1 + 1} + \sqrt{2} (- 4 \sqrt{3})}{4}}$

Step 4.22.15.3.4

Add $1$ and $1$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \cdot \sqrt{2} - 3 \sqrt{12} + 5 {\sqrt{2}}^{2} + \sqrt{2} (- 4 \sqrt{3})}{4}}$

Step 4.22.15.4

Multiply $\sqrt{2} (- 4 \sqrt{3})$ .

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Step 4.22.15.4.1

Combine using the product rule for radicals.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \cdot \sqrt{2} - 3 \sqrt{12} + 5 {\sqrt{2}}^{2} - 4 \sqrt{2 \cdot 3}}{4}}$

Step 4.22.15.4.2

Multiply $2$ by $3$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \cdot \sqrt{2} - 3 \sqrt{12} + 5 {\sqrt{2}}^{2} - 4 \sqrt{6}}{4}}$

Step 4.22.16

Simplify each term.

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Step 4.22.16.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.22.16.1.1

Factor $4$ out of $12$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 3 \sqrt{4 (3)} + 5 {\sqrt{2}}^{2} - 4 \sqrt{6}}{4}}$

Step 4.22.16.1.2

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 3 \sqrt{2^{2} \cdot 3} + 5 {\sqrt{2}}^{2} - 4 \sqrt{6}}{4}}$

Step 4.22.16.2

Pull terms out from under the radical.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 3 (2 \sqrt{3}) + 5 {\sqrt{2}}^{2} - 4 \sqrt{6}}{4}}$

Step 4.22.16.3

Multiply $2$ by $- 3$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 6 \sqrt{3} + 5 {\sqrt{2}}^{2} - 4 \sqrt{6}}{4}}$

Step 4.22.16.4

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

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Step 4.22.16.4.1

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 6 \sqrt{3} + 5 {(2^{\frac{1}{2}})}^{2} - 4 \sqrt{6}}{4}}$

Step 4.22.16.4.2

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 6 \sqrt{3} + 5 \cdot 2^{\frac{1}{2} \cdot 2} - 4 \sqrt{6}}{4}}$

Step 4.22.16.4.3

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

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 6 \sqrt{3} + 5 \cdot 2^{\frac{2}{2}} - 4 \sqrt{6}}{4}}$

Step 4.22.16.4.4

Cancel the common factor of $2$ .

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Step 4.22.16.4.4.1

Cancel the common factor.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 6 \sqrt{3} + 5 \cdot 2^{\frac{2}{2}} - 4 \sqrt{6}}{4}}$

Step 4.22.16.4.4.2

Rewrite the expression.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 6 \sqrt{3} + 5 \cdot 2^{1} - 4 \sqrt{6}}{4}}$

Step 4.22.16.4.5

Evaluate the exponent.

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 6 \sqrt{3} + 5 \cdot 2 - 4 \sqrt{6}}{4}}$

Step 4.22.16.5

Multiply $5$ by $2$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{- 8 \sqrt{6} + 18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 6 \sqrt{3} + 10 - 4 \sqrt{6}}{4}}$

Step 4.23

Subtract $4 \sqrt{6}$ from $- 8 \sqrt{6}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{18 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 6 \sqrt{3} + 10 - 12 \sqrt{6}}{4}}$

Step 4.24

Add $18$ and $10$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{28 - 10 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 6 \sqrt{3} - 12 \sqrt{6}}{4}}$

Step 4.25

Subtract $6 \sqrt{3}$ from $- 10 \sqrt{3}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 12 \sqrt{2} + 8 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.26

Add $12 \sqrt{2}$ and $8 \sqrt{2}$ .

$\sqrt{(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27

Simplify each term.

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Step 4.27.1

Expand $(4 + \sqrt{6} + \sqrt{2}) (5 - 2 \sqrt{6})$ by multiplying each term in the first expression by each term in the second expression.

$\sqrt{4 \cdot 5 + 4 (- 2 \sqrt{6}) + \sqrt{6} \cdot 5 + \sqrt{6} (- 2 \sqrt{6}) + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2

Simplify each term.

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Step 4.27.2.1

Multiply $4$ by $5$ .

$\sqrt{20 + 4 (- 2 \sqrt{6}) + \sqrt{6} \cdot 5 + \sqrt{6} (- 2 \sqrt{6}) + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.2

Multiply $- 2$ by $4$ .

$\sqrt{20 - 8 \sqrt{6} + \sqrt{6} \cdot 5 + \sqrt{6} (- 2 \sqrt{6}) + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.3

Move $5$ to the left of $\sqrt{6}$ .

$\sqrt{20 - 8 \sqrt{6} + 5 \cdot \sqrt{6} + \sqrt{6} (- 2 \sqrt{6}) + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.4

Multiply $\sqrt{6} (- 2 \sqrt{6})$ .

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Step 4.27.2.4.1

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 ({\sqrt{6}}^{1} \sqrt{6}) + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.4.2

Raise $\sqrt{6}$ to the power of $1$ .

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 ({\sqrt{6}}^{1} {\sqrt{6}}^{1}) + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.4.3

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

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 {\sqrt{6}}^{1 + 1} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.4.4

Add $1$ and $1$ .

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 {\sqrt{6}}^{2} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.5

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.27.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 {(6^{\frac{1}{2}})}^{2} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.5.2

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

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 \cdot 6^{\frac{1}{2} \cdot 2} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.5.3

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

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 \cdot 6^{\frac{2}{2}} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.5.4

Cancel the common factor of $2$ .

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Step 4.27.2.5.4.1

Cancel the common factor.

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 \cdot 6^{\frac{2}{2}} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.5.4.2

Rewrite the expression.

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 \cdot 6^{1} + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.5.5

Evaluate the exponent.

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 2 \cdot 6 + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.6

Multiply $- 2$ by $6$ .

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + \sqrt{2} \cdot 5 + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.7

Move $5$ to the left of $\sqrt{2}$ .

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \cdot \sqrt{2} + \sqrt{2} (- 2 \sqrt{6}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.8

Multiply $\sqrt{2} (- 2 \sqrt{6})$ .

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Step 4.27.2.8.1

Combine using the product rule for radicals.

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \sqrt{2} - 2 \sqrt{2 \cdot 6} + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.8.2

Multiply $2$ by $6$ .

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \sqrt{2} - 2 \sqrt{12} + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.9

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.27.2.9.1

Factor $4$ out of $12$ .

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \sqrt{2} - 2 \sqrt{4 (3)} + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.9.2

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

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \sqrt{2} - 2 \sqrt{2^{2} \cdot 3} + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.10

Pull terms out from under the radical.

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \sqrt{2} - 2 (2 \sqrt{3}) + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.2.11

Multiply $2$ by $- 2$ .

$\sqrt{20 - 8 \sqrt{6} + 5 \sqrt{6} - 12 + 5 \sqrt{2} - 4 \sqrt{3} + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.3

Subtract $12$ from $20$ .

$\sqrt{8 - 8 \sqrt{6} + 5 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.4

Add $- 8 \sqrt{6}$ and $5 \sqrt{6}$ .

$\sqrt{8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + \frac{28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.5

Cancel the common factor of $28 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}$ and $4$ .

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Step 4.27.5.1

Factor $4$ out of $28$ .

$\sqrt{8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + \frac{4 \cdot 7 - 16 \sqrt{3} + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.5.2

Factor $4$ out of $- 16 \sqrt{3}$ .

$\sqrt{8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + \frac{4 \cdot 7 + 4 (- 4 \sqrt{3}) + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.5.3

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

$\sqrt{8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + \frac{4 (7 - 4 \sqrt{3}) + 20 \sqrt{2} - 12 \sqrt{6}}{4}}$

Step 4.27.5.4

Factor $4$ out of $20 \sqrt{2}$ .

$\sqrt{8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + \frac{4 (7 - 4 \sqrt{3}) + 4 (5 \sqrt{2}) - 12 \sqrt{6}}{4}}$

Step 4.27.5.5

Factor $4$ out of $4 (7 - 4 \sqrt{3}) + 4 (5 \sqrt{2})$ .

$\sqrt{8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + \frac{4 (7 - 4 \sqrt{3} + 5 \sqrt{2}) - 12 \sqrt{6}}{4}}$

Step 4.27.5.6

Factor $4$ out of $- 12 \sqrt{6}$ .

$\sqrt{8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + \frac{4 (7 - 4 \sqrt{3} + 5 \sqrt{2}) + 4 (- 3 \sqrt{6})}{4}}$

Step 4.27.5.7

Factor $4$ out of $4 (7 - 4 \sqrt{3} + 5 \sqrt{2}) + 4 (- 3 \sqrt{6})$ .

$\sqrt{8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + \frac{4 (7 - 4 \sqrt{3} + 5 \sqrt{2} - 3 \sqrt{6})}{4}}$

Step 4.27.5.8

Cancel the common factors.

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Step 4.27.5.8.1

Factor $4$ out of $4$ .

$\sqrt{8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + \frac{4 (7 - 4 \sqrt{3} + 5 \sqrt{2} - 3 \sqrt{6})}{4 (1)}}$

Step 4.27.5.8.2

Cancel the common factor.

$\sqrt{8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + \frac{4 (7 - 4 \sqrt{3} + 5 \sqrt{2} - 3 \sqrt{6})}{4 \cdot 1}}$

Step 4.27.5.8.3

Rewrite the expression.

$\sqrt{8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + \frac{7 - 4 \sqrt{3} + 5 \sqrt{2} - 3 \sqrt{6}}{1}}$

Step 4.27.5.8.4

Divide $7 - 4 \sqrt{3} + 5 \sqrt{2} - 3 \sqrt{6}$ by $1$ .

$\sqrt{8 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} + 7 - 4 \sqrt{3} + 5 \sqrt{2} - 3 \sqrt{6}}$

Step 4.28

Add $8$ and $7$ .

$\sqrt{15 - 3 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} - 4 \sqrt{3} + 5 \sqrt{2} - 3 \sqrt{6}}$

Step 4.29

Subtract $3 \sqrt{6}$ from $- 3 \sqrt{6}$ .

$\sqrt{15 - 6 \sqrt{6} + 5 \sqrt{2} - 4 \sqrt{3} - 4 \sqrt{3} + 5 \sqrt{2}}$

Step 4.30

Add $5 \sqrt{2}$ and $5 \sqrt{2}$ .

$\sqrt{15 - 6 \sqrt{6} + 10 \sqrt{2} - 4 \sqrt{3} - 4 \sqrt{3}}$

Step 4.31

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

$\sqrt{15 - 6 \sqrt{6} + 10 \sqrt{2} - 8 \sqrt{3}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt{15 - 6 \sqrt{6} + 10 \sqrt{2} - 8 \sqrt{3}}$

Decimal Form:

$0.76732698 \dots$