Trigonometry Examples

$\csc (\frac{5 π}{24})$

Step 1

Rewrite $\frac{5 π}{24}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\csc (\frac{\frac{5 π}{12}}{2})$

Step 2

Apply the reciprocal identity to $\csc (\frac{\frac{5 π}{12}}{2})$ .

$\frac{1}{\sin (\frac{\frac{5 π}{12}}{2})}$

Step 3

Apply the sine half-angle identity.

$\frac{1}{\pm \sqrt{\frac{1 - \cos (\frac{5 π}{12})}{2}}}$

Step 4

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

$\frac{1}{\sqrt{\frac{1 - \cos (\frac{5 π}{12})}{2}}}$

Step 5

Simplify $\frac{1}{\sqrt{\frac{1 - \cos (\frac{5 π}{12})}{2}}}$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

The exact value of $\cos (\frac{5 π}{12})$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 5.1.1.1

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

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

Step 5.1.1.2

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

$\frac{1}{\sqrt{\frac{1 - (\cos (\frac{π}{6}) \cos (\frac{π}{4}) - \sin (\frac{π}{6}) \sin (\frac{π}{4}))}{2}}}$

Step 5.1.1.3

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

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

Step 5.1.1.4

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

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

Step 5.1.1.5

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{1}{\sqrt{\frac{1 - (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (\frac{π}{4}))}{2}}}$

Step 5.1.1.6

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

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

Step 5.1.1.7

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

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Step 5.1.1.7.1

Simplify each term.

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Step 5.1.1.7.1.1

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

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Step 5.1.1.7.1.1.1

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

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

Step 5.1.1.7.1.1.2

Combine using the product rule for radicals.

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

Step 5.1.1.7.1.1.3

Multiply $3$ by $2$ .

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

Step 5.1.1.7.1.1.4

Multiply $2$ by $2$ .

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

Step 5.1.1.7.1.2

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

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Step 5.1.1.7.1.2.1

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

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

Step 5.1.1.7.1.2.2

Multiply $2$ by $2$ .

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

Step 5.1.1.7.2

Combine the numerators over the common denominator.

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

Step 5.1.2

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

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

Step 5.1.3

Combine the numerators over the common denominator.

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

Step 5.1.4

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

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Step 5.1.4.1

Apply the distributive property.

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

Step 5.1.4.2

Multiply $- - \sqrt{2}$ .

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Step 5.1.4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 5.1.4.2.2

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

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

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.2

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

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Step 5.2.2.1

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

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

Step 5.2.2.2

Multiply $4$ by $2$ .

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

Step 5.2.3

Rewrite $\sqrt{\frac{4 - \sqrt{6} + \sqrt{2}}{8}}$ as $\frac{\sqrt{4 - \sqrt{6} + \sqrt{2}}}{\sqrt{8}}$ .

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

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.2.4.1.1

Factor $4$ out of $8$ .

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

Step 5.2.4.1.2

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

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

Step 5.2.4.2

Pull terms out from under the radical.

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

Step 5.2.5

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

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

Step 5.2.6

Combine and simplify the denominator.

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Step 5.2.6.1

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

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

Step 5.2.6.2

Move $\sqrt{2}$ .

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

Step 5.2.6.3

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

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

Step 5.2.6.4

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

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

Step 5.2.6.5

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

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

Step 5.2.6.6

Add $1$ and $1$ .

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

Step 5.2.6.7

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

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Step 5.2.6.7.1

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

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

Step 5.2.6.7.2

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

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

Step 5.2.6.7.3

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

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

Step 5.2.6.7.4

Cancel the common factor of $2$ .

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Step 5.2.6.7.4.1

Cancel the common factor.

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

Step 5.2.6.7.4.2

Rewrite the expression.

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

Step 5.2.6.7.5

Evaluate the exponent.

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

Step 5.2.7

Combine using the product rule for radicals.

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

Step 5.2.8

Multiply $2$ by $2$ .

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

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.4

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

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

Step 5.5

Multiply $\frac{4}{\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}$ by $\frac{\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}{\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}$ .

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

Step 5.6

Combine and simplify the denominator.

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Step 5.6.1

Multiply $\frac{4}{\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}$ by $\frac{\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}{\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}$ .

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

Step 5.6.2

Raise $\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}$ to the power of $1$ .

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

Step 5.6.3

Raise $\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}$ to the power of $1$ .

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

Step 5.6.4

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

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

Step 5.6.5

Add $1$ and $1$ .

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

Step 5.6.6

Rewrite ${\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}}^{2}$ as $(4 - \sqrt{6} + \sqrt{2}) \cdot 2$ .

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Step 5.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(4 - \sqrt{6} + \sqrt{2}) \cdot 2}$ as ${((4 - \sqrt{6} + \sqrt{2}) \cdot 2)}^{\frac{1}{2}}$ .

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

Step 5.6.6.2

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

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

Step 5.6.6.3

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

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

Step 5.6.6.4

Cancel the common factor of $2$ .

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Step 5.6.6.4.1

Cancel the common factor.

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

Step 5.6.6.4.2

Rewrite the expression.

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

Step 5.6.6.5

Simplify.

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

Step 5.7

Cancel the common factor of $4$ and $2$ .

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Step 5.7.1

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

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

Step 5.7.2

Cancel the common factors.

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Step 5.7.2.1

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

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

Step 5.7.2.2

Cancel the common factor.

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

Step 5.7.2.3

Rewrite the expression.

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

Step 5.8

Simplify the numerator.

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Step 5.8.1

Apply the distributive property.

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

Step 5.8.2

Simplify.

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Step 5.8.2.1

Multiply $4$ by $2$ .

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

Step 5.8.2.2

Multiply $2$ by $- 1$ .

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

Step 5.8.2.3

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

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

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

Step 5.9

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

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

Step 5.10

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

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

Step 5.11

Expand the denominator using the FOIL method.

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

Step 5.12

Simplify.

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

Step 5.13

Cancel the common factor of $2$ and $20 - 8 \sqrt{6}$ .

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Step 5.13.1

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

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

Step 5.13.2

Cancel the common factors.

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Step 5.13.2.1

Factor $2$ out of $20$ .

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

Step 5.13.2.2

Factor $2$ out of $- 8 \sqrt{6}$ .

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

Step 5.13.2.3

Factor $2$ out of $2 (10) + 2 (- 4 \sqrt{6})$ .

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

Step 5.13.2.4

Cancel the common factor.

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

Step 5.13.2.5

Rewrite the expression.

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

Step 5.14

Multiply $\frac{\sqrt{8 - 2 \sqrt{6} + 2 \sqrt{2}} (4 - \sqrt{6} - \sqrt{2})}{10 - 4 \sqrt{6}}$ by $\frac{10 + 4 \sqrt{6}}{10 + 4 \sqrt{6}}$ .

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

Step 5.15

Multiply $\frac{\sqrt{8 - 2 \sqrt{6} + 2 \sqrt{2}} (4 - \sqrt{6} - \sqrt{2})}{10 - 4 \sqrt{6}}$ by $\frac{10 + 4 \sqrt{6}}{10 + 4 \sqrt{6}}$ .

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

Step 5.16

Expand the denominator using the FOIL method.

$\frac{\sqrt{8 - 2 \sqrt{6} + 2 \sqrt{2}} (4 - \sqrt{6} - \sqrt{2}) (10 + 4 \sqrt{6})}{100 + 40 \sqrt{6} - 40 \sqrt{6} - 16 {\sqrt{6}}^{2}}$

Step 5.17

Simplify.

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

Step 5.18

Cancel the common factor of $10 + 4 \sqrt{6}$ and $4$ .

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Step 5.18.1

Factor $2$ out of $\sqrt{8 - 2 \sqrt{6} + 2 \sqrt{2}} (4 - \sqrt{6} - \sqrt{2}) (10 + 4 \sqrt{6})$ .

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

Step 5.18.2

Cancel the common factors.

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Step 5.18.2.1

Factor $2$ out of $4$ .

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

Step 5.18.2.2

Cancel the common factor.

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

Step 5.18.2.3

Rewrite the expression.

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

Step 5.19

Group $5 + 2 \sqrt{6}$ and $\sqrt{8 - 2 \sqrt{6} + 2 \sqrt{2}}$ together.

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

Step 5.20

Apply the distributive property.

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

Step 5.21

Combine using the product rule for radicals.

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

Step 5.22

Move $6$ to the left of $8 - 2 \sqrt{6} + 2 \sqrt{2}$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.64267963 \dots$