Precalculus Examples

$\frac{2 \tan (22.5)}{1 - \tan^{2} (22.5)}$

Step 1

The exact value of $\tan (22.5)$ is $\sqrt{3 - 2 \sqrt{2}}$ .

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Step 1.1

Rewrite $22.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\frac{2 \tan (\frac{45}{2})}{1 - \tan^{2} (22.5)}$

Step 1.2

Apply the tangent half-angle identity.

$\frac{2 (\pm \sqrt{\frac{1 - \cos (45)}{1 + \cos (45)}})}{1 - \tan^{2} (22.5)}$

Step 1.3

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

$\frac{2 \sqrt{\frac{1 - \cos (45)}{1 + \cos (45)}}}{1 - \tan^{2} (22.5)}$

Step 1.4

Simplify $\sqrt{\frac{1 - \cos (45)}{1 + \cos (45)}}$ .

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Step 1.4.1

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

$\frac{2 \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \cos (45)}}}{1 - \tan^{2} (22.5)}$

Step 1.4.2

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

$\frac{2 \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{1 + \cos (45)}}}{1 - \tan^{2} (22.5)}$

Step 1.4.3

Combine the numerators over the common denominator.

$\frac{2 \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{1 + \cos (45)}}}{1 - \tan^{2} (22.5)}$

Step 1.4.4

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

$\frac{2 \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{1 + \frac{\sqrt{2}}{2}}}}{1 - \tan^{2} (22.5)}$

Step 1.4.5

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

$\frac{2 \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{\frac{2}{2} + \frac{\sqrt{2}}{2}}}}{1 - \tan^{2} (22.5)}$

Step 1.4.6

Combine the numerators over the common denominator.

$\frac{2 \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{\frac{2 + \sqrt{2}}{2}}}}{1 - \tan^{2} (22.5)}$

Step 1.4.7

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{2}{2 + \sqrt{2}}}}{1 - \tan^{2} (22.5)}$

Step 1.4.8

Cancel the common factor of $2$ .

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Step 1.4.8.1

Cancel the common factor.

$\frac{2 \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{2}{2 + \sqrt{2}}}}{1 - \tan^{2} (22.5)}$

Step 1.4.8.2

Rewrite the expression.

$\frac{2 \sqrt{(2 - \sqrt{2}) \frac{1}{2 + \sqrt{2}}}}{1 - \tan^{2} (22.5)}$

Step 1.4.9

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

$\frac{2 \sqrt{(2 - \sqrt{2}) (\frac{1}{2 + \sqrt{2}} \cdot \frac{2 - \sqrt{2}}{2 - \sqrt{2}})}}{1 - \tan^{2} (22.5)}$

Step 1.4.10

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

$\frac{2 \sqrt{(2 - \sqrt{2}) \frac{2 - \sqrt{2}}{(2 + \sqrt{2}) (2 - \sqrt{2})}}}{1 - \tan^{2} (22.5)}$

Step 1.4.11

Expand the denominator using the FOIL method.

$\frac{2 \sqrt{(2 - \sqrt{2}) \frac{2 - \sqrt{2}}{4 - 2 \sqrt{2} + \sqrt{2} \cdot 2 - {\sqrt{2}}^{2}}}}{1 - \tan^{2} (22.5)}$

Step 1.4.12

Simplify.

$\frac{2 \sqrt{(2 - \sqrt{2}) \frac{2 - \sqrt{2}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.13

Apply the distributive property.

$\frac{2 \sqrt{2 \frac{2 - \sqrt{2}}{2} - \sqrt{2} \frac{2 - \sqrt{2}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.14

Cancel the common factor of $2$ .

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Step 1.4.14.1

Cancel the common factor.

$\frac{2 \sqrt{2 \frac{2 - \sqrt{2}}{2} - \sqrt{2} \frac{2 - \sqrt{2}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.14.2

Rewrite the expression.

$\frac{2 \sqrt{2 - \sqrt{2} - \sqrt{2} \frac{2 - \sqrt{2}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.15

Combine $\frac{2 - \sqrt{2}}{2}$ and $\sqrt{2}$ .

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{(2 - \sqrt{2}) \sqrt{2}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16

Simplify each term.

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Step 1.4.16.1

Apply the distributive property.

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - \sqrt{2} \sqrt{2}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.2

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

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Step 1.4.16.2.1

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

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - ({\sqrt{2}}^{1} \sqrt{2})}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.2.2

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

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.2.3

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

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - {\sqrt{2}}^{1 + 1}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.2.4

Add $1$ and $1$ .

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - {\sqrt{2}}^{2}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.3

Simplify each term.

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Step 1.4.16.3.1

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

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Step 1.4.16.3.1.1

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

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - {(2^{\frac{1}{2}})}^{2}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.3.1.2

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

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - 2^{\frac{1}{2} \cdot 2}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.3.1.3

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

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - 2^{\frac{2}{2}}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.3.1.4

Cancel the common factor of $2$ .

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Step 1.4.16.3.1.4.1

Cancel the common factor.

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - 2^{\frac{2}{2}}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.3.1.4.2

Rewrite the expression.

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - 2^{1}}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.3.1.5

Evaluate the exponent.

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - 1 \cdot 2}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.3.2

Multiply $- 1$ by $2$ .

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - 2}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.4

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

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Step 1.4.16.4.1

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

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 (\sqrt{2}) - 2}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.4.2

Factor $2$ out of $- 2$ .

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 (\sqrt{2}) + 2 \cdot - 1}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.4.3

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

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 (\sqrt{2} - 1)}{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.4.4

Cancel the common factors.

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Step 1.4.16.4.4.1

Factor $2$ out of $2$ .

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 (\sqrt{2} - 1)}{2 (1)}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.4.4.2

Cancel the common factor.

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{2 (\sqrt{2} - 1)}{2 \cdot 1}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.4.4.3

Rewrite the expression.

$\frac{2 \sqrt{2 - \sqrt{2} - \frac{\sqrt{2} - 1}{1}}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.4.4.4

Divide $\sqrt{2} - 1$ by $1$ .

$\frac{2 \sqrt{2 - \sqrt{2} - (\sqrt{2} - 1)}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.5

Apply the distributive property.

$\frac{2 \sqrt{2 - \sqrt{2} - \sqrt{2} - - 1}}{1 - \tan^{2} (22.5)}$

Step 1.4.16.6

Multiply $- 1$ by $- 1$ .

$\frac{2 \sqrt{2 - \sqrt{2} - \sqrt{2} + 1}}{1 - \tan^{2} (22.5)}$

Step 1.4.17

Add $2$ and $1$ .

$\frac{2 \sqrt{3 - \sqrt{2} - \sqrt{2}}}{1 - \tan^{2} (22.5)}$

Step 1.4.18

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

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - \tan^{2} (22.5)}$

Step 2

The exact value of $\tan (22.5)$ is $\sqrt{3 - 2 \sqrt{2}}$ .

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Step 2.1

Rewrite $22.5$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - \tan^{2} (\frac{45}{2})}$

Step 2.2

Apply the tangent half-angle identity.

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {(\pm \sqrt{\frac{1 - \cos (45)}{1 + \cos (45)}})}^{2}}$

Step 2.3

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

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{\frac{1 - \cos (45)}{1 + \cos (45)}}}^{2}}$

Step 2.4

Simplify $\sqrt{\frac{1 - \cos (45)}{1 + \cos (45)}}$ .

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Step 2.4.1

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

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \cos (45)}}}^{2}}$

Step 2.4.2

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

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{\frac{\frac{2}{2} - \frac{\sqrt{2}}{2}}{1 + \cos (45)}}}^{2}}$

Step 2.4.3

Combine the numerators over the common denominator.

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{1 + \cos (45)}}}^{2}}$

Step 2.4.4

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

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

Step 2.4.5

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

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

Step 2.4.6

Combine the numerators over the common denominator.

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

Step 2.4.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4.8

Cancel the common factor of $2$ .

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Step 2.4.8.1

Cancel the common factor.

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

Step 2.4.8.2

Rewrite the expression.

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

Step 2.4.9

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

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

Step 2.4.10

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

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

Step 2.4.11

Expand the denominator using the FOIL method.

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

Step 2.4.12

Simplify.

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

Step 2.4.13

Apply the distributive property.

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{2 \frac{2 - \sqrt{2}}{2} - \sqrt{2} \frac{2 - \sqrt{2}}{2}}}^{2}}$

Step 2.4.14

Cancel the common factor of $2$ .

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Step 2.4.14.1

Cancel the common factor.

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{2 \frac{2 - \sqrt{2}}{2} - \sqrt{2} \frac{2 - \sqrt{2}}{2}}}^{2}}$

Step 2.4.14.2

Rewrite the expression.

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{2 - \sqrt{2} - \sqrt{2} \frac{2 - \sqrt{2}}{2}}}^{2}}$

Step 2.4.15

Combine $\frac{2 - \sqrt{2}}{2}$ and $\sqrt{2}$ .

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

Step 2.4.16

Simplify each term.

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Step 2.4.16.1

Apply the distributive property.

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - \sqrt{2} \sqrt{2}}{2}}}^{2}}$

Step 2.4.16.2

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

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Step 2.4.16.2.1

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

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

Step 2.4.16.2.2

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

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

Step 2.4.16.2.3

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

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

Step 2.4.16.2.4

Add $1$ and $1$ .

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - {\sqrt{2}}^{2}}{2}}}^{2}}$

Step 2.4.16.3

Simplify each term.

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Step 2.4.16.3.1

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

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Step 2.4.16.3.1.1

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

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

Step 2.4.16.3.1.2

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

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

Step 2.4.16.3.1.3

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

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - 2^{\frac{2}{2}}}{2}}}^{2}}$

Step 2.4.16.3.1.4

Cancel the common factor of $2$ .

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Step 2.4.16.3.1.4.1

Cancel the common factor.

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - 2^{\frac{2}{2}}}{2}}}^{2}}$

Step 2.4.16.3.1.4.2

Rewrite the expression.

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - 2^{1}}{2}}}^{2}}$

Step 2.4.16.3.1.5

Evaluate the exponent.

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

Step 2.4.16.3.2

Multiply $- 1$ by $2$ .

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{2 - \sqrt{2} - \frac{2 \sqrt{2} - 2}{2}}}^{2}}$

Step 2.4.16.4

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

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Step 2.4.16.4.1

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

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

Step 2.4.16.4.2

Factor $2$ out of $- 2$ .

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

Step 2.4.16.4.3

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

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

Step 2.4.16.4.4

Cancel the common factors.

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Step 2.4.16.4.4.1

Factor $2$ out of $2$ .

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

Step 2.4.16.4.4.2

Cancel the common factor.

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

Step 2.4.16.4.4.3

Rewrite the expression.

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{2 - \sqrt{2} - \frac{\sqrt{2} - 1}{1}}}^{2}}$

Step 2.4.16.4.4.4

Divide $\sqrt{2} - 1$ by $1$ .

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

Step 2.4.16.5

Apply the distributive property.

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{2 - \sqrt{2} - \sqrt{2} - - 1}}^{2}}$

Step 2.4.16.6

Multiply $- 1$ by $- 1$ .

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

Step 2.4.17

Add $2$ and $1$ .

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{3 - \sqrt{2} - \sqrt{2}}}^{2}}$

Step 2.4.18

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

$\frac{2 \sqrt{3 - 2 \sqrt{2}}}{1 - {\sqrt{3 - 2 \sqrt{2}}}^{2}}$

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Simplify.

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

Step 4

Apply the distributive property.

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

Step 5

Multiply $- 1$ by $3$ .

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

Step 6

Multiply $- 2$ by $- 1$ .

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

Step 7

Subtract $3$ from $1$ .

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

Step 8

Cancel the common factor of $2$ and $- 2 + 2 \sqrt{2}$ .

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Step 8.1

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

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

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $2$ out of $- 2$ .

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

Step 8.2.2

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

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

Step 8.2.3

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

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

Step 8.2.4

Cancel the common factor.

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

Step 8.2.5

Rewrite the expression.

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

Step 9

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

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

Step 10

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

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

Step 11

Expand the denominator using the FOIL method.

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

Step 12

Simplify.

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

Step 13

Move the negative one from the denominator of $\frac{\sqrt{3 - 2 \sqrt{2}} (- 1 - \sqrt{2})}{- 1}$ .

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

Step 14

Rewrite $- 1 \cdot (\sqrt{3 - 2 \sqrt{2}} (- 1 - \sqrt{2}))$ as $- (\sqrt{3 - 2 \sqrt{2}} (- 1 - \sqrt{2}))$ .

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

Step 15

Apply the distributive property.

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

Step 16

Move $- 1$ to the left of $\sqrt{3 - 2 \sqrt{2}}$ .

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

Step 17

Combine using the product rule for radicals.

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

Step 18

Rewrite $- 1 \sqrt{3 - 2 \sqrt{2}}$ as $- \sqrt{3 - 2 \sqrt{2}}$ .

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

Step 19

Apply the distributive property.

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

Step 20

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

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Step 20.1

Multiply $- 1$ by $- 1$ .

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

Step 20.2

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

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

Step 21

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

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Step 21.1

Multiply $- 1$ by $- 1$ .

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

Step 21.2

Multiply $\sqrt{(3 - 2 \sqrt{2}) \cdot 2}$ by $1$ .

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

Step 22

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.000000000000 \dots$