Trigonometry Examples

$\sqrt{\frac{1 - (\frac{4 \sqrt{17}}{17})}{2}}$

Step 1

Write

$1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{17}{17} - \frac{4 \sqrt{17}}{17}}{2}}$

Step 2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{17 - 4 \sqrt{17}}{17}}{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{17 - 4 \sqrt{17}}{17} \cdot \frac{1}{2}}$

Step 4

Multiply

$\frac{17 - 4 \sqrt{17}}{17} \cdot \frac{1}{2}$ .

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Step 4.1

Multiply

$\frac{17 - 4 \sqrt{17}}{17}$ by

$\frac{1}{2}$ .

$\sqrt{\frac{17 - 4 \sqrt{17}}{17 \cdot 2}}$

Step 4.2

Multiply

$17$ by

$2$ .

$\sqrt{\frac{17 - 4 \sqrt{17}}{34}}$

Step 5

Rewrite

$\sqrt{\frac{17 - 4 \sqrt{17}}{34}}$ as

$\frac{\sqrt{17 - 4 \sqrt{17}}}{\sqrt{34}}$ .

$\frac{\sqrt{17 - 4 \sqrt{17}}}{\sqrt{34}}$

Step 6

Multiply

$\frac{\sqrt{17 - 4 \sqrt{17}}}{\sqrt{34}}$ by

$\frac{\sqrt{34}}{\sqrt{34}}$ .

$\frac{\sqrt{17 - 4 \sqrt{17}}}{\sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply

$\frac{\sqrt{17 - 4 \sqrt{17}}}{\sqrt{34}}$ by

$\frac{\sqrt{34}}{\sqrt{34}}$ .

$\frac{\sqrt{17 - 4 \sqrt{17}} \sqrt{34}}{\sqrt{34} \sqrt{34}}$

Step 7.2

Raise

$\sqrt{34}$ to the power of

$1$ .

$\frac{\sqrt{17 - 4 \sqrt{17}} \sqrt{34}}{{\sqrt{34}}^{1} \sqrt{34}}$

Step 7.3

Raise

$\sqrt{34}$ to the power of

$1$ .

$\frac{\sqrt{17 - 4 \sqrt{17}} \sqrt{34}}{{\sqrt{34}}^{1} {\sqrt{34}}^{1}}$

Step 7.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{17 - 4 \sqrt{17}} \sqrt{34}}{{\sqrt{34}}^{1 + 1}}$

Step 7.5

Add

$1$ and

$1$ .

$\frac{\sqrt{17 - 4 \sqrt{17}} \sqrt{34}}{{\sqrt{34}}^{2}}$

Step 7.6

Rewrite

${\sqrt{34}}^{2}$ as

$34$ .

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Step 7.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{34}$ as

$34^{\frac{1}{2}}$ .

$\frac{\sqrt{17 - 4 \sqrt{17}} \sqrt{34}}{{(34^{\frac{1}{2}})}^{2}}$

Step 7.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{17 - 4 \sqrt{17}} \sqrt{34}}{34^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{\sqrt{17 - 4 \sqrt{17}} \sqrt{34}}{34^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of

$2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{\sqrt{17 - 4 \sqrt{17}} \sqrt{34}}{34^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{\sqrt{17 - 4 \sqrt{17}} \sqrt{34}}{34^{1}}$

Step 7.6.5

Evaluate the exponent.

$\frac{\sqrt{17 - 4 \sqrt{17}} \sqrt{34}}{34}$

Step 8

Combine using the product rule for radicals.

$\frac{\sqrt{(17 - 4 \sqrt{17}) \cdot 34}}{34}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(17 - 4 \sqrt{17}) \cdot 34}}{34}$

Decimal Form:

$0.12218326 \dots$