Trigonometry Examples

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

Step 1

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

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

Step 2

To write $\frac{4}{- \sqrt{17}}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

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

Step 3

Write each expression with a common denominator of $\sqrt{17}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{4}{- \sqrt{17}}$ by $\frac{- 1}{- 1}$ .

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

Step 3.2

Multiply $- 1$ by $- 1$ .

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

Step 3.3

Multiply $\sqrt{17}$ by $1$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Multiply $4$ by $- 1$ .

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

Step 6

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

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

Step 7

Simplify terms.

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Step 7.1

Combine and simplify the denominator.

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Step 7.1.1

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

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

Step 7.1.2

Raise $\sqrt{17}$ to the power of $1$ .

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

Step 7.1.3

Raise $\sqrt{17}$ to the power of $1$ .

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

Step 7.1.4

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

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

Step 7.1.5

Add $1$ and $1$ .

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

Step 7.1.6

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 7.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

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

Step 7.1.6.2

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

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

Step 7.1.6.3

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

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

Step 7.1.6.4

Cancel the common factor of $2$ .

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Step 7.1.6.4.1

Cancel the common factor.

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

Step 7.1.6.4.2

Rewrite the expression.

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

Step 7.1.6.5

Evaluate the exponent.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Combine using the product rule for radicals.

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

Step 8

Simplify each term.

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Step 8.1

Multiply $17$ by $17$ .

$\sqrt{\frac{1}{2} \cdot \frac{\sqrt{289} - 4 \sqrt{17}}{17}}$

Step 8.2

Rewrite $289$ as $17^{2}$ .

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

Step 8.3

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

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

Step 9

Combine fractions.

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Step 9.1

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

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

Step 9.2

Multiply $2$ by $17$ .

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

Step 10

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 11

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 12

Combine and simplify the denominator.

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Step 12.1

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 12.2

Raise $\sqrt{34}$ to the power of $1$ .

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

Step 12.3

Raise $\sqrt{34}$ to the power of $1$ .

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

Step 12.4

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 12.5

Add $1$ and $1$ .

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

Step 12.6

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

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Step 12.6.1

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 12.6.2

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 12.6.3

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

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

Step 12.6.4

Cancel the common factor of $2$ .

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Step 12.6.4.1

Cancel the common factor.

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

Step 12.6.4.2

Rewrite the expression.

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

Step 12.6.5

Evaluate the exponent.

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

Step 13

Combine using the product rule for radicals.

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

Step 14

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.12218326 \dots$