Trigonometry Examples

$\sqrt{\frac{1}{72} \cdot (241 - 13 \sqrt{313})}$

Step 1

Apply the distributive property.

$\sqrt{\frac{1}{72} \cdot 241 + \frac{1}{72} (- 13 \sqrt{313})}$

Step 2

Combine $\frac{1}{72}$ and $241$ .

$\sqrt{\frac{241}{72} + \frac{1}{72} (- 13 \sqrt{313})}$

Step 3

Multiply $\frac{1}{72} (- 13 \sqrt{313})$ .

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Step 3.1

Combine $- 13$ and $\frac{1}{72}$ .

$\sqrt{\frac{241}{72} + \frac{- 13}{72} \sqrt{313}}$

Step 3.2

Combine $\frac{- 13}{72}$ and $\sqrt{313}$ .

$\sqrt{\frac{241}{72} + \frac{- 13 \sqrt{313}}{72}}$

Step 4

Move the negative in front of the fraction.

$\sqrt{\frac{241}{72} - \frac{13 \sqrt{313}}{72}}$

Step 5

Combine the numerators over the common denominator.

$\sqrt{\frac{241 - 13 \sqrt{313}}{72}}$

Step 6

Rewrite $\sqrt{\frac{241 - 13 \sqrt{313}}{72}}$ as $\frac{\sqrt{241 - 13 \sqrt{313}}}{\sqrt{72}}$ .

$\frac{\sqrt{241 - 13 \sqrt{313}}}{\sqrt{72}}$

Step 7

Simplify the denominator.

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Step 7.1

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 7.1.1

Factor $36$ out of $72$ .

$\frac{\sqrt{241 - 13 \sqrt{313}}}{\sqrt{36 (2)}}$

Step 7.1.2

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{241 - 13 \sqrt{313}}}{\sqrt{6^{2} \cdot 2}}$

Step 7.2

Pull terms out from under the radical.

$\frac{\sqrt{241 - 13 \sqrt{313}}}{6 \sqrt{2}}$

Step 8

Multiply $\frac{\sqrt{241 - 13 \sqrt{313}}}{6 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{241 - 13 \sqrt{313}}}{6 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{\sqrt{241 - 13 \sqrt{313}}}{6 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{241 - 13 \sqrt{313}} \sqrt{2}}{6 \sqrt{2} \sqrt{2}}$

Step 9.2

Move $\sqrt{2}$ .

$\frac{\sqrt{241 - 13 \sqrt{313}} \sqrt{2}}{6 (\sqrt{2} \sqrt{2})}$

Step 9.3

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

$\frac{\sqrt{241 - 13 \sqrt{313}} \sqrt{2}}{6 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 9.4

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

$\frac{\sqrt{241 - 13 \sqrt{313}} \sqrt{2}}{6 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 9.5

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

$\frac{\sqrt{241 - 13 \sqrt{313}} \sqrt{2}}{6 {\sqrt{2}}^{1 + 1}}$

Step 9.6

Add $1$ and $1$ .

$\frac{\sqrt{241 - 13 \sqrt{313}} \sqrt{2}}{6 {\sqrt{2}}^{2}}$

Step 9.7

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

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Step 9.7.1

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

$\frac{\sqrt{241 - 13 \sqrt{313}} \sqrt{2}}{6 {(2^{\frac{1}{2}})}^{2}}$

Step 9.7.2

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

$\frac{\sqrt{241 - 13 \sqrt{313}} \sqrt{2}}{6 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 9.7.3

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

$\frac{\sqrt{241 - 13 \sqrt{313}} \sqrt{2}}{6 \cdot 2^{\frac{2}{2}}}$

Step 9.7.4

Cancel the common factor of $2$ .

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Step 9.7.4.1

Cancel the common factor.

$\frac{\sqrt{241 - 13 \sqrt{313}} \sqrt{2}}{6 \cdot 2^{\frac{2}{2}}}$

Step 9.7.4.2

Rewrite the expression.

$\frac{\sqrt{241 - 13 \sqrt{313}} \sqrt{2}}{6 \cdot 2^{1}}$

Step 9.7.5

Evaluate the exponent.

$\frac{\sqrt{241 - 13 \sqrt{313}} \sqrt{2}}{6 \cdot 2}$

Step 10

Combine using the product rule for radicals.

$\frac{\sqrt{(241 - 13 \sqrt{313}) \cdot 2}}{6 \cdot 2}$

Step 11

Multiply $6$ by $2$ .

$\frac{\sqrt{(241 - 13 \sqrt{313}) \cdot 2}}{12}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(241 - 13 \sqrt{313}) \cdot 2}}{12}$

Decimal Form:

$0.39098383 \dots$