Trigonometry Examples

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

Step 1

Multiply $\frac{6}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

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

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{6}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

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

Step 2.2

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 2.3

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 3

Cancel the common factor of $6$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Divide $\sqrt{6}$ by $1$ .

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

Step 4

Multiply $- - \sqrt{6}$ .

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Multiply $\sqrt{6}$ by $1$ .

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

Step 5

Rewrite $\sqrt{\frac{1 + \sqrt{6}}{2}}$ as $\frac{\sqrt{1 + \sqrt{6}}}{\sqrt{2}}$ .

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

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

Step 7.6.5

Evaluate the exponent.

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

Step 8

Combine using the product rule for radicals.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.31329542 \dots$