Trigonometry Examples

$\sqrt{\frac{6 + 5 \sqrt{61}}{122}}$

Step 1

Rewrite $\sqrt{\frac{6 + 5 \sqrt{61}}{122}}$ as $\frac{\sqrt{6 + 5 \sqrt{61}}}{\sqrt{122}}$ .

$\frac{\sqrt{6 + 5 \sqrt{61}}}{\sqrt{122}}$

Step 2

Multiply $\frac{\sqrt{6 + 5 \sqrt{61}}}{\sqrt{122}}$ by $\frac{\sqrt{122}}{\sqrt{122}}$ .

$\frac{\sqrt{6 + 5 \sqrt{61}}}{\sqrt{122}} \cdot \frac{\sqrt{122}}{\sqrt{122}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt{6 + 5 \sqrt{61}}}{\sqrt{122}}$ by $\frac{\sqrt{122}}{\sqrt{122}}$ .

$\frac{\sqrt{6 + 5 \sqrt{61}} \sqrt{122}}{\sqrt{122} \sqrt{122}}$

Step 3.2

Raise $\sqrt{122}$ to the power of $1$ .

$\frac{\sqrt{6 + 5 \sqrt{61}} \sqrt{122}}{{\sqrt{122}}^{1} \sqrt{122}}$

Step 3.3

Raise $\sqrt{122}$ to the power of $1$ .

$\frac{\sqrt{6 + 5 \sqrt{61}} \sqrt{122}}{{\sqrt{122}}^{1} {\sqrt{122}}^{1}}$

Step 3.4

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

$\frac{\sqrt{6 + 5 \sqrt{61}} \sqrt{122}}{{\sqrt{122}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{\sqrt{6 + 5 \sqrt{61}} \sqrt{122}}{{\sqrt{122}}^{2}}$

Step 3.6

Rewrite ${\sqrt{122}}^{2}$ as $122$ .

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Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{122}$ as $122^{\frac{1}{2}}$ .

$\frac{\sqrt{6 + 5 \sqrt{61}} \sqrt{122}}{{(122^{\frac{1}{2}})}^{2}}$

Step 3.6.2

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

$\frac{\sqrt{6 + 5 \sqrt{61}} \sqrt{122}}{122^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

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

$\frac{\sqrt{6 + 5 \sqrt{61}} \sqrt{122}}{122^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{\sqrt{6 + 5 \sqrt{61}} \sqrt{122}}{122^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{\sqrt{6 + 5 \sqrt{61}} \sqrt{122}}{122^{1}}$

Step 3.6.5

Evaluate the exponent.

$\frac{\sqrt{6 + 5 \sqrt{61}} \sqrt{122}}{122}$

Step 4

Combine using the product rule for radicals.

$\frac{\sqrt{(6 + 5 \sqrt{61}) \cdot 122}}{122}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(6 + 5 \sqrt{61}) \cdot 122}}{122}$

Decimal Form:

$0.60767798 \dots$