Trigonometry Examples

$\sqrt{\frac{1 - (- \frac{9 \sqrt{82}}{82})}{2}}$

Step 1

Multiply $- (- \frac{9 \sqrt{82}}{82})$ .

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Step 1.1

Multiply $- 1$ by $- 1$ .

$\sqrt{\frac{1 + 1 \frac{9 \sqrt{82}}{82}}{2}}$

Step 1.2

Multiply $\frac{9 \sqrt{82}}{82}$ by $1$ .

$\sqrt{\frac{1 + \frac{9 \sqrt{82}}{82}}{2}}$

Step 2

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

$\sqrt{\frac{\frac{82}{82} + \frac{9 \sqrt{82}}{82}}{2}}$

Step 3

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{82 + 9 \sqrt{82}}{82}}{2}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{82 + 9 \sqrt{82}}{82} \cdot \frac{1}{2}}$

Step 5

Multiply $\frac{82 + 9 \sqrt{82}}{82} \cdot \frac{1}{2}$ .

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Step 5.1

Multiply $\frac{82 + 9 \sqrt{82}}{82}$ by $\frac{1}{2}$ .

$\sqrt{\frac{82 + 9 \sqrt{82}}{82 \cdot 2}}$

Step 5.2

Multiply $82$ by $2$ .

$\sqrt{\frac{82 + 9 \sqrt{82}}{164}}$

Step 6

Rewrite $\sqrt{\frac{82 + 9 \sqrt{82}}{164}}$ as $\frac{\sqrt{82 + 9 \sqrt{82}}}{\sqrt{164}}$ .

$\frac{\sqrt{82 + 9 \sqrt{82}}}{\sqrt{164}}$

Step 7

Simplify the denominator.

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Step 7.1

Rewrite $164$ as $2^{2} \cdot 41$ .

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Step 7.1.1

Factor $4$ out of $164$ .

$\frac{\sqrt{82 + 9 \sqrt{82}}}{\sqrt{4 (41)}}$

Step 7.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{82 + 9 \sqrt{82}}}{\sqrt{2^{2} \cdot 41}}$

Step 7.2

Pull terms out from under the radical.

$\frac{\sqrt{82 + 9 \sqrt{82}}}{2 \sqrt{41}}$

Step 8

Multiply $\frac{\sqrt{82 + 9 \sqrt{82}}}{2 \sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\frac{\sqrt{82 + 9 \sqrt{82}}}{2 \sqrt{41}} \cdot \frac{\sqrt{41}}{\sqrt{41}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{\sqrt{82 + 9 \sqrt{82}}}{2 \sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\frac{\sqrt{82 + 9 \sqrt{82}} \sqrt{41}}{2 \sqrt{41} \sqrt{41}}$

Step 9.2

Move $\sqrt{41}$ .

$\frac{\sqrt{82 + 9 \sqrt{82}} \sqrt{41}}{2 (\sqrt{41} \sqrt{41})}$

Step 9.3

Raise $\sqrt{41}$ to the power of $1$ .

$\frac{\sqrt{82 + 9 \sqrt{82}} \sqrt{41}}{2 ({\sqrt{41}}^{1} \sqrt{41})}$

Step 9.4

Raise $\sqrt{41}$ to the power of $1$ .

$\frac{\sqrt{82 + 9 \sqrt{82}} \sqrt{41}}{2 ({\sqrt{41}}^{1} {\sqrt{41}}^{1})}$

Step 9.5

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

$\frac{\sqrt{82 + 9 \sqrt{82}} \sqrt{41}}{2 {\sqrt{41}}^{1 + 1}}$

Step 9.6

Add $1$ and $1$ .

$\frac{\sqrt{82 + 9 \sqrt{82}} \sqrt{41}}{2 {\sqrt{41}}^{2}}$

Step 9.7

Rewrite ${\sqrt{41}}^{2}$ as $41$ .

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Step 9.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{41}$ as $41^{\frac{1}{2}}$ .

$\frac{\sqrt{82 + 9 \sqrt{82}} \sqrt{41}}{2 {(41^{\frac{1}{2}})}^{2}}$

Step 9.7.2

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

$\frac{\sqrt{82 + 9 \sqrt{82}} \sqrt{41}}{2 \cdot 41^{\frac{1}{2} \cdot 2}}$

Step 9.7.3

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

$\frac{\sqrt{82 + 9 \sqrt{82}} \sqrt{41}}{2 \cdot 41^{\frac{2}{2}}}$

Step 9.7.4

Cancel the common factor of $2$ .

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Step 9.7.4.1

Cancel the common factor.

$\frac{\sqrt{82 + 9 \sqrt{82}} \sqrt{41}}{2 \cdot 41^{\frac{2}{2}}}$

Step 9.7.4.2

Rewrite the expression.

$\frac{\sqrt{82 + 9 \sqrt{82}} \sqrt{41}}{2 \cdot 41^{1}}$

Step 9.7.5

Evaluate the exponent.

$\frac{\sqrt{82 + 9 \sqrt{82}} \sqrt{41}}{2 \cdot 41}$

Step 10

Combine using the product rule for radicals.

$\frac{\sqrt{(82 + 9 \sqrt{82}) \cdot 41}}{2 \cdot 41}$

Step 11

Multiply $2$ by $41$ .

$\frac{\sqrt{(82 + 9 \sqrt{82}) \cdot 41}}{82}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(82 + 9 \sqrt{82}) \cdot 41}}{82}$

Decimal Form:

$0.99846976 \dots$