Trigonometry Examples

$\sqrt{\frac{\frac{\sqrt{41} - 5}{\sqrt{41}}}{2}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{\sqrt{41} - 5}{\sqrt{41}} \cdot \frac{1}{2}}$

Step 2

Multiply $\frac{\sqrt{41} - 5}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\sqrt{\frac{\sqrt{41} - 5}{\sqrt{41}} \cdot \frac{\sqrt{41}}{\sqrt{41}} \frac{1}{2}}$

Step 3

Simplify terms.

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Step 3.1

Combine and simplify the denominator.

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Step 3.1.1

Multiply $\frac{\sqrt{41} - 5}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\sqrt{\frac{(\sqrt{41} - 5) \sqrt{41}}{\sqrt{41} \sqrt{41}} \cdot \frac{1}{2}}$

Step 3.1.2

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

$\sqrt{\frac{(\sqrt{41} - 5) \sqrt{41}}{{\sqrt{41}}^{1} \sqrt{41}} \cdot \frac{1}{2}}$

Step 3.1.3

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

$\sqrt{\frac{(\sqrt{41} - 5) \sqrt{41}}{{\sqrt{41}}^{1} {\sqrt{41}}^{1}} \cdot \frac{1}{2}}$

Step 3.1.4

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

$\sqrt{\frac{(\sqrt{41} - 5) \sqrt{41}}{{\sqrt{41}}^{1 + 1}} \cdot \frac{1}{2}}$

Step 3.1.5

Add $1$ and $1$ .

$\sqrt{\frac{(\sqrt{41} - 5) \sqrt{41}}{{\sqrt{41}}^{2}} \cdot \frac{1}{2}}$

Step 3.1.6

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

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Step 3.1.6.1

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

$\sqrt{\frac{(\sqrt{41} - 5) \sqrt{41}}{{(41^{\frac{1}{2}})}^{2}} \cdot \frac{1}{2}}$

Step 3.1.6.2

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

$\sqrt{\frac{(\sqrt{41} - 5) \sqrt{41}}{41^{\frac{1}{2} \cdot 2}} \cdot \frac{1}{2}}$

Step 3.1.6.3

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

$\sqrt{\frac{(\sqrt{41} - 5) \sqrt{41}}{41^{\frac{2}{2}}} \cdot \frac{1}{2}}$

Step 3.1.6.4

Cancel the common factor of $2$ .

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Step 3.1.6.4.1

Cancel the common factor.

$\sqrt{\frac{(\sqrt{41} - 5) \sqrt{41}}{41^{\frac{2}{2}}} \cdot \frac{1}{2}}$

Step 3.1.6.4.2

Rewrite the expression.

$\sqrt{\frac{(\sqrt{41} - 5) \sqrt{41}}{41^{1}} \cdot \frac{1}{2}}$

Step 3.1.6.5

Evaluate the exponent.

$\sqrt{\frac{(\sqrt{41} - 5) \sqrt{41}}{41} \cdot \frac{1}{2}}$

Step 3.2

Apply the distributive property.

$\sqrt{\frac{\sqrt{41} \sqrt{41} - 5 \sqrt{41}}{41} \cdot \frac{1}{2}}$

Step 3.3

Combine using the product rule for radicals.

$\sqrt{\frac{\sqrt{41 \cdot 41} - 5 \sqrt{41}}{41} \cdot \frac{1}{2}}$

Step 4

Simplify each term.

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Step 4.1

Multiply $41$ by $41$ .

$\sqrt{\frac{\sqrt{1681} - 5 \sqrt{41}}{41} \cdot \frac{1}{2}}$

Step 4.2

Rewrite $1681$ as $41^{2}$ .

$\sqrt{\frac{\sqrt{41^{2}} - 5 \sqrt{41}}{41} \cdot \frac{1}{2}}$

Step 4.3

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

$\sqrt{\frac{41 - 5 \sqrt{41}}{41} \cdot \frac{1}{2}}$

Step 5

Multiply $\frac{41 - 5 \sqrt{41}}{41} \cdot \frac{1}{2}$ .

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Step 5.1

Multiply $\frac{41 - 5 \sqrt{41}}{41}$ by $\frac{1}{2}$ .

$\sqrt{\frac{41 - 5 \sqrt{41}}{41 \cdot 2}}$

Step 5.2

Multiply $41$ by $2$ .

$\sqrt{\frac{41 - 5 \sqrt{41}}{82}}$

Step 6

Rewrite $\sqrt{\frac{41 - 5 \sqrt{41}}{82}}$ as $\frac{\sqrt{41 - 5 \sqrt{41}}}{\sqrt{82}}$ .

$\frac{\sqrt{41 - 5 \sqrt{41}}}{\sqrt{82}}$

Step 7

Multiply $\frac{\sqrt{41 - 5 \sqrt{41}}}{\sqrt{82}}$ by $\frac{\sqrt{82}}{\sqrt{82}}$ .

$\frac{\sqrt{41 - 5 \sqrt{41}}}{\sqrt{82}} \cdot \frac{\sqrt{82}}{\sqrt{82}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{\sqrt{41 - 5 \sqrt{41}}}{\sqrt{82}}$ by $\frac{\sqrt{82}}{\sqrt{82}}$ .

$\frac{\sqrt{41 - 5 \sqrt{41}} \sqrt{82}}{\sqrt{82} \sqrt{82}}$

Step 8.2

Raise $\sqrt{82}$ to the power of $1$ .

$\frac{\sqrt{41 - 5 \sqrt{41}} \sqrt{82}}{{\sqrt{82}}^{1} \sqrt{82}}$

Step 8.3

Raise $\sqrt{82}$ to the power of $1$ .

$\frac{\sqrt{41 - 5 \sqrt{41}} \sqrt{82}}{{\sqrt{82}}^{1} {\sqrt{82}}^{1}}$

Step 8.4

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

$\frac{\sqrt{41 - 5 \sqrt{41}} \sqrt{82}}{{\sqrt{82}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$\frac{\sqrt{41 - 5 \sqrt{41}} \sqrt{82}}{{\sqrt{82}}^{2}}$

Step 8.6

Rewrite ${\sqrt{82}}^{2}$ as $82$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{82}$ as $82^{\frac{1}{2}}$ .

$\frac{\sqrt{41 - 5 \sqrt{41}} \sqrt{82}}{{(82^{\frac{1}{2}})}^{2}}$

Step 8.6.2

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

$\frac{\sqrt{41 - 5 \sqrt{41}} \sqrt{82}}{82^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

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

$\frac{\sqrt{41 - 5 \sqrt{41}} \sqrt{82}}{82^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{\sqrt{41 - 5 \sqrt{41}} \sqrt{82}}{82^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$\frac{\sqrt{41 - 5 \sqrt{41}} \sqrt{82}}{82^{1}}$

Step 8.6.5

Evaluate the exponent.

$\frac{\sqrt{41 - 5 \sqrt{41}} \sqrt{82}}{82}$

Step 9

Combine using the product rule for radicals.

$\frac{\sqrt{(41 - 5 \sqrt{41}) \cdot 82}}{82}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(41 - 5 \sqrt{41}) \cdot 82}}{82}$

Decimal Form:

$0.33100694 \dots$