Trigonometry Examples

$\sqrt{\frac{\sqrt{41} - 4}{\frac{\sqrt{41}}{2}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{(\sqrt{41} - 4) \frac{2}{\sqrt{41}}}$

Step 2

Multiply $\frac{2}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\sqrt{(\sqrt{41} - 4) (\frac{2}{\sqrt{41}} \cdot \frac{\sqrt{41}}{\sqrt{41}})}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{2}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\sqrt{(\sqrt{41} - 4) \frac{2 \sqrt{41}}{\sqrt{41} \sqrt{41}}}$

Step 3.2

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

$\sqrt{(\sqrt{41} - 4) \frac{2 \sqrt{41}}{{\sqrt{41}}^{1} \sqrt{41}}}$

Step 3.3

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

$\sqrt{(\sqrt{41} - 4) \frac{2 \sqrt{41}}{{\sqrt{41}}^{1} {\sqrt{41}}^{1}}}$

Step 3.4

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

$\sqrt{(\sqrt{41} - 4) \frac{2 \sqrt{41}}{{\sqrt{41}}^{1 + 1}}}$

Step 3.5

Add $1$ and $1$ .

$\sqrt{(\sqrt{41} - 4) \frac{2 \sqrt{41}}{{\sqrt{41}}^{2}}}$

Step 3.6

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

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Step 3.6.1

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

$\sqrt{(\sqrt{41} - 4) \frac{2 \sqrt{41}}{{(41^{\frac{1}{2}})}^{2}}}$

Step 3.6.2

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

$\sqrt{(\sqrt{41} - 4) \frac{2 \sqrt{41}}{41^{\frac{1}{2} \cdot 2}}}$

Step 3.6.3

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

$\sqrt{(\sqrt{41} - 4) \frac{2 \sqrt{41}}{41^{\frac{2}{2}}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\sqrt{(\sqrt{41} - 4) \frac{2 \sqrt{41}}{41^{\frac{2}{2}}}}$

Step 3.6.4.2

Rewrite the expression.

$\sqrt{(\sqrt{41} - 4) \frac{2 \sqrt{41}}{41^{1}}}$

Step 3.6.5

Evaluate the exponent.

$\sqrt{(\sqrt{41} - 4) \frac{2 \sqrt{41}}{41}}$

Step 4

Apply the distributive property.

$\sqrt{\sqrt{41} \frac{2 \sqrt{41}}{41} - 4 \frac{2 \sqrt{41}}{41}}$

Step 5

Multiply $\sqrt{41} \frac{2 \sqrt{41}}{41}$ .

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Step 5.1

Combine $\sqrt{41}$ and $\frac{2 \sqrt{41}}{41}$ .

$\sqrt{\frac{\sqrt{41} (2 \sqrt{41})}{41} - 4 \frac{2 \sqrt{41}}{41}}$

Step 5.2

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

$\sqrt{\frac{2 ({\sqrt{41}}^{1} \sqrt{41})}{41} - 4 \frac{2 \sqrt{41}}{41}}$

Step 5.3

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

$\sqrt{\frac{2 ({\sqrt{41}}^{1} {\sqrt{41}}^{1})}{41} - 4 \frac{2 \sqrt{41}}{41}}$

Step 5.4

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

$\sqrt{\frac{2 {\sqrt{41}}^{1 + 1}}{41} - 4 \frac{2 \sqrt{41}}{41}}$

Step 5.5

Add $1$ and $1$ .

$\sqrt{\frac{2 {\sqrt{41}}^{2}}{41} - 4 \frac{2 \sqrt{41}}{41}}$

Step 6

Multiply $- 4 \frac{2 \sqrt{41}}{41}$ .

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Step 6.1

Combine $- 4$ and $\frac{2 \sqrt{41}}{41}$ .

$\sqrt{\frac{2 {\sqrt{41}}^{2}}{41} + \frac{- 4 (2 \sqrt{41})}{41}}$

Step 6.2

Multiply $2$ by $- 4$ .

$\sqrt{\frac{2 {\sqrt{41}}^{2}}{41} + \frac{- 8 \sqrt{41}}{41}}$

Step 7

Combine the numerators over the common denominator.

$\sqrt{\frac{2 {\sqrt{41}}^{2} - 8 \sqrt{41}}{41}}$

Step 8

Simplify the numerator.

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Step 8.1

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

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Step 8.1.1

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

$\sqrt{\frac{2 {(41^{\frac{1}{2}})}^{2} - 8 \sqrt{41}}{41}}$

Step 8.1.2

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

$\sqrt{\frac{2 \cdot 41^{\frac{1}{2} \cdot 2} - 8 \sqrt{41}}{41}}$

Step 8.1.3

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

$\sqrt{\frac{2 \cdot 41^{\frac{2}{2}} - 8 \sqrt{41}}{41}}$

Step 8.1.4

Cancel the common factor of $2$ .

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Step 8.1.4.1

Cancel the common factor.

$\sqrt{\frac{2 \cdot 41^{\frac{2}{2}} - 8 \sqrt{41}}{41}}$

Step 8.1.4.2

Rewrite the expression.

$\sqrt{\frac{2 \cdot 41^{1} - 8 \sqrt{41}}{41}}$

Step 8.1.5

Evaluate the exponent.

$\sqrt{\frac{2 \cdot 41 - 8 \sqrt{41}}{41}}$

Step 8.2

Multiply $2$ by $41$ .

$\sqrt{\frac{82 - 8 \sqrt{41}}{41}}$

Step 9

Rewrite $\sqrt{\frac{82 - 8 \sqrt{41}}{41}}$ as $\frac{\sqrt{82 - 8 \sqrt{41}}}{\sqrt{41}}$ .

$\frac{\sqrt{82 - 8 \sqrt{41}}}{\sqrt{41}}$

Step 10

Multiply $\frac{\sqrt{82 - 8 \sqrt{41}}}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\frac{\sqrt{82 - 8 \sqrt{41}}}{\sqrt{41}} \cdot \frac{\sqrt{41}}{\sqrt{41}}$

Step 11

Combine and simplify the denominator.

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Step 11.1

Multiply $\frac{\sqrt{82 - 8 \sqrt{41}}}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\frac{\sqrt{82 - 8 \sqrt{41}} \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Step 11.2

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

$\frac{\sqrt{82 - 8 \sqrt{41}} \sqrt{41}}{{\sqrt{41}}^{1} \sqrt{41}}$

Step 11.3

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

$\frac{\sqrt{82 - 8 \sqrt{41}} \sqrt{41}}{{\sqrt{41}}^{1} {\sqrt{41}}^{1}}$

Step 11.4

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

$\frac{\sqrt{82 - 8 \sqrt{41}} \sqrt{41}}{{\sqrt{41}}^{1 + 1}}$

Step 11.5

Add $1$ and $1$ .

$\frac{\sqrt{82 - 8 \sqrt{41}} \sqrt{41}}{{\sqrt{41}}^{2}}$

Step 11.6

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

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Step 11.6.1

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

$\frac{\sqrt{82 - 8 \sqrt{41}} \sqrt{41}}{{(41^{\frac{1}{2}})}^{2}}$

Step 11.6.2

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

$\frac{\sqrt{82 - 8 \sqrt{41}} \sqrt{41}}{41^{\frac{1}{2} \cdot 2}}$

Step 11.6.3

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

$\frac{\sqrt{82 - 8 \sqrt{41}} \sqrt{41}}{41^{\frac{2}{2}}}$

Step 11.6.4

Cancel the common factor of $2$ .

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Step 11.6.4.1

Cancel the common factor.

$\frac{\sqrt{82 - 8 \sqrt{41}} \sqrt{41}}{41^{\frac{2}{2}}}$

Step 11.6.4.2

Rewrite the expression.

$\frac{\sqrt{82 - 8 \sqrt{41}} \sqrt{41}}{41^{1}}$

Step 11.6.5

Evaluate the exponent.

$\frac{\sqrt{82 - 8 \sqrt{41}} \sqrt{41}}{41}$

Step 12

Combine using the product rule for radicals.

$\frac{\sqrt{(82 - 8 \sqrt{41}) \cdot 41}}{41}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(82 - 8 \sqrt{41}) \cdot 41}}{41}$

Decimal Form:

$0.86637746 \dots$