Trigonometry Examples

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

Step 1

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

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Add $4$ and $1$ .

$\sqrt{\frac{\frac{5}{4}}{{(\frac{1}{4})}^{2}}}$

Step 4

Apply the product rule to $\frac{1}{4}$ .

$\sqrt{\frac{\frac{5}{4}}{\frac{1^{2}}{4^{2}}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{5}{4} \cdot \frac{4^{2}}{1^{2}}}$

Step 6

Combine.

$\sqrt{\frac{5 \cdot 4^{2}}{4 \cdot 1^{2}}}$

Step 7

Cancel the common factor of $4^{2}$ and $4$ .

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Step 7.1

Factor $4$ out of $5 \cdot 4^{2}$ .

$\sqrt{\frac{4 (5 \cdot 4)}{4 \cdot 1^{2}}}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $4$ out of $4 \cdot 1^{2}$ .

$\sqrt{\frac{4 (5 \cdot 4)}{4 (1^{2})}}$

Step 7.2.2

Cancel the common factor.

$\sqrt{\frac{4 (5 \cdot 4)}{4 \cdot 1^{2}}}$

Step 7.2.3

Rewrite the expression.

$\sqrt{\frac{5 \cdot 4}{1^{2}}}$

Step 8

Simplify the expression.

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Step 8.1

Multiply $5$ by $4$ .

$\sqrt{\frac{20}{1^{2}}}$

Step 8.2

One to any power is one.

$\sqrt{\frac{20}{1}}$

Step 8.3

Divide $20$ by $1$ .

$\sqrt{20}$

Step 9

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 9.1

Factor $4$ out of $20$ .

$\sqrt{4 (5)}$

Step 9.2

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

$\sqrt{2^{2} \cdot 5}$

Step 10

Pull terms out from under the radical.

$2 \sqrt{5}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{5}$

Decimal Form:

$4.47213595 \dots$