Trigonometry Examples

$\sqrt{\frac{1 - \frac{15}{25}}{2}}$

Step 1

Cancel the common factor of $15$ and $25$ .

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Step 1.1

Factor $5$ out of $15$ .

$\sqrt{\frac{1 - \frac{5 (3)}{25}}{2}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $5$ out of $25$ .

$\sqrt{\frac{1 - \frac{5 \cdot 3}{5 \cdot 5}}{2}}$

Step 1.2.2

Cancel the common factor.

$\sqrt{\frac{1 - \frac{5 \cdot 3}{5 \cdot 5}}{2}}$

Step 1.2.3

Rewrite the expression.

$\sqrt{\frac{1 - \frac{3}{5}}{2}}$

Step 2

Simplify the expression.

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Step 2.1

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

$\sqrt{\frac{\frac{5}{5} - \frac{3}{5}}{2}}$

Step 2.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{5 - 3}{5}}{2}}$

Step 2.3

Subtract $3$ from $5$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

$\sqrt{\frac{1}{5}}$

Step 5

Rewrite $\sqrt{\frac{1}{5}}$ as $\frac{\sqrt{1}}{\sqrt{5}}$ .

$\frac{\sqrt{1}}{\sqrt{5}}$

Step 6

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{5}}$

Step 7

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 8.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 8.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 8.4

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

$\frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Evaluate the exponent.

$\frac{\sqrt{5}}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5}}{5}$

Decimal Form:

$0.44721359 \dots$