Trigonometry Examples

$\sqrt{\frac{1 - \frac{3}{\sqrt{10}}}{2}}$

Step 1

Multiply $\frac{3}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sqrt{\frac{1 - (\frac{3}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})}{2}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{3}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sqrt{\frac{1 - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}}{2}}$

Step 2.2

Raise $\sqrt{10}$ to the power of $1$ .

$\sqrt{\frac{1 - \frac{3 \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}}{2}}$

Step 2.3

Raise $\sqrt{10}$ to the power of $1$ .

$\sqrt{\frac{1 - \frac{3 \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}}{2}}$

Step 2.4

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

$\sqrt{\frac{1 - \frac{3 \sqrt{10}}{{\sqrt{10}}^{1 + 1}}}{2}}$

Step 2.5

Add $1$ and $1$ .

$\sqrt{\frac{1 - \frac{3 \sqrt{10}}{{\sqrt{10}}^{2}}}{2}}$

Step 2.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sqrt{\frac{1 - \frac{3 \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}}{2}}$

Step 2.6.2

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

$\sqrt{\frac{1 - \frac{3 \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}}{2}}$

Step 2.6.3

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

$\sqrt{\frac{1 - \frac{3 \sqrt{10}}{10^{\frac{2}{2}}}}{2}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{1 - \frac{3 \sqrt{10}}{10^{\frac{2}{2}}}}{2}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{1 - \frac{3 \sqrt{10}}{10^{1}}}{2}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{1 - \frac{3 \sqrt{10}}{10}}{2}}$

Step 3

Simplify the expression.

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Step 3.1

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

$\sqrt{\frac{\frac{10}{10} - \frac{3 \sqrt{10}}{10}}{2}}$

Step 3.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{10 - 3 \sqrt{10}}{10}}{2}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{10 - 3 \sqrt{10}}{10} \cdot \frac{1}{2}}$

Step 5

Multiply $\frac{10 - 3 \sqrt{10}}{10} \cdot \frac{1}{2}$ .

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Step 5.1

Multiply $\frac{10 - 3 \sqrt{10}}{10}$ by $\frac{1}{2}$ .

$\sqrt{\frac{10 - 3 \sqrt{10}}{10 \cdot 2}}$

Step 5.2

Multiply $10$ by $2$ .

$\sqrt{\frac{10 - 3 \sqrt{10}}{20}}$

Step 6

Rewrite $\sqrt{\frac{10 - 3 \sqrt{10}}{20}}$ as $\frac{\sqrt{10 - 3 \sqrt{10}}}{\sqrt{20}}$ .

$\frac{\sqrt{10 - 3 \sqrt{10}}}{\sqrt{20}}$

Step 7

Simplify the denominator.

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Step 7.1

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

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Step 7.1.1

Factor $4$ out of $20$ .

$\frac{\sqrt{10 - 3 \sqrt{10}}}{\sqrt{4 (5)}}$

Step 7.1.2

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

$\frac{\sqrt{10 - 3 \sqrt{10}}}{\sqrt{2^{2} \cdot 5}}$

Step 7.2

Pull terms out from under the radical.

$\frac{\sqrt{10 - 3 \sqrt{10}}}{2 \sqrt{5}}$

Step 8

Multiply $\frac{\sqrt{10 - 3 \sqrt{10}}}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{10 - 3 \sqrt{10}}}{2 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{\sqrt{10 - 3 \sqrt{10}}}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{10 - 3 \sqrt{10}} \sqrt{5}}{2 \sqrt{5} \sqrt{5}}$

Step 9.2

Move $\sqrt{5}$ .

$\frac{\sqrt{10 - 3 \sqrt{10}} \sqrt{5}}{2 (\sqrt{5} \sqrt{5})}$

Step 9.3

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

$\frac{\sqrt{10 - 3 \sqrt{10}} \sqrt{5}}{2 ({\sqrt{5}}^{1} \sqrt{5})}$

Step 9.4

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

$\frac{\sqrt{10 - 3 \sqrt{10}} \sqrt{5}}{2 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}$

Step 9.5

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

$\frac{\sqrt{10 - 3 \sqrt{10}} \sqrt{5}}{2 {\sqrt{5}}^{1 + 1}}$

Step 9.6

Add $1$ and $1$ .

$\frac{\sqrt{10 - 3 \sqrt{10}} \sqrt{5}}{2 {\sqrt{5}}^{2}}$

Step 9.7

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

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Step 9.7.1

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

$\frac{\sqrt{10 - 3 \sqrt{10}} \sqrt{5}}{2 {(5^{\frac{1}{2}})}^{2}}$

Step 9.7.2

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

$\frac{\sqrt{10 - 3 \sqrt{10}} \sqrt{5}}{2 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 9.7.3

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

$\frac{\sqrt{10 - 3 \sqrt{10}} \sqrt{5}}{2 \cdot 5^{\frac{2}{2}}}$

Step 9.7.4

Cancel the common factor of $2$ .

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Step 9.7.4.1

Cancel the common factor.

$\frac{\sqrt{10 - 3 \sqrt{10}} \sqrt{5}}{2 \cdot 5^{\frac{2}{2}}}$

Step 9.7.4.2

Rewrite the expression.

$\frac{\sqrt{10 - 3 \sqrt{10}} \sqrt{5}}{2 \cdot 5^{1}}$

Step 9.7.5

Evaluate the exponent.

$\frac{\sqrt{10 - 3 \sqrt{10}} \sqrt{5}}{2 \cdot 5}$

Step 10

Combine using the product rule for radicals.

$\frac{\sqrt{(10 - 3 \sqrt{10}) \cdot 5}}{2 \cdot 5}$

Step 11

Multiply $2$ by $5$ .

$\frac{\sqrt{(10 - 3 \sqrt{10}) \cdot 5}}{10}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(10 - 3 \sqrt{10}) \cdot 5}}{10}$

Decimal Form:

$0.16018224 \dots$