Trigonometry Examples

$\sqrt{\frac{\frac{\sqrt{145} - 9}{\sqrt{145}}}{2}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{\sqrt{145} - 9}{\sqrt{145}} \cdot \frac{1}{2}}$

Step 2

Multiply $\frac{\sqrt{145} - 9}{\sqrt{145}}$ by $\frac{\sqrt{145}}{\sqrt{145}}$ .

$\sqrt{\frac{\sqrt{145} - 9}{\sqrt{145}} \cdot \frac{\sqrt{145}}{\sqrt{145}} \frac{1}{2}}$

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 $\frac{\sqrt{145} - 9}{\sqrt{145}}$ by $\frac{\sqrt{145}}{\sqrt{145}}$ .

$\sqrt{\frac{(\sqrt{145} - 9) \sqrt{145}}{\sqrt{145} \sqrt{145}} \cdot \frac{1}{2}}$

Step 3.1.2

Raise $\sqrt{145}$ to the power of $1$ .

$\sqrt{\frac{(\sqrt{145} - 9) \sqrt{145}}{{\sqrt{145}}^{1} \sqrt{145}} \cdot \frac{1}{2}}$

Step 3.1.3

Raise $\sqrt{145}$ to the power of $1$ .

$\sqrt{\frac{(\sqrt{145} - 9) \sqrt{145}}{{\sqrt{145}}^{1} {\sqrt{145}}^{1}} \cdot \frac{1}{2}}$

Step 3.1.4

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

$\sqrt{\frac{(\sqrt{145} - 9) \sqrt{145}}{{\sqrt{145}}^{1 + 1}} \cdot \frac{1}{2}}$

Step 3.1.5

Add $1$ and $1$ .

$\sqrt{\frac{(\sqrt{145} - 9) \sqrt{145}}{{\sqrt{145}}^{2}} \cdot \frac{1}{2}}$

Step 3.1.6

Rewrite ${\sqrt{145}}^{2}$ as $145$ .

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Step 3.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{145}$ as $145^{\frac{1}{2}}$ .

$\sqrt{\frac{(\sqrt{145} - 9) \sqrt{145}}{{(145^{\frac{1}{2}})}^{2}} \cdot \frac{1}{2}}$

Step 3.1.6.2

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

$\sqrt{\frac{(\sqrt{145} - 9) \sqrt{145}}{145^{\frac{1}{2} \cdot 2}} \cdot \frac{1}{2}}$

Step 3.1.6.3

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

$\sqrt{\frac{(\sqrt{145} - 9) \sqrt{145}}{145^{\frac{2}{2}}} \cdot \frac{1}{2}}$

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.

$\sqrt{\frac{(\sqrt{145} - 9) \sqrt{145}}{145^{\frac{2}{2}}} \cdot \frac{1}{2}}$

Step 3.1.6.4.2

Rewrite the expression.

$\sqrt{\frac{(\sqrt{145} - 9) \sqrt{145}}{145^{1}} \cdot \frac{1}{2}}$

Step 3.1.6.5

Evaluate the exponent.

$\sqrt{\frac{(\sqrt{145} - 9) \sqrt{145}}{145} \cdot \frac{1}{2}}$

Step 3.2

Apply the distributive property.

$\sqrt{\frac{\sqrt{145} \sqrt{145} - 9 \sqrt{145}}{145} \cdot \frac{1}{2}}$

Step 3.3

Combine using the product rule for radicals.

$\sqrt{\frac{\sqrt{145 \cdot 145} - 9 \sqrt{145}}{145} \cdot \frac{1}{2}}$

Step 4

Simplify each term.

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Step 4.1

Multiply $145$ by $145$ .

$\sqrt{\frac{\sqrt{21025} - 9 \sqrt{145}}{145} \cdot \frac{1}{2}}$

Step 4.2

Rewrite $21025$ as $145^{2}$ .

$\sqrt{\frac{\sqrt{145^{2}} - 9 \sqrt{145}}{145} \cdot \frac{1}{2}}$

Step 4.3

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

$\sqrt{\frac{145 - 9 \sqrt{145}}{145} \cdot \frac{1}{2}}$

Step 5

Multiply $\frac{145 - 9 \sqrt{145}}{145} \cdot \frac{1}{2}$ .

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Step 5.1

Multiply $\frac{145 - 9 \sqrt{145}}{145}$ by $\frac{1}{2}$ .

$\sqrt{\frac{145 - 9 \sqrt{145}}{145 \cdot 2}}$

Step 5.2

Multiply $145$ by $2$ .

$\sqrt{\frac{145 - 9 \sqrt{145}}{290}}$

Step 6

Rewrite $\sqrt{\frac{145 - 9 \sqrt{145}}{290}}$ as $\frac{\sqrt{145 - 9 \sqrt{145}}}{\sqrt{290}}$ .

$\frac{\sqrt{145 - 9 \sqrt{145}}}{\sqrt{290}}$

Step 7

Multiply $\frac{\sqrt{145 - 9 \sqrt{145}}}{\sqrt{290}}$ by $\frac{\sqrt{290}}{\sqrt{290}}$ .

$\frac{\sqrt{145 - 9 \sqrt{145}}}{\sqrt{290}} \cdot \frac{\sqrt{290}}{\sqrt{290}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{\sqrt{145 - 9 \sqrt{145}}}{\sqrt{290}}$ by $\frac{\sqrt{290}}{\sqrt{290}}$ .

$\frac{\sqrt{145 - 9 \sqrt{145}} \sqrt{290}}{\sqrt{290} \sqrt{290}}$

Step 8.2

Raise $\sqrt{290}$ to the power of $1$ .

$\frac{\sqrt{145 - 9 \sqrt{145}} \sqrt{290}}{{\sqrt{290}}^{1} \sqrt{290}}$

Step 8.3

Raise $\sqrt{290}$ to the power of $1$ .

$\frac{\sqrt{145 - 9 \sqrt{145}} \sqrt{290}}{{\sqrt{290}}^{1} {\sqrt{290}}^{1}}$

Step 8.4

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

$\frac{\sqrt{145 - 9 \sqrt{145}} \sqrt{290}}{{\sqrt{290}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$\frac{\sqrt{145 - 9 \sqrt{145}} \sqrt{290}}{{\sqrt{290}}^{2}}$

Step 8.6

Rewrite ${\sqrt{290}}^{2}$ as $290$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{290}$ as $290^{\frac{1}{2}}$ .

$\frac{\sqrt{145 - 9 \sqrt{145}} \sqrt{290}}{{(290^{\frac{1}{2}})}^{2}}$

Step 8.6.2

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

$\frac{\sqrt{145 - 9 \sqrt{145}} \sqrt{290}}{290^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

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

$\frac{\sqrt{145 - 9 \sqrt{145}} \sqrt{290}}{290^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{\sqrt{145 - 9 \sqrt{145}} \sqrt{290}}{290^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$\frac{\sqrt{145 - 9 \sqrt{145}} \sqrt{290}}{290^{1}}$

Step 8.6.5

Evaluate the exponent.

$\frac{\sqrt{145 - 9 \sqrt{145}} \sqrt{290}}{290}$

Step 9

Combine using the product rule for radicals.

$\frac{\sqrt{(145 - 9 \sqrt{145}) \cdot 290}}{290}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(145 - 9 \sqrt{145}) \cdot 290}}{290}$

Decimal Form:

$0.35538055 \dots$