Trigonometry Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Multiply $\frac{2}{\sqrt{145}}$ by $\frac{\sqrt{145}}{\sqrt{145}}$ .

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{2}{\sqrt{145}}$ by $\frac{\sqrt{145}}{\sqrt{145}}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Evaluate the exponent.

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

Step 4

Apply the distributive property.

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

Step 5

Multiply $\sqrt{145} \frac{2 \sqrt{145}}{145}$ .

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Step 5.1

Combine $\sqrt{145}$ and $\frac{2 \sqrt{145}}{145}$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 6

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

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Step 6.1

Combine $- 9$ and $\frac{2 \sqrt{145}}{145}$ .

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

Step 6.2

Multiply $2$ by $- 9$ .

$\sqrt{\frac{2 {\sqrt{145}}^{2}}{145} + \frac{- 18 \sqrt{145}}{145}}$

Step 7

Combine the numerators over the common denominator.

$\sqrt{\frac{2 {\sqrt{145}}^{2} - 18 \sqrt{145}}{145}}$

Step 8

Simplify the numerator.

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Step 8.1

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

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Step 8.1.1

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

$\sqrt{\frac{2 {(145^{\frac{1}{2}})}^{2} - 18 \sqrt{145}}{145}}$

Step 8.1.2

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

$\sqrt{\frac{2 \cdot 145^{\frac{1}{2} \cdot 2} - 18 \sqrt{145}}{145}}$

Step 8.1.3

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

$\sqrt{\frac{2 \cdot 145^{\frac{2}{2}} - 18 \sqrt{145}}{145}}$

Step 8.1.4

Cancel the common factor of $2$ .

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Step 8.1.4.1

Cancel the common factor.

$\sqrt{\frac{2 \cdot 145^{\frac{2}{2}} - 18 \sqrt{145}}{145}}$

Step 8.1.4.2

Rewrite the expression.

$\sqrt{\frac{2 \cdot 145^{1} - 18 \sqrt{145}}{145}}$

Step 8.1.5

Evaluate the exponent.

$\sqrt{\frac{2 \cdot 145 - 18 \sqrt{145}}{145}}$

Step 8.2

Multiply $2$ by $145$ .

$\sqrt{\frac{290 - 18 \sqrt{145}}{145}}$

Step 9

Rewrite $\sqrt{\frac{290 - 18 \sqrt{145}}{145}}$ as $\frac{\sqrt{290 - 18 \sqrt{145}}}{\sqrt{145}}$ .

$\frac{\sqrt{290 - 18 \sqrt{145}}}{\sqrt{145}}$

Step 10

Multiply $\frac{\sqrt{290 - 18 \sqrt{145}}}{\sqrt{145}}$ by $\frac{\sqrt{145}}{\sqrt{145}}$ .

$\frac{\sqrt{290 - 18 \sqrt{145}}}{\sqrt{145}} \cdot \frac{\sqrt{145}}{\sqrt{145}}$

Step 11

Combine and simplify the denominator.

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Step 11.1

Multiply $\frac{\sqrt{290 - 18 \sqrt{145}}}{\sqrt{145}}$ by $\frac{\sqrt{145}}{\sqrt{145}}$ .

$\frac{\sqrt{290 - 18 \sqrt{145}} \sqrt{145}}{\sqrt{145} \sqrt{145}}$

Step 11.2

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

$\frac{\sqrt{290 - 18 \sqrt{145}} \sqrt{145}}{{\sqrt{145}}^{1} \sqrt{145}}$

Step 11.3

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

$\frac{\sqrt{290 - 18 \sqrt{145}} \sqrt{145}}{{\sqrt{145}}^{1} {\sqrt{145}}^{1}}$

Step 11.4

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

$\frac{\sqrt{290 - 18 \sqrt{145}} \sqrt{145}}{{\sqrt{145}}^{1 + 1}}$

Step 11.5

Add $1$ and $1$ .

$\frac{\sqrt{290 - 18 \sqrt{145}} \sqrt{145}}{{\sqrt{145}}^{2}}$

Step 11.6

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

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Step 11.6.1

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

$\frac{\sqrt{290 - 18 \sqrt{145}} \sqrt{145}}{{(145^{\frac{1}{2}})}^{2}}$

Step 11.6.2

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

$\frac{\sqrt{290 - 18 \sqrt{145}} \sqrt{145}}{145^{\frac{1}{2} \cdot 2}}$

Step 11.6.3

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

$\frac{\sqrt{290 - 18 \sqrt{145}} \sqrt{145}}{145^{\frac{2}{2}}}$

Step 11.6.4

Cancel the common factor of $2$ .

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Step 11.6.4.1

Cancel the common factor.

$\frac{\sqrt{290 - 18 \sqrt{145}} \sqrt{145}}{145^{\frac{2}{2}}}$

Step 11.6.4.2

Rewrite the expression.

$\frac{\sqrt{290 - 18 \sqrt{145}} \sqrt{145}}{145^{1}}$

Step 11.6.5

Evaluate the exponent.

$\frac{\sqrt{290 - 18 \sqrt{145}} \sqrt{145}}{145}$

Step 12

Combine using the product rule for radicals.

$\frac{\sqrt{(290 - 18 \sqrt{145}) \cdot 145}}{145}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(290 - 18 \sqrt{145}) \cdot 145}}{145}$

Decimal Form:

$0.71076111 \dots$