Trigonometry Examples

$\sqrt{\frac{\sqrt{145} - 7}{\sqrt{\frac{145}{2}}}}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145}}{\sqrt{2}}}}$

Step 1.2

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

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}}}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

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

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145} \sqrt{2}}{\sqrt{2} \sqrt{2}}}}$

Step 1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}}}$

Step 1.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}}}$

Step 1.3.4

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

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}}}$

Step 1.3.5

Add $1$ and $1$ .

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145} \sqrt{2}}{{\sqrt{2}}^{2}}}}$

Step 1.3.6

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

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Step 1.3.6.1

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

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}}}$

Step 1.3.6.2

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

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}}}$

Step 1.3.6.3

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

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145} \sqrt{2}}{2^{\frac{2}{2}}}}}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145} \sqrt{2}}{2^{\frac{2}{2}}}}}$

Step 1.3.6.4.2

Rewrite the expression.

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145} \sqrt{2}}{2^{1}}}}$

Step 1.3.6.5

Evaluate the exponent.

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145} \sqrt{2}}{2}}}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{145 \cdot 2}}{2}}}$

Step 1.4.2

Multiply $145$ by $2$ .

$\sqrt{\frac{\sqrt{145} - 7}{\frac{\sqrt{290}}{2}}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{(\sqrt{145} - 7) \frac{2}{\sqrt{290}}}$

Step 3

Multiply $\frac{2}{\sqrt{290}}$ by $\frac{\sqrt{290}}{\sqrt{290}}$ .

$\sqrt{(\sqrt{145} - 7) (\frac{2}{\sqrt{290}} \cdot \frac{\sqrt{290}}{\sqrt{290}})}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{2}{\sqrt{290}}$ by $\frac{\sqrt{290}}{\sqrt{290}}$ .

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{\sqrt{290} \sqrt{290}}}$

Step 4.2

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

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{{\sqrt{290}}^{1} \sqrt{290}}}$

Step 4.3

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

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{{\sqrt{290}}^{1} {\sqrt{290}}^{1}}}$

Step 4.4

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

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{{\sqrt{290}}^{1 + 1}}}$

Step 4.5

Add $1$ and $1$ .

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{{\sqrt{290}}^{2}}}$

Step 4.6

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

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Step 4.6.1

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

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{{(290^{\frac{1}{2}})}^{2}}}$

Step 4.6.2

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

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{290^{\frac{1}{2} \cdot 2}}}$

Step 4.6.3

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

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{290^{\frac{2}{2}}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{290^{\frac{2}{2}}}}$

Step 4.6.4.2

Rewrite the expression.

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{290^{1}}}$

Step 4.6.5

Evaluate the exponent.

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{290}}$

Step 5

Cancel the common factor of $2$ and $290$ .

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Step 5.1

Factor $2$ out of $2 \sqrt{290}$ .

$\sqrt{(\sqrt{145} - 7) \frac{2 (\sqrt{290})}{290}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $290$ .

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{2 \cdot 145}}$

Step 5.2.2

Cancel the common factor.

$\sqrt{(\sqrt{145} - 7) \frac{2 \sqrt{290}}{2 \cdot 145}}$

Step 5.2.3

Rewrite the expression.

$\sqrt{(\sqrt{145} - 7) \frac{\sqrt{290}}{145}}$

Step 6

Apply the distributive property.

$\sqrt{\sqrt{145} \frac{\sqrt{290}}{145} - 7 \frac{\sqrt{290}}{145}}$

Step 7

Multiply $\sqrt{145} \frac{\sqrt{290}}{145}$ .

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Step 7.1

Combine $\sqrt{145}$ and $\frac{\sqrt{290}}{145}$ .

$\sqrt{\frac{\sqrt{145} \sqrt{290}}{145} - 7 \frac{\sqrt{290}}{145}}$

Step 7.2

Combine using the product rule for radicals.

$\sqrt{\frac{\sqrt{145 \cdot 290}}{145} - 7 \frac{\sqrt{290}}{145}}$

Step 7.3

Multiply $145$ by $290$ .

$\sqrt{\frac{\sqrt{42050}}{145} - 7 \frac{\sqrt{290}}{145}}$

Step 8

Combine fractions.

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Step 8.1

Combine $- 7$ and $\frac{\sqrt{290}}{145}$ .

$\sqrt{\frac{\sqrt{42050}}{145} + \frac{- 7 \sqrt{290}}{145}}$

Step 8.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\sqrt{42050} - 7 \sqrt{290}}{145}}$

Step 9

Simplify each term.

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Step 9.1

Rewrite $42050$ as $145^{2} \cdot 2$ .

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Step 9.1.1

Factor $21025$ out of $42050$ .

$\sqrt{\frac{\sqrt{21025 (2)} - 7 \sqrt{290}}{145}}$

Step 9.1.2

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

$\sqrt{\frac{\sqrt{145^{2} \cdot 2} - 7 \sqrt{290}}{145}}$

Step 9.2

Pull terms out from under the radical.

$\sqrt{\frac{145 \sqrt{2} - 7 \sqrt{290}}{145}}$

Step 10

Rewrite $\sqrt{\frac{145 \sqrt{2} - 7 \sqrt{290}}{145}}$ as $\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}}}{\sqrt{145}}$ .

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}}}{\sqrt{145}}$

Step 11

Multiply $\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}}}{\sqrt{145}}$ by $\frac{\sqrt{145}}{\sqrt{145}}$ .

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}}}{\sqrt{145}} \cdot \frac{\sqrt{145}}{\sqrt{145}}$

Step 12

Combine and simplify the denominator.

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Step 12.1

Multiply $\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}}}{\sqrt{145}}$ by $\frac{\sqrt{145}}{\sqrt{145}}$ .

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}} \sqrt{145}}{\sqrt{145} \sqrt{145}}$

Step 12.2

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

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}} \sqrt{145}}{{\sqrt{145}}^{1} \sqrt{145}}$

Step 12.3

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

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}} \sqrt{145}}{{\sqrt{145}}^{1} {\sqrt{145}}^{1}}$

Step 12.4

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

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}} \sqrt{145}}{{\sqrt{145}}^{1 + 1}}$

Step 12.5

Add $1$ and $1$ .

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}} \sqrt{145}}{{\sqrt{145}}^{2}}$

Step 12.6

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

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Step 12.6.1

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

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}} \sqrt{145}}{{(145^{\frac{1}{2}})}^{2}}$

Step 12.6.2

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

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}} \sqrt{145}}{145^{\frac{1}{2} \cdot 2}}$

Step 12.6.3

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

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}} \sqrt{145}}{145^{\frac{2}{2}}}$

Step 12.6.4

Cancel the common factor of $2$ .

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Step 12.6.4.1

Cancel the common factor.

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}} \sqrt{145}}{145^{\frac{2}{2}}}$

Step 12.6.4.2

Rewrite the expression.

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}} \sqrt{145}}{145^{1}}$

Step 12.6.5

Evaluate the exponent.

$\frac{\sqrt{145 \sqrt{2} - 7 \sqrt{290}} \sqrt{145}}{145}$

Step 13

Combine using the product rule for radicals.

$\frac{\sqrt{(145 \sqrt{2} - 7 \sqrt{290}) \cdot 145}}{145}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(145 \sqrt{2} - 7 \sqrt{290}) \cdot 145}}{145}$

Decimal Form:

$0.76948375 \dots$