Trigonometry Examples

\sqrt{\frac{1 - (\frac{3}{\sqrt{73}})}{2}}

$\sqrt{\frac{1 - (\frac{3}{\sqrt{73}})}{2}}$

Step 1

Multiply

\frac{3}{\sqrt{73}}

$\frac{3}{\sqrt{73}}$ by

\frac{\sqrt{73}}{\sqrt{73}}

$\frac{\sqrt{73}}{\sqrt{73}}$ .

\sqrt{\frac{1 - (\frac{3}{\sqrt{73}} \cdot \frac{\sqrt{73}}{\sqrt{73}})}{2}}

$\sqrt{\frac{1 - (\frac{3}{\sqrt{73}} \cdot \frac{\sqrt{73}}{\sqrt{73}})}{2}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply

\frac{3}{\sqrt{73}}

$\frac{3}{\sqrt{73}}$ by

\frac{\sqrt{73}}{\sqrt{73}}

$\frac{\sqrt{73}}{\sqrt{73}}$ .

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{\sqrt{73} \sqrt{73}}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{\sqrt{73} \sqrt{73}}}{2}}$

Step 2.2

Raise

\sqrt{73}

$\sqrt{73}$ to the power of

1

$1$ .

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{{\sqrt{73}}^{1} \sqrt{73}}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{{\sqrt{73}}^{1} \sqrt{73}}}{2}}$

Step 2.3

Raise

\sqrt{73}

$\sqrt{73}$ to the power of

1

$1$ .

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{{\sqrt{73}}^{1} {\sqrt{73}}^{1}}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{{\sqrt{73}}^{1} {\sqrt{73}}^{1}}}{2}}$

Step 2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{{\sqrt{73}}^{1 + 1}}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{{\sqrt{73}}^{1 + 1}}}{2}}$

Step 2.5

Add

1

$1$ and

1

$1$ .

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{{\sqrt{73}}^{2}}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{{\sqrt{73}}^{2}}}{2}}$

Step 2.6

Rewrite

{\sqrt{73}}^{2}

${\sqrt{73}}^{2}$ as

73

$73$ .

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Step 2.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{73}

$\sqrt{73}$ as

73^{\frac{1}{2}}

$73^{\frac{1}{2}}$ .

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{{(73^{\frac{1}{2}})}^{2}}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{{(73^{\frac{1}{2}})}^{2}}}{2}}$

Step 2.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73^{\frac{1}{2} \cdot 2}}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73^{\frac{1}{2} \cdot 2}}}{2}}$

Step 2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73^{\frac{2}{2}}}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73^{\frac{2}{2}}}}{2}}$

Step 2.6.4

Cancel the common factor of

2

$2$ .

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Step 2.6.4.1

Cancel the common factor.

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73^{\frac{2}{2}}}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73^{\frac{2}{2}}}}{2}}$

Step 2.6.4.2

Rewrite the expression.

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73^{1}}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73^{1}}}{2}}$

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73^{1}}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73^{1}}}{2}}$

Step 2.6.5

Evaluate the exponent.

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73}}{2}}$

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73}}{2}}$

\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73}}{2}}

$\sqrt{\frac{1 - \frac{3 \sqrt{73}}{73}}{2}}$

Step 3

Simplify the expression.

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Step 3.1

Write

1

$1$ as a fraction with a common denominator.

\sqrt{\frac{\frac{73}{73} - \frac{3 \sqrt{73}}{73}}{2}}

$\sqrt{\frac{\frac{73}{73} - \frac{3 \sqrt{73}}{73}}{2}}$

Step 3.2

Combine the numerators over the common denominator.

\sqrt{\frac{\frac{73 - 3 \sqrt{73}}{73}}{2}}

$\sqrt{\frac{\frac{73 - 3 \sqrt{73}}{73}}{2}}$

\sqrt{\frac{\frac{73 - 3 \sqrt{73}}{73}}{2}}

$\sqrt{\frac{\frac{73 - 3 \sqrt{73}}{73}}{2}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

\sqrt{\frac{73 - 3 \sqrt{73}}{73} \cdot \frac{1}{2}}

$\sqrt{\frac{73 - 3 \sqrt{73}}{73} \cdot \frac{1}{2}}$

Step 5

Multiply

\frac{73 - 3 \sqrt{73}}{73} \cdot \frac{1}{2}

$\frac{73 - 3 \sqrt{73}}{73} \cdot \frac{1}{2}$ .

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Step 5.1

Multiply

\frac{73 - 3 \sqrt{73}}{73}

$\frac{73 - 3 \sqrt{73}}{73}$ by

\frac{1}{2}

$\frac{1}{2}$ .

\sqrt{\frac{73 - 3 \sqrt{73}}{73 \cdot 2}}

$\sqrt{\frac{73 - 3 \sqrt{73}}{73 \cdot 2}}$

Step 5.2

Multiply

73

$73$ by

2

$2$ .

\sqrt{\frac{73 - 3 \sqrt{73}}{146}}

$\sqrt{\frac{73 - 3 \sqrt{73}}{146}}$

\sqrt{\frac{73 - 3 \sqrt{73}}{146}}

$\sqrt{\frac{73 - 3 \sqrt{73}}{146}}$

Step 6

Rewrite

\sqrt{\frac{73 - 3 \sqrt{73}}{146}}

$\sqrt{\frac{73 - 3 \sqrt{73}}{146}}$ as

\frac{\sqrt{73 - 3 \sqrt{73}}}{\sqrt{146}}

$\frac{\sqrt{73 - 3 \sqrt{73}}}{\sqrt{146}}$ .

\frac{\sqrt{73 - 3 \sqrt{73}}}{\sqrt{146}}

$\frac{\sqrt{73 - 3 \sqrt{73}}}{\sqrt{146}}$

Step 7

Multiply

\frac{\sqrt{73 - 3 \sqrt{73}}}{\sqrt{146}}

$\frac{\sqrt{73 - 3 \sqrt{73}}}{\sqrt{146}}$ by

\frac{\sqrt{146}}{\sqrt{146}}

$\frac{\sqrt{146}}{\sqrt{146}}$ .

\frac{\sqrt{73 - 3 \sqrt{73}}}{\sqrt{146}} \cdot \frac{\sqrt{146}}{\sqrt{146}}

$\frac{\sqrt{73 - 3 \sqrt{73}}}{\sqrt{146}} \cdot \frac{\sqrt{146}}{\sqrt{146}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply

\frac{\sqrt{73 - 3 \sqrt{73}}}{\sqrt{146}}

$\frac{\sqrt{73 - 3 \sqrt{73}}}{\sqrt{146}}$ by

\frac{\sqrt{146}}{\sqrt{146}}

$\frac{\sqrt{146}}{\sqrt{146}}$ .

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{\sqrt{146} \sqrt{146}}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{\sqrt{146} \sqrt{146}}$

Step 8.2

Raise

\sqrt{146}

$\sqrt{146}$ to the power of

1

$1$ .

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{{\sqrt{146}}^{1} \sqrt{146}}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{{\sqrt{146}}^{1} \sqrt{146}}$

Step 8.3

Raise

\sqrt{146}

$\sqrt{146}$ to the power of

1

$1$ .

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{{\sqrt{146}}^{1} {\sqrt{146}}^{1}}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{{\sqrt{146}}^{1} {\sqrt{146}}^{1}}$

Step 8.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{{\sqrt{146}}^{1 + 1}}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{{\sqrt{146}}^{1 + 1}}$

Step 8.5

Add

1

$1$ and

1

$1$ .

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{{\sqrt{146}}^{2}}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{{\sqrt{146}}^{2}}$

Step 8.6

Rewrite

{\sqrt{146}}^{2}

${\sqrt{146}}^{2}$ as

146

$146$ .

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Step 8.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{146}

$\sqrt{146}$ as

146^{\frac{1}{2}}

$146^{\frac{1}{2}}$ .

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{{(146^{\frac{1}{2}})}^{2}}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{{(146^{\frac{1}{2}})}^{2}}$

Step 8.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146^{\frac{1}{2} \cdot 2}}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146^{\frac{2}{2}}}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of

2

$2$ .

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Step 8.6.4.1

Cancel the common factor.

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146^{\frac{2}{2}}}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146^{1}}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146^{1}}$

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146^{1}}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146^{1}}$

Step 8.6.5

Evaluate the exponent.

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146}$

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146}$

\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146}

$\frac{\sqrt{73 - 3 \sqrt{73}} \sqrt{146}}{146}$

Step 9

Combine using the product rule for radicals.

\frac{\sqrt{(73 - 3 \sqrt{73}) \cdot 146}}{146}

$\frac{\sqrt{(73 - 3 \sqrt{73}) \cdot 146}}{146}$

Step 10

The result can be shown in multiple forms.

Exact Form:

\frac{\sqrt{(73 - 3 \sqrt{73}) \cdot 146}}{146}

$\frac{\sqrt{(73 - 3 \sqrt{73}) \cdot 146}}{146}$

Decimal Form:

0.56959483 \dots

$0.56959483 \dots$