Trigonometry Examples

\sqrt{\frac{\frac{\sqrt{61} - 6}{\sqrt{61}}}{2}}

$\sqrt{\frac{\frac{\sqrt{61} - 6}{\sqrt{61}}}{2}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

\sqrt{\frac{\sqrt{61} - 6}{\sqrt{61}} \cdot \frac{1}{2}}

$\sqrt{\frac{\sqrt{61} - 6}{\sqrt{61}} \cdot \frac{1}{2}}$

Step 2

Multiply

\frac{\sqrt{61} - 6}{\sqrt{61}}

$\frac{\sqrt{61} - 6}{\sqrt{61}}$ by

\frac{\sqrt{61}}{\sqrt{61}}

$\frac{\sqrt{61}}{\sqrt{61}}$ .

\sqrt{\frac{\sqrt{61} - 6}{\sqrt{61}} \cdot \frac{\sqrt{61}}{\sqrt{61}} \frac{1}{2}}

$\sqrt{\frac{\sqrt{61} - 6}{\sqrt{61}} \cdot \frac{\sqrt{61}}{\sqrt{61}} \frac{1}{2}}$

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{61} - 6}{\sqrt{61}}

$\frac{\sqrt{61} - 6}{\sqrt{61}}$ by

\frac{\sqrt{61}}{\sqrt{61}}

$\frac{\sqrt{61}}{\sqrt{61}}$ .

\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{\sqrt{61} \sqrt{61}} \cdot \frac{1}{2}}

$\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{\sqrt{61} \sqrt{61}} \cdot \frac{1}{2}}$

Step 3.1.2

Raise

\sqrt{61}

$\sqrt{61}$ to the power of

1

$1$ .

\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{{\sqrt{61}}^{1} \sqrt{61}} \cdot \frac{1}{2}}

$\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{{\sqrt{61}}^{1} \sqrt{61}} \cdot \frac{1}{2}}$

Step 3.1.3

Raise

\sqrt{61}

$\sqrt{61}$ to the power of

1

$1$ .

\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{{\sqrt{61}}^{1} {\sqrt{61}}^{1}} \cdot \frac{1}{2}}

$\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{{\sqrt{61}}^{1} {\sqrt{61}}^{1}} \cdot \frac{1}{2}}$

Step 3.1.4

Use the power rule

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

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

\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{{\sqrt{61}}^{1 + 1}} \cdot \frac{1}{2}}

$\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{{\sqrt{61}}^{1 + 1}} \cdot \frac{1}{2}}$

Step 3.1.5

Add

1

$1$ and

1

$1$ .

\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{{\sqrt{61}}^{2}} \cdot \frac{1}{2}}

$\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{{\sqrt{61}}^{2}} \cdot \frac{1}{2}}$

Step 3.1.6

Rewrite

{\sqrt{61}}^{2}

${\sqrt{61}}^{2}$ as

61

$61$ .

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Step 3.1.6.1

Use

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

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

\sqrt{61}

$\sqrt{61}$ as

61^{\frac{1}{2}}

$61^{\frac{1}{2}}$ .

\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{{(61^{\frac{1}{2}})}^{2}} \cdot \frac{1}{2}}

$\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{{(61^{\frac{1}{2}})}^{2}} \cdot \frac{1}{2}}$

Step 3.1.6.2

Apply the power rule and multiply exponents,

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

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

\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{61^{\frac{1}{2} \cdot 2}} \cdot \frac{1}{2}}

$\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{61^{\frac{1}{2} \cdot 2}} \cdot \frac{1}{2}}$

Step 3.1.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{61^{\frac{2}{2}}} \cdot \frac{1}{2}}

$\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{61^{\frac{2}{2}}} \cdot \frac{1}{2}}$

Step 3.1.6.4

Cancel the common factor of

2

$2$ .

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Step 3.1.6.4.1

Cancel the common factor.

$\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{61^{\frac{2}{2}}} \cdot \frac{1}{2}}$

Step 3.1.6.4.2

Rewrite the expression.

$\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{61^{1}} \cdot \frac{1}{2}}$

Step 3.1.6.5

Evaluate the exponent.

$\sqrt{\frac{(\sqrt{61} - 6) \sqrt{61}}{61} \cdot \frac{1}{2}}$

Step 3.2

Apply the distributive property.

$\sqrt{\frac{\sqrt{61} \sqrt{61} - 6 \sqrt{61}}{61} \cdot \frac{1}{2}}$

Step 3.3

Combine using the product rule for radicals.

$\sqrt{\frac{\sqrt{61 \cdot 61} - 6 \sqrt{61}}{61} \cdot \frac{1}{2}}$

Step 4

Simplify each term.

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Step 4.1

Multiply

$61$ by

$61$ .

$\sqrt{\frac{\sqrt{3721} - 6 \sqrt{61}}{61} \cdot \frac{1}{2}}$

Step 4.2

Rewrite

$3721$ as

$61^{2}$ .

$\sqrt{\frac{\sqrt{61^{2}} - 6 \sqrt{61}}{61} \cdot \frac{1}{2}}$

Step 4.3

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

$\sqrt{\frac{61 - 6 \sqrt{61}}{61} \cdot \frac{1}{2}}$

Step 5

Multiply

$\frac{61 - 6 \sqrt{61}}{61} \cdot \frac{1}{2}$ .

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Step 5.1

Multiply

$\frac{61 - 6 \sqrt{61}}{61}$ by

$\frac{1}{2}$ .

$\sqrt{\frac{61 - 6 \sqrt{61}}{61 \cdot 2}}$

Step 5.2

Multiply

$61$ by

$2$ .

$\sqrt{\frac{61 - 6 \sqrt{61}}{122}}$

Step 6

Rewrite

$\sqrt{\frac{61 - 6 \sqrt{61}}{122}}$ as

$\frac{\sqrt{61 - 6 \sqrt{61}}}{\sqrt{122}}$ .

$\frac{\sqrt{61 - 6 \sqrt{61}}}{\sqrt{122}}$

Step 7

Multiply

$\frac{\sqrt{61 - 6 \sqrt{61}}}{\sqrt{122}}$ by

$\frac{\sqrt{122}}{\sqrt{122}}$ .

$\frac{\sqrt{61 - 6 \sqrt{61}}}{\sqrt{122}} \cdot \frac{\sqrt{122}}{\sqrt{122}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply

$\frac{\sqrt{61 - 6 \sqrt{61}}}{\sqrt{122}}$ by

$\frac{\sqrt{122}}{\sqrt{122}}$ .

$\frac{\sqrt{61 - 6 \sqrt{61}} \sqrt{122}}{\sqrt{122} \sqrt{122}}$

Step 8.2

Raise

$\sqrt{122}$ to the power of

$1$ .

$\frac{\sqrt{61 - 6 \sqrt{61}} \sqrt{122}}{{\sqrt{122}}^{1} \sqrt{122}}$

Step 8.3

Raise

$\sqrt{122}$ to the power of

$1$ .

$\frac{\sqrt{61 - 6 \sqrt{61}} \sqrt{122}}{{\sqrt{122}}^{1} {\sqrt{122}}^{1}}$

Step 8.4

Use the power rule

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

$\frac{\sqrt{61 - 6 \sqrt{61}} \sqrt{122}}{{\sqrt{122}}^{1 + 1}}$

Step 8.5

Add

$1$ and

$1$ .

$\frac{\sqrt{61 - 6 \sqrt{61}} \sqrt{122}}{{\sqrt{122}}^{2}}$

Step 8.6

Rewrite

${\sqrt{122}}^{2}$ as

$122$ .

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Step 8.6.1

Use

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

$\sqrt{122}$ as

$122^{\frac{1}{2}}$ .

$\frac{\sqrt{61 - 6 \sqrt{61}} \sqrt{122}}{{(122^{\frac{1}{2}})}^{2}}$

Step 8.6.2

Apply the power rule and multiply exponents,

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

$\frac{\sqrt{61 - 6 \sqrt{61}} \sqrt{122}}{122^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{\sqrt{61 - 6 \sqrt{61}} \sqrt{122}}{122^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of

$2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{\sqrt{61 - 6 \sqrt{61}} \sqrt{122}}{122^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$\frac{\sqrt{61 - 6 \sqrt{61}} \sqrt{122}}{122^{1}}$

Step 8.6.5

Evaluate the exponent.

$\frac{\sqrt{61 - 6 \sqrt{61}} \sqrt{122}}{122}$

Step 9

Combine using the product rule for radicals.

$\frac{\sqrt{(61 - 6 \sqrt{61}) \cdot 122}}{122}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(61 - 6 \sqrt{61}) \cdot 122}}{122}$

Decimal Form:

$0.34042526 \dots$