Trigonometry Examples

\sqrt{\frac{\frac{3 - \sqrt{5}}{3}}{2}}

$\sqrt{\frac{\frac{3 - \sqrt{5}}{3}}{2}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

\sqrt{\frac{3 - \sqrt{5}}{3} \cdot \frac{1}{2}}

$\sqrt{\frac{3 - \sqrt{5}}{3} \cdot \frac{1}{2}}$

Step 2

Multiply

\frac{3 - \sqrt{5}}{3} \cdot \frac{1}{2}

$\frac{3 - \sqrt{5}}{3} \cdot \frac{1}{2}$ .

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Step 2.1

Multiply

\frac{3 - \sqrt{5}}{3}

$\frac{3 - \sqrt{5}}{3}$ by

\frac{1}{2}

$\frac{1}{2}$ .

\sqrt{\frac{3 - \sqrt{5}}{3 \cdot 2}}

$\sqrt{\frac{3 - \sqrt{5}}{3 \cdot 2}}$

Step 2.2

Multiply

3

$3$ by

2

$2$ .

\sqrt{\frac{3 - \sqrt{5}}{6}}

$\sqrt{\frac{3 - \sqrt{5}}{6}}$

\sqrt{\frac{3 - \sqrt{5}}{6}}

$\sqrt{\frac{3 - \sqrt{5}}{6}}$

Step 3

Rewrite

\sqrt{\frac{3 - \sqrt{5}}{6}}

$\sqrt{\frac{3 - \sqrt{5}}{6}}$ as

\frac{\sqrt{3 - \sqrt{5}}}{\sqrt{6}}

$\frac{\sqrt{3 - \sqrt{5}}}{\sqrt{6}}$ .

\frac{\sqrt{3 - \sqrt{5}}}{\sqrt{6}}

$\frac{\sqrt{3 - \sqrt{5}}}{\sqrt{6}}$

Step 4

Multiply

\frac{\sqrt{3 - \sqrt{5}}}{\sqrt{6}}

$\frac{\sqrt{3 - \sqrt{5}}}{\sqrt{6}}$ by

\frac{\sqrt{6}}{\sqrt{6}}

$\frac{\sqrt{6}}{\sqrt{6}}$ .

\frac{\sqrt{3 - \sqrt{5}}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}

$\frac{\sqrt{3 - \sqrt{5}}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply

\frac{\sqrt{3 - \sqrt{5}}}{\sqrt{6}}

$\frac{\sqrt{3 - \sqrt{5}}}{\sqrt{6}}$ by

\frac{\sqrt{6}}{\sqrt{6}}

$\frac{\sqrt{6}}{\sqrt{6}}$ .

\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{\sqrt{6} \sqrt{6}}

$\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 5.2

Raise

\sqrt{6}

$\sqrt{6}$ to the power of

1

$1$ .

\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}

$\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 5.3

Raise

\sqrt{6}

$\sqrt{6}$ to the power of

1

$1$ .

\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}

$\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 5.4

Use the power rule

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

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

\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}

$\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 5.5

Add

1

$1$ and

1

$1$ .

\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{{\sqrt{6}}^{2}}

$\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 5.6

Rewrite

{\sqrt{6}}^{2}

${\sqrt{6}}^{2}$ as

6

$6$ .

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Step 5.6.1

Use

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

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

\sqrt{6}

$\sqrt{6}$ as

6^{\frac{1}{2}}

$6^{\frac{1}{2}}$ .

\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}

$\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 5.6.2

Apply the power rule and multiply exponents,

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

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

\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}

$\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 5.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{6^{\frac{2}{2}}}

$\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 5.6.4

Cancel the common factor of

2

$2$ .

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Step 5.6.4.1

Cancel the common factor.

$\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 5.6.4.2

Rewrite the expression.

$\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{6^{1}}$

Step 5.6.5

Evaluate the exponent.

$\frac{\sqrt{3 - \sqrt{5}} \sqrt{6}}{6}$

Step 6

Combine using the product rule for radicals.

$\frac{\sqrt{(3 - \sqrt{5}) \cdot 6}}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{(3 - \sqrt{5}) \cdot 6}}{6}$

Decimal Form:

$0.35682208 \dots$