Trigonometry Examples

\frac{\sqrt{3}}{\sqrt{67}}

$\frac{\sqrt{3}}{\sqrt{67}}$

Step 1

Multiply

\frac{\sqrt{3}}{\sqrt{67}}

$\frac{\sqrt{3}}{\sqrt{67}}$ by

\frac{\sqrt{67}}{\sqrt{67}}

$\frac{\sqrt{67}}{\sqrt{67}}$ .

\frac{\sqrt{3}}{\sqrt{67}} \cdot \frac{\sqrt{67}}{\sqrt{67}}

$\frac{\sqrt{3}}{\sqrt{67}} \cdot \frac{\sqrt{67}}{\sqrt{67}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply

\frac{\sqrt{3}}{\sqrt{67}}

$\frac{\sqrt{3}}{\sqrt{67}}$ by

\frac{\sqrt{67}}{\sqrt{67}}

$\frac{\sqrt{67}}{\sqrt{67}}$ .

\frac{\sqrt{3} \sqrt{67}}{\sqrt{67} \sqrt{67}}

$\frac{\sqrt{3} \sqrt{67}}{\sqrt{67} \sqrt{67}}$

Step 2.2

Raise

\sqrt{67}

$\sqrt{67}$ to the power of

1

$1$ .

\frac{\sqrt{3} \sqrt{67}}{{\sqrt{67}}^{1} \sqrt{67}}

$\frac{\sqrt{3} \sqrt{67}}{{\sqrt{67}}^{1} \sqrt{67}}$

Step 2.3

Raise

\sqrt{67}

$\sqrt{67}$ to the power of

1

$1$ .

\frac{\sqrt{3} \sqrt{67}}{{\sqrt{67}}^{1} {\sqrt{67}}^{1}}

$\frac{\sqrt{3} \sqrt{67}}{{\sqrt{67}}^{1} {\sqrt{67}}^{1}}$

Step 2.4

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{67}}{{\sqrt{67}}^{1 + 1}}

$\frac{\sqrt{3} \sqrt{67}}{{\sqrt{67}}^{1 + 1}}$

Step 2.5

Add

1

$1$ and

1

$1$ .

\frac{\sqrt{3} \sqrt{67}}{{\sqrt{67}}^{2}}

$\frac{\sqrt{3} \sqrt{67}}{{\sqrt{67}}^{2}}$

Step 2.6

Rewrite

{\sqrt{67}}^{2}

${\sqrt{67}}^{2}$ as

67

$67$ .

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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{67}

$\sqrt{67}$ as

67^{\frac{1}{2}}

$67^{\frac{1}{2}}$ .

\frac{\sqrt{3} \sqrt{67}}{{(67^{\frac{1}{2}})}^{2}}

$\frac{\sqrt{3} \sqrt{67}}{{(67^{\frac{1}{2}})}^{2}}$

Step 2.6.2

Apply the power rule and multiply exponents,

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

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

\frac{\sqrt{3} \sqrt{67}}{67^{\frac{1}{2} \cdot 2}}

$\frac{\sqrt{3} \sqrt{67}}{67^{\frac{1}{2} \cdot 2}}$

Step 2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{\sqrt{3} \sqrt{67}}{67^{\frac{2}{2}}}

$\frac{\sqrt{3} \sqrt{67}}{67^{\frac{2}{2}}}$

Step 2.6.4

Cancel the common factor of

2

$2$ .

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Step 2.6.4.1

Cancel the common factor.

$\frac{\sqrt{3} \sqrt{67}}{67^{\frac{2}{2}}}$

Step 2.6.4.2

Rewrite the expression.

$\frac{\sqrt{3} \sqrt{67}}{67^{1}}$

Step 2.6.5

Evaluate the exponent.

$\frac{\sqrt{3} \sqrt{67}}{67}$

Step 3

Simplify the numerator.

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Step 3.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 67}}{67}$

Step 3.2

Multiply

$3$ by

$67$ .

$\frac{\sqrt{201}}{67}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{201}}{67}$

Decimal Form:

$0.21160368 \dots$