Trigonometry Examples

$\frac{2}{\sqrt{54}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $54$ as $3^{2} \cdot 6$ .

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Step 1.1.1

Factor $9$ out of $54$ .

$\frac{2}{\sqrt{9 (6)}}$

Step 1.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{2}{\sqrt{3^{2} \cdot 6}}$

Step 1.2

Pull terms out from under the radical.

$\frac{2}{3 \sqrt{6}}$

Step 2

Multiply $\frac{2}{3 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{2}{3 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{2}{3 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{2 \sqrt{6}}{3 \sqrt{6} \sqrt{6}}$

Step 3.2

Move $\sqrt{6}$ .

$\frac{2 \sqrt{6}}{3 (\sqrt{6} \sqrt{6})}$

Step 3.3

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 3.4

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 3.5

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

$\frac{2 \sqrt{6}}{3 {\sqrt{6}}^{1 + 1}}$

Step 3.6

Add $1$ and $1$ .

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

Step 3.7

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

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Step 3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 3.7.2

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

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

Step 3.7.3

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

$\frac{2 \sqrt{6}}{3 \cdot 6^{\frac{2}{2}}}$

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

$\frac{2 \sqrt{6}}{3 \cdot 6^{\frac{2}{2}}}$

Step 3.7.4.2

Rewrite the expression.

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

Step 3.7.5

Evaluate the exponent.

$\frac{2 \sqrt{6}}{3 \cdot 6}$

Step 4

Cancel the common factor of $2$ and $6$ .

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Step 4.1

Factor $2$ out of $2 \sqrt{6}$ .

$\frac{2 (\sqrt{6})}{3 \cdot 6}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $2$ out of $3 \cdot 6$ .

$\frac{2 (\sqrt{6})}{2 (3 \cdot 3)}$

Step 4.2.2

Cancel the common factor.

$\frac{2 \sqrt{6}}{2 (3 \cdot 3)}$

Step 4.2.3

Rewrite the expression.

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

Step 5

Multiply $3$ by $3$ .

$\frac{\sqrt{6}}{9}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6}}{9}$

Decimal Form:

$0.27216552 \dots$