Trigonometry Examples

$\sqrt[3]{\frac{3}{54}}$

Step 1

Cancel the common factor of $3$ and $54$ .

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Step 1.1

Factor $3$ out of $3$ .

$\sqrt[3]{\frac{3 (1)}{54}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $3$ out of $54$ .

$\sqrt[3]{\frac{3 \cdot 1}{3 \cdot 18}}$

Step 1.2.2

Cancel the common factor.

$\sqrt[3]{\frac{3 \cdot 1}{3 \cdot 18}}$

Step 1.2.3

Rewrite the expression.

$\sqrt[3]{\frac{1}{18}}$

Step 2

Rewrite $\sqrt[3]{\frac{1}{18}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{18}}$ .

$\frac{\sqrt[3]{1}}{\sqrt[3]{18}}$

Step 3

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[3]{18}}$

Step 4

Multiply $\frac{1}{\sqrt[3]{18}}$ by $\frac{{\sqrt[3]{18}}^{2}}{{\sqrt[3]{18}}^{2}}$ .

$\frac{1}{\sqrt[3]{18}} \cdot \frac{{\sqrt[3]{18}}^{2}}{{\sqrt[3]{18}}^{2}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{1}{\sqrt[3]{18}}$ by $\frac{{\sqrt[3]{18}}^{2}}{{\sqrt[3]{18}}^{2}}$ .

$\frac{{\sqrt[3]{18}}^{2}}{\sqrt[3]{18} {\sqrt[3]{18}}^{2}}$

Step 5.2

Raise $\sqrt[3]{18}$ to the power of $1$ .

$\frac{{\sqrt[3]{18}}^{2}}{{\sqrt[3]{18}}^{1} {\sqrt[3]{18}}^{2}}$

Step 5.3

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

$\frac{{\sqrt[3]{18}}^{2}}{{\sqrt[3]{18}}^{1 + 2}}$

Step 5.4

Add $1$ and $2$ .

$\frac{{\sqrt[3]{18}}^{2}}{{\sqrt[3]{18}}^{3}}$

Step 5.5

Rewrite ${\sqrt[3]{18}}^{3}$ as $18$ .

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Step 5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{18}$ as $18^{\frac{1}{3}}$ .

$\frac{{\sqrt[3]{18}}^{2}}{{(18^{\frac{1}{3}})}^{3}}$

Step 5.5.2

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

$\frac{{\sqrt[3]{18}}^{2}}{18^{\frac{1}{3} \cdot 3}}$

Step 5.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{{\sqrt[3]{18}}^{2}}{18^{\frac{3}{3}}}$

Step 5.5.4

Cancel the common factor of $3$ .

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Step 5.5.4.1

Cancel the common factor.

$\frac{{\sqrt[3]{18}}^{2}}{18^{\frac{3}{3}}}$

Step 5.5.4.2

Rewrite the expression.

$\frac{{\sqrt[3]{18}}^{2}}{18^{1}}$

Step 5.5.5

Evaluate the exponent.

$\frac{{\sqrt[3]{18}}^{2}}{18}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite ${\sqrt[3]{18}}^{2}$ as $\sqrt[3]{18^{2}}$ .

$\frac{\sqrt[3]{18^{2}}}{18}$

Step 6.2

Raise $18$ to the power of $2$ .

$\frac{\sqrt[3]{324}}{18}$

Step 6.3

Rewrite $324$ as $3^{3} \cdot 12$ .

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Step 6.3.1

Factor $27$ out of $324$ .

$\frac{\sqrt[3]{27 (12)}}{18}$

Step 6.3.2

Rewrite $27$ as $3^{3}$ .

$\frac{\sqrt[3]{3^{3} \cdot 12}}{18}$

Step 6.4

Pull terms out from under the radical.

$\frac{3 \sqrt[3]{12}}{18}$

Step 7

Cancel the common factor of $3$ and $18$ .

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Step 7.1

Factor $3$ out of $3 \sqrt[3]{12}$ .

$\frac{3 (\sqrt[3]{12})}{18}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $3$ out of $18$ .

$\frac{3 \sqrt[3]{12}}{3 \cdot 6}$

Step 7.2.2

Cancel the common factor.

$\frac{3 \sqrt[3]{12}}{3 \cdot 6}$

Step 7.2.3

Rewrite the expression.

$\frac{\sqrt[3]{12}}{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[3]{12}}{6}$

Decimal Form:

$0.38157141 \dots$