Trigonometry Examples

$\sqrt[3]{- \frac{5}{16}}$

Step 1

Rewrite $- \frac{5}{16}$ as ${(- \frac{1}{2})}^{3} \frac{5}{2}$ .

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Step 1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \frac{5}{16}}$

Step 1.2

Factor the perfect power $1^{3}$ out of $5$ .

$\sqrt[3]{{(- 1)}^{3} \frac{1^{3} \cdot 5}{16}}$

Step 1.3

Factor the perfect power $2^{3}$ out of $16$ .

$\sqrt[3]{{(- 1)}^{3} \frac{1^{3} \cdot 5}{2^{3} \cdot 2}}$

Step 1.4

Rearrange the fraction $\frac{1^{3} \cdot 5}{2^{3} \cdot 2}$ .

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

Step 1.5

Rewrite ${(- 1)}^{3} {(\frac{1}{2})}^{3}$ as ${(- \frac{1}{2})}^{3}$ .

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

Step 2

Pull terms out from under the radical.

$- \frac{1}{2} \sqrt[3]{\frac{5}{2}}$

Step 3

Rewrite $\sqrt[3]{\frac{5}{2}}$ as $\frac{\sqrt[3]{5}}{\sqrt[3]{2}}$ .

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

Step 4

Multiply $\frac{\sqrt[3]{5}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

$- \frac{1}{2} (\frac{\sqrt[3]{5}}{\sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt[3]{5}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

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

Step 5.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

Step 5.3

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

$- \frac{1}{2} \cdot \frac{\sqrt[3]{5} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Step 5.4

Add $1$ and $2$ .

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

Step 5.5

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

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Step 5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

$- \frac{1}{2} \cdot \frac{\sqrt[3]{5} {\sqrt[3]{2}}^{2}}{{(2^{\frac{1}{3}})}^{3}}$

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.5.4

Cancel the common factor of $3$ .

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Step 5.5.4.1

Cancel the common factor.

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

Step 5.5.4.2

Rewrite the expression.

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

Step 5.5.5

Evaluate the exponent.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

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

$- \frac{1}{2} \cdot \frac{\sqrt[3]{5} \sqrt[3]{4}}{2}$

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

$- \frac{1}{2} \cdot \frac{\sqrt[3]{5 \cdot 4}}{2}$

Step 7.2

Multiply $5$ by $4$ .

$- \frac{1}{2} \cdot \frac{\sqrt[3]{20}}{2}$

Step 8

Multiply $- \frac{1}{2} \cdot \frac{\sqrt[3]{20}}{2}$ .

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Step 8.1

Multiply $\frac{\sqrt[3]{20}}{2}$ by $\frac{1}{2}$ .

$- \frac{\sqrt[3]{20}}{2 \cdot 2}$

Step 8.2

Multiply $2$ by $2$ .

$- \frac{\sqrt[3]{20}}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt[3]{20}}{4}$

Decimal Form:

$- 0.67860440 \dots$