Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $2$ .

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

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

Step 3.5.5

Evaluate the exponent.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Simplify the numerator.

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Step 5.1

Combine using the product rule for radicals.

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

Step 5.2

Multiply $5$ by $4$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.35720880 \dots$