Algebra Examples

$\sqrt[3]{\frac{3}{125}}$

Step 1

Rewrite $\sqrt[3]{\frac{3}{125}}$ as $\frac{\sqrt[3]{3}}{\sqrt[3]{125}}$ .

$\frac{\sqrt[3]{3}}{\sqrt[3]{125}}$

Step 2

Multiply to rationalize the numerator.

$\frac{\sqrt[3]{3} {\sqrt[3]{3}}^{2}}{\sqrt[3]{125} {\sqrt[3]{3}}^{2}}$

Step 3

Simplify.

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Step 3.1

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

$\frac{{\sqrt[3]{3}}^{1} {\sqrt[3]{3}}^{2}}{\sqrt[3]{125} {\sqrt[3]{3}}^{2}}$

Step 3.2

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

$\frac{{\sqrt[3]{3}}^{1 + 2}}{\sqrt[3]{125} {\sqrt[3]{3}}^{2}}$

Step 3.3

Add $1$ and $2$ .

$\frac{{\sqrt[3]{3}}^{3}}{\sqrt[3]{125} {\sqrt[3]{3}}^{2}}$

Step 3.4

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

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Step 3.4.1

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

$\frac{{(3^{\frac{1}{3}})}^{3}}{\sqrt[3]{125} {\sqrt[3]{3}}^{2}}$

Step 3.4.2

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

$\frac{3^{\frac{1}{3} \cdot 3}}{\sqrt[3]{125} {\sqrt[3]{3}}^{2}}$

Step 3.4.3

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

$\frac{3^{\frac{3}{3}}}{\sqrt[3]{125} {\sqrt[3]{3}}^{2}}$

Step 3.4.4

Cancel the common factor of $3$ .

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Step 3.4.4.1

Cancel the common factor.

$\frac{3^{\frac{3}{3}}}{\sqrt[3]{125} {\sqrt[3]{3}}^{2}}$

Step 3.4.4.2

Rewrite the expression.

$\frac{3^{1}}{\sqrt[3]{125} {\sqrt[3]{3}}^{2}}$

Step 3.4.5

Evaluate the exponent.

$\frac{3}{\sqrt[3]{125} {\sqrt[3]{3}}^{2}}$

Step 3.5

Start simplifying.

$\frac{3}{\sqrt[3]{125} \sqrt[3]{3^{2}}}$

Step 3.6

Rewrite $125$ as $5^{3}$ .

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

Step 3.7

Pull terms out from under the radical, assuming real numbers.

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

Step 3.8

Raise $3$ to the power of $2$ .

$\frac{3}{5 \sqrt[3]{9}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{5 \sqrt[3]{9}}$

Decimal Form:

$0.28844991 \dots$