Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $2$ .

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

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

Step 3.5.5

Simplify.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

Apply the product rule to $3 m$ .

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

Step 4.3

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

$\frac{\sqrt[3]{5} \sqrt[3]{9 m^{2}}}{3 m}$

Step 5

Simplify the numerator.

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Step 5.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{5 (9 m^{2})}}{3 m}$

Step 5.2

Multiply $5$ by $9$ .

$\frac{\sqrt[3]{45 m^{2}}}{3 m}$