Algebra Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Add $1$ and $2$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

Apply the product rule to $2 x$ .

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

Step 3.3

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

$\frac{\sqrt[3]{3} \sqrt[3]{4 x^{2}}}{2 x}$

Step 4

Simplify the numerator.

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Step 4.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{3 (4 x^{2})}}{2 x}$

Step 4.2

Multiply $3$ by $4$ .

$\frac{\sqrt[3]{12 x^{2}}}{2 x}$