Algebra Examples

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

Step 1

Reduce the expression $\frac{4 x}{5 x^{2}}$ by cancelling the common factors.

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Step 1.1

Factor $x$ out of $4 x$ .

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

Step 1.2

Factor $x$ out of $5 x^{2}$ .

$\sqrt[3]{\frac{x \cdot 4}{x (5 x)}}$

Step 1.3

Cancel the common factor.

$\sqrt[3]{\frac{x \cdot 4}{x (5 x)}}$

Step 1.4

Rewrite the expression.

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

Step 2

Rewrite $\sqrt[3]{\frac{4}{5 x}}$ as $\frac{\sqrt[3]{4}}{\sqrt[3]{5 x}}$ .

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

Step 3

Multiply $\frac{\sqrt[3]{4}}{\sqrt[3]{5 x}}$ by $\frac{{\sqrt[3]{5 x}}^{2}}{{\sqrt[3]{5 x}}^{2}}$ .

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt[3]{4}}{\sqrt[3]{5 x}}$ by $\frac{{\sqrt[3]{5 x}}^{2}}{{\sqrt[3]{5 x}}^{2}}$ .

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

Step 4.2

Raise $\sqrt[3]{5 x}$ to the power of $1$ .

$\frac{\sqrt[3]{4} {\sqrt[3]{5 x}}^{2}}{{\sqrt[3]{5 x}}^{1} {\sqrt[3]{5 x}}^{2}}$

Step 4.3

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

$\frac{\sqrt[3]{4} {\sqrt[3]{5 x}}^{2}}{{\sqrt[3]{5 x}}^{1 + 2}}$

Step 4.4

Add $1$ and $2$ .

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

Step 4.5

Rewrite ${\sqrt[3]{5 x}}^{3}$ as $5 x$ .

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Step 4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 x}$ as ${(5 x)}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{4} {\sqrt[3]{5 x}}^{2}}{{({(5 x)}^{\frac{1}{3}})}^{3}}$

Step 4.5.2

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

$\frac{\sqrt[3]{4} {\sqrt[3]{5 x}}^{2}}{{(5 x)}^{\frac{1}{3} \cdot 3}}$

Step 4.5.3

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

$\frac{\sqrt[3]{4} {\sqrt[3]{5 x}}^{2}}{{(5 x)}^{\frac{3}{3}}}$

Step 4.5.4

Cancel the common factor of $3$ .

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Step 4.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{4} {\sqrt[3]{5 x}}^{2}}{{(5 x)}^{\frac{3}{3}}}$

Step 4.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{4} {\sqrt[3]{5 x}}^{2}}{{(5 x)}^{1}}$

Step 4.5.5

Simplify.

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

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[3]{5 x}}^{2}$ as $\sqrt[3]{{(5 x)}^{2}}$ .

$\frac{\sqrt[3]{4} \sqrt[3]{{(5 x)}^{2}}}{5 x}$

Step 5.2

Apply the product rule to $5 x$ .

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

Step 5.3

Raise $5$ to the power of $2$ .

$\frac{\sqrt[3]{4} \sqrt[3]{25 x^{2}}}{5 x}$

Step 6

Simplify the numerator.

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Step 6.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{4 (25 x^{2})}}{5 x}$

Step 6.2

Multiply $4$ by $25$ .

$\frac{\sqrt[3]{100 x^{2}}}{5 x}$