Algebra Examples

$\frac{4 x \sqrt[3]{6}}{5 \sqrt[3]{9 x}}$

Step 1

Combine $\sqrt[3]{6}$ and $\sqrt[3]{9 x}$ into a single radical.

$\frac{4 x \sqrt[3]{\frac{6}{9 x}}}{5}$

Step 2

Cancel the common factor of $6$ and $9$ .

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Step 2.1

Factor $3$ out of $6$ .

$\frac{4 x \sqrt[3]{\frac{3 (2)}{9 x}}}{5}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $3$ out of $9 x$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Add $1$ and $2$ .

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

Step 3.3.5

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

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Step 3.3.5.1

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

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

Step 3.3.5.2

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

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

Step 3.3.5.3

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

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

Step 3.3.5.4

Cancel the common factor of $3$ .

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Step 3.3.5.4.1

Cancel the common factor.

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

Step 3.3.5.4.2

Rewrite the expression.

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

Step 3.3.5.5

Simplify.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

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

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

Step 3.4.2

Apply the product rule to $3 x$ .

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

Step 3.4.3

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

$\frac{4 x \frac{\sqrt[3]{2} \sqrt[3]{9 x^{2}}}{3 x}}{5}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Combine using the product rule for radicals.

$\frac{4 x \frac{\sqrt[3]{2 (9 x^{2})}}{3 x}}{5}$

Step 3.5.2

Multiply $2$ by $9$ .

$\frac{4 x \frac{\sqrt[3]{18 x^{2}}}{3 x}}{5}$

Step 3.6

Combine exponents.

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Step 3.6.1

Combine $4$ and $\frac{\sqrt[3]{18 x^{2}}}{3 x}$ .

$\frac{x \frac{4 \sqrt[3]{18 x^{2}}}{3 x}}{5}$

Step 3.6.2

Combine $x$ and $\frac{4 \sqrt[3]{18 x^{2}}}{3 x}$ .

$\frac{\frac{x (4 \sqrt[3]{18 x^{2}})}{3 x}}{5}$

Step 3.7

Remove unnecessary parentheses.

$\frac{\frac{x \cdot 4 \sqrt[3]{18 x^{2}}}{3 x}}{5}$

Step 3.8

Reduce the expression $\frac{x \cdot 4 \sqrt[3]{18 x^{2}}}{3 x}$ by cancelling the common factors.

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Step 3.8.1

Cancel the common factor.

$\frac{\frac{x \cdot 4 \sqrt[3]{18 x^{2}}}{3 x}}{5}$

Step 3.8.2

Rewrite the expression.

$\frac{\frac{4 \sqrt[3]{18 x^{2}}}{3}}{5}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 \sqrt[3]{18 x^{2}}}{3} \cdot \frac{1}{5}$

Step 5

Combine.

$\frac{4 \sqrt[3]{18 x^{2}} \cdot 1}{3 \cdot 5}$

Step 6

Multiply.

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Step 6.1

Multiply $4$ by $1$ .

$\frac{4 \sqrt[3]{18 x^{2}}}{3 \cdot 5}$

Step 6.2

Multiply $3$ by $5$ .

$\frac{4 \sqrt[3]{18 x^{2}}}{15}$