Simplify the Radical Expression cube root of (24x^3)/(36x)

$\sqrt[3]{\frac{24 x^{3}}{36 x}}$

Step 1

Reduce the expression $\frac{24 x^{3}}{36 x}$ by cancelling the common factors.

Factor $12$ out of $24 x^{3}$ .

$\sqrt[3]{\frac{12 (2 x^{3})}{36 x}}$

Factor $12$ out of $36 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

Cancel the common factor of $x^{3}$ and $x$ .

Factor $x$ out of $2 x^{3}$ .

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

Cancel the common factors.

Factor $x$ out of $3 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 6

Simplify the numerator.

Rewrite ${\sqrt[3]{3}}^{2}$ as ${(3^{2})}^{\frac{1}{3}}$ .

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

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

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

Step 7

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $9$ by $2$ .

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

Simplify the Radical Expression cube root of (24x^3)/(36x)

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