Simplify the Radical Expression 8/( cube root of 3x^2)

$\frac{8}{\sqrt[3]{3 x^{2}}}$

Step 1

Multiply $\frac{8}{\sqrt[3]{3 x^{2}}}$ by $\frac{{\sqrt[3]{3 x^{2}}}^{2}}{{\sqrt[3]{3 x^{2}}}^{2}}$ .

$\frac{8}{\sqrt[3]{3 x^{2}}} \cdot \frac{{\sqrt[3]{3 x^{2}}}^{2}}{{\sqrt[3]{3 x^{2}}}^{2}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{8}{\sqrt[3]{3 x^{2}}}$ by $\frac{{\sqrt[3]{3 x^{2}}}^{2}}{{\sqrt[3]{3 x^{2}}}^{2}}$ .

$\frac{8 {\sqrt[3]{3 x^{2}}}^{2}}{\sqrt[3]{3 x^{2}} {\sqrt[3]{3 x^{2}}}^{2}}$

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

$\frac{8 {\sqrt[3]{3 x^{2}}}^{2}}{{\sqrt[3]{3 x^{2}}}^{1} {\sqrt[3]{3 x^{2}}}^{2}}$

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

$\frac{8 {\sqrt[3]{3 x^{2}}}^{2}}{{\sqrt[3]{3 x^{2}}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{8 {\sqrt[3]{3 x^{2}}}^{2}}{{\sqrt[3]{3 x^{2}}}^{3}}$

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

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

$\frac{8 {\sqrt[3]{3 x^{2}}}^{2}}{{({(3 x^{2})}^{\frac{1}{3}})}^{3}}$

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

$\frac{8 {\sqrt[3]{3 x^{2}}}^{2}}{{(3 x^{2})}^{\frac{1}{3} \cdot 3}}$

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

$\frac{8 {\sqrt[3]{3 x^{2}}}^{2}}{{(3 x^{2})}^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{8 {\sqrt[3]{3 x^{2}}}^{2}}{{(3 x^{2})}^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{8 {\sqrt[3]{3 x^{2}}}^{2}}{{(3 x^{2})}^{1}}$

Simplify.

$\frac{8 {\sqrt[3]{3 x^{2}}}^{2}}{3 x^{2}}$

Step 3

Simplify the numerator.

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

$\frac{8 \sqrt[3]{{(3 x^{2})}^{2}}}{3 x^{2}}$

Apply the product rule to $3 x^{2}$ .

$\frac{8 \sqrt[3]{3^{2} {(x^{2})}^{2}}}{3 x^{2}}$

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

$\frac{8 \sqrt[3]{9 {(x^{2})}^{2}}}{3 x^{2}}$

Multiply the exponents in ${(x^{2})}^{2}$ .

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

$\frac{8 \sqrt[3]{9 x^{2 \cdot 2}}}{3 x^{2}}$

Multiply $2$ by $2$ .

$\frac{8 \sqrt[3]{9 x^{4}}}{3 x^{2}}$

Rewrite $9 x^{4}$ as $x^{3} \cdot (9 x)$ .

Factor out $x^{3}$ .

$\frac{8 \sqrt[3]{9 (x^{3} x)}}{3 x^{2}}$

Reorder $9$ and $x^{3}$ .

$\frac{8 \sqrt[3]{x^{3} \cdot 9 x}}{3 x^{2}}$

Add parentheses.

$\frac{8 \sqrt[3]{x^{3} \cdot (9 x)}}{3 x^{2}}$

Pull terms out from under the radical.

$\frac{8 x \sqrt[3]{9 x}}{3 x^{2}}$

Step 4

Cancel the common factor of $x$ and $x^{2}$ .

Factor $x$ out of $8 x \sqrt[3]{9 x}$ .

$\frac{x (8 \sqrt[3]{9 x})}{3 x^{2}}$

Cancel the common factors.

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

$\frac{x (8 \sqrt[3]{9 x})}{x (3 x)}$

Cancel the common factor.

$\frac{x (8 \sqrt[3]{9 x})}{x (3 x)}$

Rewrite the expression.

$\frac{8 \sqrt[3]{9 x}}{3 x}$

Simplify the Radical Expression 8/( cube root of 3x^2)

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