Simplify cube root of (3x)/(7x^2)

$\sqrt[3]{\frac{3 x}{7 x^{2}}}$

Step 1

Reduce the expression $\frac{3 x}{7 x^{2}}$ by cancelling the common factors.

Factor $x$ out of $3 x$ .

$\sqrt[3]{\frac{x \cdot 3}{7 x^{2}}}$

Factor $x$ out of $7 x^{2}$ .

$\sqrt[3]{\frac{x \cdot 3}{x (7 x)}}$

Cancel the common factor.

$\sqrt[3]{\frac{x \cdot 3}{x (7 x)}}$

Rewrite the expression.

$\sqrt[3]{\frac{3}{7 x}}$

Step 2

Rewrite $\sqrt[3]{\frac{3}{7 x}}$ as $\frac{\sqrt[3]{3}}{\sqrt[3]{7 x}}$ .

$\frac{\sqrt[3]{3}}{\sqrt[3]{7 x}}$

Step 3

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{7 x}}$ by $\frac{{\sqrt[3]{7 x}}^{2}}{{\sqrt[3]{7 x}}^{2}}$ .

$\frac{\sqrt[3]{3}}{\sqrt[3]{7 x}} \cdot \frac{{\sqrt[3]{7 x}}^{2}}{{\sqrt[3]{7 x}}^{2}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{7 x}}$ by $\frac{{\sqrt[3]{7 x}}^{2}}{{\sqrt[3]{7 x}}^{2}}$ .

$\frac{\sqrt[3]{3} {\sqrt[3]{7 x}}^{2}}{\sqrt[3]{7 x} {\sqrt[3]{7 x}}^{2}}$

Raise $\sqrt[3]{7 x}$ to the power of $1$ .

$\frac{\sqrt[3]{3} {\sqrt[3]{7 x}}^{2}}{{\sqrt[3]{7 x}}^{1} {\sqrt[3]{7 x}}^{2}}$

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

$\frac{\sqrt[3]{3} {\sqrt[3]{7 x}}^{2}}{{\sqrt[3]{7 x}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{\sqrt[3]{3} {\sqrt[3]{7 x}}^{2}}{{\sqrt[3]{7 x}}^{3}}$

Rewrite ${\sqrt[3]{7 x}}^{3}$ as $7 x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7 x}$ as ${(7 x)}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{3} {\sqrt[3]{7 x}}^{2}}{{({(7 x)}^{\frac{1}{3}})}^{3}}$

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

$\frac{\sqrt[3]{3} {\sqrt[3]{7 x}}^{2}}{{(7 x)}^{\frac{1}{3} \cdot 3}}$

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

$\frac{\sqrt[3]{3} {\sqrt[3]{7 x}}^{2}}{{(7 x)}^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\sqrt[3]{3} {\sqrt[3]{7 x}}^{2}}{{(7 x)}^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{\sqrt[3]{3} {\sqrt[3]{7 x}}^{2}}{{(7 x)}^{1}}$

Simplify.

$\frac{\sqrt[3]{3} {\sqrt[3]{7 x}}^{2}}{7 x}$

Step 5

Simplify the numerator.

Rewrite ${\sqrt[3]{7 x}}^{2}$ as ${({(7 x)}^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{3} \sqrt[3]{{(7 x)}^{2}}}{7 x}$

Apply the product rule to $7 x$ .

$\frac{\sqrt[3]{3} \sqrt[3]{7^{2} x^{2}}}{7 x}$

Raise $7$ to the power of $2$ .

$\frac{\sqrt[3]{3} \sqrt[3]{49 x^{2}}}{7 x}$

Step 6

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{\sqrt[3]{3 (49 x^{2})}}{7 x}$

Multiply $3$ by $49$ .

$\frac{\sqrt[3]{147 x^{2}}}{7 x}$

Simplify cube root of (3x)/(7x^2)

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