Simplify ( cube root of 14)/( cube root of 7x^2y)

$\frac{\sqrt[3]{14}}{\sqrt[3]{7 x^{2} y}}$

Step 1

Combine $\sqrt[3]{14}$ and $\sqrt[3]{7 x^{2} y}$ into a single radical.

$\sqrt[3]{\frac{14}{7 x^{2} y}}$

Step 2

Reduce the expression $\frac{14}{7 x^{2} y}$ by cancelling the common factors.

Factor $7$ out of $14$ .

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 6

Simplify the numerator.

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

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

Rewrite $x^{4} y^{2}$ as $x^{3} (x y^{2})$ .

Factor out $x^{3}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

Combine using the product rule for radicals.

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

Step 7

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

Factor $x$ out of $x \sqrt[3]{2 x y^{2}}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify ( cube root of 14)/( cube root of 7x^2y)

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