Simplify ( cube root of 7xy^2)/( cube root of 4x^2)

$\frac{\sqrt[3]{7 x y^{2}}}{\sqrt[3]{4 x^{2}}}$

Step 1

Combine $\sqrt[3]{7 x y^{2}}$ and $\sqrt[3]{4 x^{2}}$ into a single radical.

$\sqrt[3]{\frac{7 x y^{2}}{4 x^{2}}}$

Step 2

Reduce the expression $\frac{7 x y^{2}}{4 x^{2}}$ by cancelling the common factors.

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

$\sqrt[3]{\frac{x (7 y^{2})}{4 x^{2}}}$

Factor $x$ out of $4 x^{2}$ .

$\sqrt[3]{\frac{x (7 y^{2})}{x (4 x)}}$

Cancel the common factor.

$\sqrt[3]{\frac{x (7 y^{2})}{x (4 x)}}$

Rewrite the expression.

$\sqrt[3]{\frac{7 y^{2}}{4 x}}$

Step 3

Rewrite $\sqrt[3]{\frac{7 y^{2}}{4 x}}$ as $\frac{\sqrt[3]{7 y^{2}}}{\sqrt[3]{4 x}}$ .

$\frac{\sqrt[3]{7 y^{2}}}{\sqrt[3]{4 x}}$

Step 4

Multiply $\frac{\sqrt[3]{7 y^{2}}}{\sqrt[3]{4 x}}$ by $\frac{{\sqrt[3]{4 x}}^{2}}{{\sqrt[3]{4 x}}^{2}}$ .

$\frac{\sqrt[3]{7 y^{2}}}{\sqrt[3]{4 x}} \cdot \frac{{\sqrt[3]{4 x}}^{2}}{{\sqrt[3]{4 x}}^{2}}$

Step 5

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{7 y^{2}}}{\sqrt[3]{4 x}}$ by $\frac{{\sqrt[3]{4 x}}^{2}}{{\sqrt[3]{4 x}}^{2}}$ .

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{4 x}}^{2}}{\sqrt[3]{4 x} {\sqrt[3]{4 x}}^{2}}$

Raise $\sqrt[3]{4 x}$ to the power of $1$ .

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{4 x}}^{2}}{{\sqrt[3]{4 x}}^{1} {\sqrt[3]{4 x}}^{2}}$

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

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{4 x}}^{2}}{{\sqrt[3]{4 x}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{4 x}}^{2}}{{\sqrt[3]{4 x}}^{3}}$

Rewrite ${\sqrt[3]{4 x}}^{3}$ as $4 x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 x}$ as ${(4 x)}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{4 x}}^{2}}{{({(4 x)}^{\frac{1}{3}})}^{3}}$

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

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{4 x}}^{2}}{{(4 x)}^{\frac{1}{3} \cdot 3}}$

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

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{4 x}}^{2}}{{(4 x)}^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{4 x}}^{2}}{{(4 x)}^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{4 x}}^{2}}{{(4 x)}^{1}}$

Simplify.

$\frac{\sqrt[3]{7 y^{2}} {\sqrt[3]{4 x}}^{2}}{4 x}$

Step 6

Simplify the numerator.

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

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{{(4 x)}^{2}}}{4 x}$

Apply the product rule to $4 x$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{4^{2} x^{2}}}{4 x}$

Raise $4$ to the power of $2$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{16 x^{2}}}{4 x}$

Rewrite $16 x^{2}$ as $2^{3} \cdot (2 x^{2})$ .

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{8 (2) x^{2}}}{4 x}$

Rewrite $8$ as $2^{3}$ .

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{2^{3} \cdot 2 x^{2}}}{4 x}$

Add parentheses.

$\frac{\sqrt[3]{7 y^{2}} \sqrt[3]{2^{3} \cdot (2 x^{2})}}{4 x}$

Pull terms out from under the radical.

$\frac{\sqrt[3]{7 y^{2}} \cdot 2 \sqrt[3]{2 x^{2}}}{4 x}$

Combine exponents.

Combine using the product rule for radicals.

$\frac{2 \sqrt[3]{7 y^{2} (2 x^{2})}}{4 x}$

Multiply $2$ by $7$ .

$\frac{2 \sqrt[3]{14 y^{2} x^{2}}}{4 x}$

Step 7

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 \sqrt[3]{14 y^{2} x^{2}}$ .

$\frac{2 (\sqrt[3]{14 y^{2} x^{2}})}{4 x}$

Cancel the common factors.

Factor $2$ out of $4 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify ( cube root of 7xy^2)/( cube root of 4x^2)

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