Simplify ( cube root of 48x^3y^2)/( cube root of 6x^4y)

$\frac{\sqrt[3]{48 x^{3} y^{2}}}{\sqrt[3]{6 x^{4} y}}$

Step 1

Combine $\sqrt[3]{48 x^{3} y^{2}}$ and $\sqrt[3]{6 x^{4} y}$ into a single radical.

$\sqrt[3]{\frac{48 x^{3} y^{2}}{6 x^{4} y}}$

Step 2

Reduce the expression $\frac{48 x^{3} y^{2}}{6 x^{4} y}$ by cancelling the common factors.

Factor $6$ out of $48 x^{3} y^{2}$ .

$\sqrt[3]{\frac{6 (8 x^{3} y^{2})}{6 x^{4} y}}$

Factor $6$ out of $6 x^{4} y$ .

$\sqrt[3]{\frac{6 (8 x^{3} y^{2})}{6 (x^{4} y)}}$

Cancel the common factor.

$\sqrt[3]{\frac{6 (8 x^{3} y^{2})}{6 (x^{4} y)}}$

Rewrite the expression.

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

Step 3

Cancel the common factor of $x^{3}$ and $x^{4}$ .

Factor $x^{3}$ out of $8 x^{3} y^{2}$ .

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

Cancel the common factors.

Factor $x^{3}$ out of $x^{4} y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

Factor $y$ out of $8 y^{2}$ .

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

Cancel the common factors.

Factor $y$ out of $x y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Rewrite $\sqrt[3]{\frac{8 y}{x}}$ as $\frac{\sqrt[3]{8 y}}{\sqrt[3]{x}}$ .

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

Step 6

Simplify the numerator.

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

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

Pull terms out from under the radical.

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 9

Simplify the numerator.

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

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

Combine using the product rule for radicals.

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

Simplify ( cube root of 48x^3y^2)/( cube root of 6x^4y)

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