Simplify cube root of (12x^2)/(16y)

$\sqrt[3]{\frac{12 x^{2}}{16 y}}$

Step 1

Reduce the expression $\frac{12 x^{2}}{16 y}$ by cancelling the common factors.

Factor $4$ out of $12 x^{2}$ .

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

Factor $4$ out of $16 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 5

Simplify the numerator.

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

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

Apply the product rule to $4 y$ .

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

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

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

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

Factor $8$ out of $16$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Combine exponents.

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

Step 6

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

Factor $2$ out of $2 \sqrt[3]{6 x^{2} y^{2}}$ .

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

Cancel the common factors.

Factor $2$ out of $4 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify cube root of (12x^2)/(16y)

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