Simplify cube root of (3x^2)/(16y)

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

Step 1

Rewrite $\frac{3 x^{2}}{16 y}$ as ${(\frac{1}{2})}^{3} \frac{3 x^{2}}{2 y}$ .

Factor the perfect power $1^{3}$ out of $3 x^{2}$ .

$\sqrt[3]{\frac{1^{3} (3 x^{2})}{16 y}}$

Factor the perfect power $2^{3}$ out of $16 y$ .

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

Rearrange the fraction $\frac{1^{3} (3 x^{2})}{2^{3} \cdot (2 y)}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

Combine.

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

Step 5

Multiply $\sqrt[3]{3 x^{2}}$ by $1$ .

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

Step 6

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

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

Step 7

Combine and simplify the denominator.

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

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

Move $\sqrt[3]{2 y}$ .

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 8

Simplify the numerator.

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

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

Apply the product rule to $2 y$ .

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

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

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

Step 9

Multiply $2$ by $2$ .

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

Step 10

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $4$ by $3$ .

$\frac{\sqrt[3]{12 x^{2} y^{2}}}{4 y}$

Simplify cube root of (3x^2)/(16y)

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