Simplify square root of (3x)/(2y^3)

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

Step 1

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

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

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

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

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

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

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

Step 2

Pull terms out from under the radical.

$\frac{1}{y} \sqrt{\frac{3 x}{2 y}}$

Step 3

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

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{2 y}}^{2}$ as $2 y$ .

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

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

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

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 6

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

$\frac{1}{y} \cdot \frac{\sqrt{6 x y}}{2 y}$

Step 7

Multiply $\frac{1}{y} \cdot \frac{\sqrt{6 x y}}{2 y}$ .

Multiply $\frac{1}{y}$ by $\frac{\sqrt{6 x y}}{2 y}$ .

$\frac{\sqrt{6 x y}}{y (2 y)}$

Raise $y$ to the power of $1$ .

$\frac{\sqrt{6 x y}}{2 (y^{1} y)}$

Raise $y$ to the power of $1$ .

$\frac{\sqrt{6 x y}}{2 (y^{1} y^{1})}$

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

$\frac{\sqrt{6 x y}}{2 y^{1 + 1}}$

Add $1$ and $1$ .

$\frac{\sqrt{6 x y}}{2 y^{2}}$

Simplify square root of (3x)/(2y^3)

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