Simplify ( square root of x)/( square root of 6y^3)

$\frac{\sqrt{x}}{\sqrt{6 y^{3}}}$

Step 1

Simplify the denominator.

Rewrite $6 y^{3}$ as $y^{2} \cdot (6 y)$ .

Factor out $y^{2}$ .

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

Reorder $6$ and $y^{2}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

Move $\sqrt{6 y}$ .

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

Raise $\sqrt{6 y}$ to the power of $1$ .

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

Raise $\sqrt{6 y}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{6 y}}^{2}$ as $6 y$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 y}$ as ${(6 y)}^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 4

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

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

Multiply $y$ by $y$ .

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

Step 5

Combine using the product rule for radicals.

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

Step 6

Simplify the expression.

Move $6$ to the left of $y^{2}$ .

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

Reorder factors in $\frac{\sqrt{x \cdot 6 y}}{6 y^{2}}$ .

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

Simplify ( square root of x)/( square root of 6y^3)

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