Simplify ( square root of 9xy)/( square root of 2x)

$\frac{\sqrt{9 x y}}{\sqrt{2 x}}$

Combine $\sqrt{9 x y}$ and $\sqrt{2 x}$ into a single radical.

$\sqrt{\frac{9 x y}{2 x}}$

Reduce the expression $\frac{9 x y}{2 x}$ by cancelling the common factors.

Cancel the common factor.

$\sqrt{\frac{9 x y}{2 x}}$

Rewrite the expression.

$\sqrt{\frac{9 y}{2}}$

Rewrite $\sqrt{\frac{9 y}{2}}$ as $\frac{\sqrt{9 y}}{\sqrt{2}}$ .

$\frac{\sqrt{9 y}}{\sqrt{2}}$

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical.

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Combine using the product rule for radicals.

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