Questions: Simplify. Assume all variables are positive. [ sqrtfrac9 x^32 y^2= ]

Simplify the expression \(\sqrt{\frac{9 x^{3}}{2 y^{2}}}\).

Separate the square root into numerator and denominator.

\[ \sqrt{\frac{9 x^{3}}{2 y^{2}}} = \frac{\sqrt{9 x^{3}}}{\sqrt{2 y^{2}}} \]

Simplify the square roots in the numerator and denominator.

\[ \frac{\sqrt{9 x^{3}}}{\sqrt{2 y^{2}}} = \frac{\sqrt{9} \cdot \sqrt{x^{3}}}{\sqrt{2} \cdot \sqrt{y^{2}}} \]

Evaluate the square roots of constants and variables.

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

Express \(x^{3/2}\) as \(x \cdot \sqrt{x}\).

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

Rationalize the denominator by multiplying numerator and denominator by \(\sqrt{2}\).

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

Simplify the expression.

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

The simplified form of the expression is \(\boxed{\frac{3 x \sqrt{2x}}{2 y}}\).

The simplified form of the expression is \(\boxed{\frac{3 x \sqrt{2x}}{2 y}}\).