We can use the following properties of radicals:
[tex]\begin{gathered} \sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}\Rightarrow\text{ Product property} \\ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\Rightarrow\text{ Quotient property} \end{gathered}[/tex]Then, we have:
[tex]\begin{gathered} \text{ Apply the product property} \\ \sqrt[]{\frac{3x}{2}}\cdot\sqrt[]{\frac{x}{6}}=\sqrt[]{\frac{3x\cdot x}{2\cdot6}} \\ \sqrt[]{\frac{3x}{2}}\cdot\sqrt[]{\frac{x}{6}}=\sqrt[]{\frac{3x^2}{12}} \\ \text{ Simplify the expression inside the radical} \\ \sqrt[]{\frac{3x}{2}}\cdot\sqrt[]{\frac{x}{6}}=\sqrt[]{\frac{3x^2}{3\cdot4}} \\ \sqrt[]{\frac{3x}{2}}\cdot\sqrt[]{\frac{x}{6}}=\sqrt[]{\frac{x^2}{4}} \\ \text{ Apply the quotient property} \\ \sqrt[]{\frac{3x}{2}}\cdot\sqrt[]{\frac{x}{6}}=\frac{\sqrt[]{x^2}}{\sqrt[]{4}} \\ \sqrt[]{\frac{3x}{2}}\cdot\sqrt[]{\frac{x}{6}}=\frac{x}{2} \end{gathered}[/tex]Therefore, the choice that is equivalent to the given product when x > 0 is:
[tex]\frac{x}{2}[/tex]