Given:
[tex]\sqrt[]{\frac{3x}{2}}\cdot\sqrt[]{\frac{x}{6}}[/tex]
To determine the equivalent to the product when x>0, we apply multiplication distribution property as shown below:
[tex](xy)^a=x^ay^a[/tex]
So,
[tex]\begin{gathered} \sqrt[]{\frac{3x}{2}}\cdot\sqrt[]{\frac{x}{6}} \\ =\frac{\sqrt[]{3}\sqrt[]{x}}{\sqrt[]{2}}\cdot\frac{\sqrt[]{x}}{\sqrt[]{6}} \\ \text{Simplify} \\ =\frac{\sqrt[]{3}\sqrt[]{x}\sqrt[]{x}}{\sqrt[]{2}\sqrt[]{6}} \\ =\frac{\sqrt[]{3}x}{\sqrt[]{2(6)}} \\ =\frac{\sqrt[]{3}x}{\sqrt[]{12}} \\ =\frac{\sqrt[]{3}x}{2\sqrt[]{3}} \\ =\frac{x}{2} \end{gathered}[/tex]
Therefore, the answer is:
C. x/2
