To find the equivalent expression for the product [tex]\(\sqrt{\frac{3x}{2}} \cdot \sqrt{\frac{x}{6}}\)[/tex], we will simplify the expression step-by-step.
First, we use the property of square roots that states:
[tex]\[
\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}
\][/tex]
Applying this property to the given expression:
[tex]\[
\sqrt{\frac{3x}{2}} \cdot \sqrt{\frac{x}{6}} = \sqrt{\left( \frac{3x}{2} \right) \cdot \left( \frac{x}{6} \right)}
\][/tex]
Next, we multiply the fractions inside the square root:
[tex]\[
\left( \frac{3x}{2} \right) \cdot \left( \frac{x}{6} \right) = \frac{3x \cdot x}{2 \cdot 6} = \frac{3x^2}{12}
\][/tex]
We can simplify the fraction:
[tex]\[
\frac{3x^2}{12} = \frac{x^2}{4}
\][/tex]
Now, we take the square root of the simplified fraction:
[tex]\[
\sqrt{\frac{x^2}{4}} = \frac{\sqrt{x^2}}{\sqrt{4}} = \frac{x}{2}
\][/tex]
Thus, the equivalent expression for [tex]\(\sqrt{\frac{3x}{2}} \cdot \sqrt{\frac{x}{6}}\)[/tex] is:
[tex]\[
\frac{x}{2}
\][/tex]
So, the correct choice is:
[tex]\[
\boxed{\text{B}}
\][/tex]
