Sure, let's simplify the expression step by step:
Given the expression:
[tex]\[
\sqrt{\frac{2x}{6}} \cdot \sqrt{\frac{x}{3}}
\][/tex]
First, let's simplify the fractions inside the square roots separately.
For the first square root:
[tex]\[
\sqrt{\frac{2x}{6}}
\][/tex]
We can simplify \(\frac{2x}{6}\):
[tex]\[
\frac{2x}{6} = \frac{2}{6} \cdot x = \frac{1}{3} \cdot x = \frac{x}{3}
\][/tex]
So the expression inside the first square root simplifies to:
[tex]\[
\sqrt{\frac{x}{3}}
\][/tex]
Next, let's look at the second square root:
[tex]\[
\sqrt{\frac{x}{3}}
\][/tex]
Since both expressions inside the square roots have now become the same, the given expression is:
[tex]\[
\sqrt{\frac{x}{3}} \cdot \sqrt{\frac{x}{3}}
\][/tex]
We can use the property of square roots that \(\sqrt{a} \cdot \sqrt{a} = a\):
[tex]\[
\sqrt{\frac{x}{3}} \cdot \sqrt{\frac{x}{3}} = \frac{x}{3}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\boxed{\frac{x}{3}}
\][/tex]
