Let's analyze the problem step by step. The equation given in the problem is:

$\sqrt[3]{\sqrt{2} \times 2 \times \sqrt{3}} = \frac{3\sqrt{2}\sqrt{3}}{6\sqrt{2}}.$

We need to find the value of $\sqrt[3]{\frac{1}{3}x}$.

Step 1: Simplify the left-hand side (LHS)

The LHS is $\sqrt[3]{\sqrt{2} \times 2 \times \sqrt{3}}$.

First, let's simplify the expression inside the cube root:

$\sqrt{2} \times 2 \times \sqrt{3} = 2 \times \sqrt{2 \times 3} = 2\sqrt{6}.$

Thus, the LHS becomes:

$\sqrt[3]{2\sqrt{6}}.$

Step 2: Simplify the right-hand side (RHS)

The RHS is $\frac{3\sqrt{2}\sqrt{3}}{6\sqrt{2}}$.

We can cancel out $\sqrt{2}$ from both the numerator and denominator, resulting in:

$\frac{3\sqrt{3}}{6}.$

Now simplify:

$\frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2}.$

Thus, we have:

$\sqrt[3]{2\sqrt{6}} = \frac{\sqrt{3}}{2}.$

Step 3: Find the value of $\sqrt[3]{\frac{1}{3} x}$

From this simplified equation, we see that:

$x = 2\sqrt{6}.$

Thus, $\frac{1}{3}x = \frac{2\sqrt{6}}{3}$, and we need to find:

$\sqrt[3]{\frac{2\sqrt{6}}{3}}.$

Step 4: Compare with the options

Among the given options, the closest match to $\sqrt[3]{\frac{2\sqrt{6}}{3}}$ is option D, which is $\sqrt[3]{\frac{\sqrt{2}}{3}}$.

Final Answer: Option D $\boxed{\sqrt[3]{\frac{\sqrt{2}}{3}}}$.

Do you need more details, or is there anything else you'd like to explore? Here are 5 related questions to expand your understanding:

1. How do cube roots affect the simplification of radical expressions?
2. What are some tips for simplifying expressions involving multiple radicals?
3. How do you identify the best approach when simplifying complex radical equations?
4. How would the problem change if it used a square root instead of a cube root?
5. Can this method be applied to equations involving higher-order roots, like fourth or fifth roots?

Tip: Always simplify both sides of the equation as much as possible before solving for unknown variables.