To simplify the expression $\sqrt{\frac{2}{9}} \times \sqrt{\frac{1}{3}}$, we can follow these steps:

Step 1: Multiply the square roots

We can combine the square roots under one single square root: $\sqrt{\frac{2}{9}} \times \sqrt{\frac{1}{3}} = \sqrt{\frac{2}{9} \times \frac{1}{3}} = \sqrt{\frac{2 \times 1}{9 \times 3}} = \sqrt{\frac{2}{27}}.$

Step 2: Simplify the square root

Now, we simplify the square root of the fraction: $\sqrt{\frac{2}{27}} = \frac{\sqrt{2}}{\sqrt{27}}.$ Since $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$, the expression becomes: $\frac{\sqrt{2}}{3\sqrt{3}}.$

Step 3: Rationalize the denominator

To rationalize the denominator (get rid of the square root in the denominator), multiply both the numerator and denominator by $\sqrt{3}$: $\frac{\sqrt{2}}{3\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{2} \times \sqrt{3}}{3 \times 3} = \frac{\sqrt{6}}{9}.$

Final Answer:

$\frac{\sqrt{6}}{9}.$

Would you like any further details or explanations on this? Here's a tip for simplifying square roots and fractions:

Tip: When simplifying products of square roots, first combine them under a single square root, then simplify the fraction if possible.

Here are some follow-up questions for further exploration:

1. How do you simplify a square root of a product like $\sqrt{a \times b}$?
2. Can you explain why we rationalize denominators when simplifying square roots?
3. What are the steps for simplifying $\sqrt{\frac{a}{b}}$ if the numbers are large?
4. How does simplifying square roots help in solving equations involving square roots?
5. What other methods exist for simplifying expressions involving fractions and square roots?