We are given the expression:

$\frac{\sqrt{a} \cdot \sqrt{b}}{3\sqrt{ab}}$

Let's simplify this step by step.

Step 1: Simplify the numerator

The numerator is $\sqrt{a} \cdot \sqrt{b}$, which can be combined under one square root:

$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} = \sqrt{ab}$

Step 2: Rewrite the expression

Now the expression becomes:

$\frac{\sqrt{ab}}{3\sqrt{ab}}$

Step 3: Cancel out $\sqrt{ab}$

Since $\sqrt{ab}$ appears in both the numerator and denominator, we can cancel it out:

$\frac{\sqrt{ab}}{3\sqrt{ab}} = \frac{1}{3}$

Thus, the simplified expression is:

$\boxed{\frac{1}{3}}$

Would you like more details or have any questions?

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Here are 5 related questions to extend the concept:

1. How would you simplify $\frac{\sqrt{a} \cdot \sqrt{b} \cdot \sqrt{c}}{3\sqrt{abc}}$?
2. What happens if you have cube roots instead of square roots in a similar expression?
3. How would you approach $\frac{\sqrt{x}}{x\sqrt{y}}$?
4. What is the general rule for combining square roots like $\sqrt{a} \cdot \sqrt{b}$?
5. How do you rationalize a denominator that contains a square root?

Tip: When simplifying square roots, always check if the terms inside can be combined under a single root sign to simplify the expression further.