To simplify the expression $\frac{9\sqrt{12}}{3\sqrt{6}}$:

Step 1: Simplify the square roots individually.

1. Numerator: $9\sqrt{12}$

   • Break it down: $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$
   • So, $9\sqrt{12} = 9 \times 2\sqrt{3} = 18\sqrt{3}$

2. Denominator: $3\sqrt{6}$

   • This is already in its simplest form.

Step 2: Simplify the expression.

Now, divide the simplified numerator by the simplified denominator:

$\frac{18\sqrt{3}}{3\sqrt{6}}$

Step 3: Simplify the coefficient and the square roots.

1. Coefficient:

   • $\frac{18}{3} = 6$

2. Square Roots:

   • $\frac{\sqrt{3}}{\sqrt{6}} = \frac{\sqrt{3}}{\sqrt{2 \times 3}} = \frac{\sqrt{3}}{\sqrt{3}\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ (by rationalizing the denominator).

Final Expression:

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

Final Answer:

The simplified expression is $3\sqrt{2}$.

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Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you rationalize a denominator with square roots?
2. What are some techniques for simplifying more complex radicals?
3. How do you handle cube roots or higher in expressions?
4. What is the difference between simplifying an expression and solving an equation?
5. How does simplifying help in solving algebraic equations?

Tip: When dealing with radicals, always check if you can simplify them before performing any operations, as it makes the process easier and the result cleaner.