To simplify the expression $3\sqrt{12}$, we start by rewriting $\sqrt{12}$ as $2\sqrt{3}$. Then, we multiply this result by $3$, yielding $6\sqrt{3}$.

Let’s break down the steps more clearly. First, we need to simplify the square root of $12$. The number $12$ can be expressed in terms of its prime factors:

Utilizing the property of square roots that states $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can rewrite $\sqrt{12}$ as follows:

This expression can be simplified further:

since $\sqrt{2^2} = 2$.

Next, we multiply this result by the $3$ that is outside the square root:

Thus, the simplified form of $3\sqrt{12}$ is

This process involves recognizing the factors of the number inside the square root, simplifying the square root, and then performing the multiplication. Such methods are invaluable for simplifying expressions with square roots and facilitate easier calculations.