To simplify the expression $3\sqrt{18} + 3\sqrt{12} + 2\sqrt{27}$, let's break it down step by step.

Step 1: Simplify each square root term.

1. $\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$
2. $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$
3. $\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}$

Step 2: Substitute back into the expression.

$3\sqrt{18} = 3 \times 3\sqrt{2} = 9\sqrt{2}$ $3\sqrt{12} = 3 \times 2\sqrt{3} = 6\sqrt{3}$ $2\sqrt{27} = 2 \times 3\sqrt{3} = 6\sqrt{3}$

Step 3: Combine like terms.

The expression becomes: $9\sqrt{2} + 6\sqrt{3} + 6\sqrt{3}$

Combine the $\sqrt{3}$ terms: $9\sqrt{2} + 12\sqrt{3}$

Final Answer:

The simplified expression is $9\sqrt{2} + 12\sqrt{3}$.

Would you like more details on any of these steps? Here are some related questions:

1. How do you factor square roots when simplifying expressions?
2. Can all square roots be simplified similarly?
3. What are the properties of square roots that allow simplification?
4. How do you add and subtract like radical terms?
5. What are some practical applications of simplifying radicals?

Tip: When simplifying square roots, always look for the largest perfect square factor to make the process easier.