We can simplify $3\sqrt{12} + 4\sqrt{27}$ by breaking down the square roots into their prime factors:

Step 1: Simplify $\sqrt{12}$ and $\sqrt{27}$

• $\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$
• $\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}$

Step 2: Substitute back into the expression

$3\sqrt{12} + 4\sqrt{27} = 3(2\sqrt{3}) + 4(3\sqrt{3})$

Step 3: Simplify

$3(2\sqrt{3}) = 6\sqrt{3}, \quad 4(3\sqrt{3}) = 12\sqrt{3}$ Adding these: $6\sqrt{3} + 12\sqrt{3} = 18\sqrt{3}$

Final Answer:

$18\sqrt{3}$

Would you like a detailed breakdown of any step?

Related Questions:

1. How do you simplify expressions involving square roots?
2. Can we add or subtract square roots directly?
3. What are the properties of square roots that help in simplifications?
4. How do coefficients interact with square roots during addition or subtraction?
5. How to simplify expressions with different radicals?

Tip:

Always look for perfect squares in the radicand (number under the square root) to simplify square root expressions efficiently