Let's simplify the expression \( 3\sqrt{12} - 2\sqrt{75} \).

### Step 1: Simplify the square roots
First, we simplify each square root:

- \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \)
- \( \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \)

### Step 2: Substitute the simplified square roots into the expression
Now, substitute these into the original expression:

\[
3\sqrt{12} - 2\sqrt{75} = 3(2\sqrt{3}) - 2(5\sqrt{3})
\]

### Step 3: Multiply and combine like terms
Next, perform the multiplication:

\[
6\sqrt{3} - 10\sqrt{3}
\]

Now, combine the like terms:

\[
(6 - 10)\sqrt{3} = -4\sqrt{3}
\]

### Final Answer
The simplified expression is:

\[
-4\sqrt{3}
\]

Would you like more details or have any questions? Here are some related questions:

1. How do you simplify more complex square roots?
2. What are the properties of square roots that allow for simplification?
3. How do you handle expressions with square roots that are not perfect squares?
4. How do you add or subtract expressions involving square roots?
5. Can you always simplify square roots in the same way?

**Tip:** Always look for factors that are perfect squares when simplifying square roots to make the process easier.

This detailed guide simplifies the expression 3\sqrt{12} - 2\sqrt{75} using algebraic techniques. Learn how to handle square roots and combine like terms, making it perfect for students in Grades 8-10.

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