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The problem in the image asks to express the number \( 4\sqrt{3} - \sqrt{12} \) in the form \( a\sqrt{b} \), where \( a \) and \( b \) are integers, and \( a \) is as large as possible. Let's solve this step by step.

### Step 1: Simplify \( \sqrt{12} \)
The number \( \sqrt{12} \) can be simplified by factoring:
\[
\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}
\]

### Step 2: Rewrite the original expression
Now, replace \( \sqrt{12} \) with \( 2\sqrt{3} \) in the original expression:
\[
4\sqrt{3} - \sqrt{12} = 4\sqrt{3} - 2\sqrt{3}
\]

### Step 3: Combine like terms
Both terms have \( \sqrt{3} \), so you can factor out \( \sqrt{3} \):
\[
4\sqrt{3} - 2\sqrt{3} = (4 - 2)\sqrt{3} = 2\sqrt{3}
\]

### Step 4: Identify \( a \) and \( b \)
The expression \( 2\sqrt{3} \) is already in the form \( a\sqrt{b} \), where \( a = 2 \) and \( b = 3 \).

### Final Answer:
- \( a = 2 \)
- \( b = 3 \)

Would you like further explanation on this, or have any questions? 

Here are five related questions to practice:
1. Simplify \( 5\sqrt{8} - 2\sqrt{18} \).
2. Express \( 3\sqrt{20} + \sqrt{45} \) in its simplest form.
3. Factor and simplify \( 6\sqrt{50} - 2\sqrt{8} \).
4. Combine \( 7\sqrt{2} + 5\sqrt{8} \) into a simplified form.
5. Rewrite \( \sqrt{72} + 2\sqrt{18} \) as \( a\sqrt{b} \) for integers \( a \) and \( b \).

**Tip:** Always try to simplify square roots by finding perfect squares inside the radicand to make combining terms easier!

The number 4√3 − √12 can be written in the form a√b where a and b are integers and a is as large as possible. What are the values of a and b?

This problem demonstrates how to simplify and combine terms under square roots to find integers a and b such that 4√3 − √12 can be expressed as a√b. This exercise is suitable for students from middle school to high school, strengthening understanding of root simplification and term combination.

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