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The problem in the image is asking for the simplified form of the expression:

\[
\frac{4 \sqrt{3}}{\sqrt{7} + 2\sqrt{12}}
\]

### Step 1: Simplify the denominator
First, simplify \( 2\sqrt{12} \). Since \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \), we get:
\[
2\sqrt{12} = 2 \times 2\sqrt{3} = 4\sqrt{3}
\]

Now, the denominator becomes:
\[
\sqrt{7} + 4\sqrt{3}
\]

### Step 2: Rationalize the denominator
To simplify the expression, we need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( \sqrt{7} + 4\sqrt{3} \) is \( \sqrt{7} - 4\sqrt{3} \).

Multiply the numerator and the denominator by \( \sqrt{7} - 4\sqrt{3} \):

\[
\frac{4\sqrt{3}}{\sqrt{7} + 4\sqrt{3}} \times \frac{\sqrt{7} - 4\sqrt{3}}{\sqrt{7} - 4\sqrt{3}} = \frac{4\sqrt{3}(\sqrt{7} - 4\sqrt{3})}{(\sqrt{7} + 4\sqrt{3})(\sqrt{7} - 4\sqrt{3})}
\]

### Step 3: Simplify the denominator
Use the difference of squares formula:
\[
(a + b)(a - b) = a^2 - b^2
\]
Here, \( a = \sqrt{7} \) and \( b = 4\sqrt{3} \), so the denominator becomes:
\[
(\sqrt{7})^2 - (4\sqrt{3})^2 = 7 - 16 \times 3 = 7 - 48 = -41
\]

### Step 4: Simplify the numerator
Now expand the numerator:
\[
4\sqrt{3} \times (\sqrt{7} - 4\sqrt{3}) = 4\sqrt{3}\sqrt{7} - 4\sqrt{3} \times 4\sqrt{3}
\]
\[
= 4\sqrt{21} - 16 \times 3 = 4\sqrt{21} - 48
\]

### Step 5: Final simplified expression
The final expression is:
\[
\frac{4\sqrt{21} - 48}{-41}
\]
This can be rewritten as:
\[
\boxed{\frac{48 - 4\sqrt{21}}{41}}
\]

Would you like further details or explanations?

Here are five related questions for further practice:
1. How do you rationalize a denominator involving square roots?
2. Can you simplify \( \frac{5 \sqrt{2}}{\sqrt{3} + \sqrt{5}} \) by using the same method?
3. What is the result of \( \sqrt{12} \) simplified?
4. Why does rationalizing the denominator work for simplifying expressions?
5. How would you rationalize a denominator with more than two terms?

**Tip:** Always check if square roots in expressions can be simplified before attempting further steps.

Bentuk sederhana dari \( \frac{4 \sqrt{3}}{\sqrt{7} + 2\sqrt{12}} \)

Learn how to simplify the radical expression \( \frac{4 \sqrt{3}}{\sqrt{7} + 2\sqrt{12}} \) by rationalizing the denominator using the difference of squares. This problem covers key concepts like simplifying square roots and rationalizing the denominator, suitable for students in Grades 9-11.

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