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To rationalize the denominator of the expression \(\frac{3}{\sqrt{3} - 2}\), follow these steps:

### Step 1: Multiply by the conjugate
To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{3} - 2\) is \(\sqrt{3} + 2\). 

\[
\frac{3}{\sqrt{3} - 2} \times \frac{\sqrt{3} + 2}{\sqrt{3} + 2} = \frac{3(\sqrt{3} + 2)}{(\sqrt{3} - 2)(\sqrt{3} + 2)}
\]

### Step 2: Simplify the denominator
The denominator is a difference of squares:

\[
(\sqrt{3})^2 - 2^2 = 3 - 4 = -1
\]

### Step 3: Simplify the numerator
Distribute the \(3\) across the terms in the numerator:

\[
3(\sqrt{3} + 2) = 3\sqrt{3} + 6
\]

### Step 4: Combine the results
Now, put it all together:

\[
\frac{3\sqrt{3} + 6}{-1} = -3\sqrt{3} - 6
\]

So, the simplified expression is \(\boxed{-3\sqrt{3} - 6}\).

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

1. What is the importance of using the conjugate when rationalizing denominators?
2. How do you apply the difference of squares formula in different contexts?
3. Can you rationalize the denominator of \( \frac{5}{\sqrt{2} - 1} \)?
4. What are the properties of conjugates and how do they help in simplifications?
5. How would the process change if the denominator was \( \sqrt{3} + 2 \) instead?

**Tip:** Rationalizing denominators helps in simplifying expressions and making them easier to work with, especially when dealing with square roots!

Learn how to rationalize the denominator and simplify expressions with a detailed step-by-step guide. This problem involves using the conjugate method and understanding the difference of squares. Suitable for high school students and anyone interested in simplifying algebraic expressions.

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