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We are tasked with rationalizing the denominator of the expression:

\[
\frac{3}{\sqrt{5} + \sqrt{6}}
\]

### Step 1: Multiply by the conjugate
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of \(\sqrt{5} + \sqrt{6}\), which is \(\sqrt{5} - \sqrt{6}\). This will eliminate the square roots in the denominator.

\[
\frac{3}{\sqrt{5} + \sqrt{6}} \times \frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} - \sqrt{6}} = \frac{3(\sqrt{5} - \sqrt{6})}{(\sqrt{5} + \sqrt{6})(\sqrt{5} - \sqrt{6})}
\]

### Step 2: Simplify the denominator
The denominator simplifies using the difference of squares formula:

\[
(\sqrt{5} + \sqrt{6})(\sqrt{5} - \sqrt{6}) = (\sqrt{5})^2 - (\sqrt{6})^2 = 5 - 6 = -1
\]

Thus, the expression becomes:

\[
\frac{3(\sqrt{5} - \sqrt{6})}{-1}
\]

### Step 3: Simplify the overall expression
Dividing by \(-1\) changes the signs in the numerator:

\[
-3(\sqrt{5} - \sqrt{6}) = -3\sqrt{5} + 3\sqrt{6}
\]

### Final Answer:
The simplified expression is:

\[
-3\sqrt{5} + 3\sqrt{6}
\]

Would you like further details on any part of this process?

Here are some related questions to further your understanding:
1. What are conjugates and why are they used for rationalizing denominators?
2. Can you provide a real-world example where rationalizing a denominator might be useful?
3. How would you rationalize a denominator with three terms (e.g., \(\sqrt{2} + \sqrt{3} + \sqrt{5}\))?
4. What is the difference of squares, and how is it applied in various math problems?
5. How would this method change if there were a cube root instead of a square root?

**Tip:** When rationalizing, always remember to multiply by the conjugate, as it effectively eliminates square roots in the denominator.

Rationalize the denominator of the expression: \(\frac{3}{\sqrt{5} + \sqrt{6}}\), then simplify your answer.

Learn how to rationalize the denominator of the expression 3 / (sqrt(5) + sqrt(6)) and simplify it. This solution involves key algebraic concepts like radicals, conjugates, and the difference of squares. Suitable for high school students in Grades 10-12.

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