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.