To rationalize the denominator of the expression $\frac{1}{\sqrt{5} + \sqrt{6} - \sqrt{2}}$, we follow these steps:

Step 1: Multiply by the conjugate

The first step in rationalizing a denominator with square roots is to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{5} + \sqrt{6} - \sqrt{2}$ is $\sqrt{5} + \sqrt{6} + \sqrt{2}$.

So we multiply:

$\frac{1}{\sqrt{5} + \sqrt{6} - \sqrt{2}} \times \frac{\sqrt{5} + \sqrt{6} + \sqrt{2}}{\sqrt{5} + \sqrt{6} + \sqrt{2}}$

Step 2: Expand the denominator

The denominator becomes the difference of squares:

$(\sqrt{5} + \sqrt{6} - \sqrt{2})(\sqrt{5} + \sqrt{6} + \sqrt{2})$

This simplifies to:

$(\sqrt{5} + \sqrt{6})^2 - (\sqrt{2})^2$

First, expand $(\sqrt{5} + \sqrt{6})^2$:

$(\sqrt{5} + \sqrt{6})^2 = 5 + 6 + 2\sqrt{30} = 11 + 2\sqrt{30}$

Now subtract $(\sqrt{2})^2 = 2$:

$(11 + 2\sqrt{30}) - 2 = 9 + 2\sqrt{30}$

Thus, the denominator becomes $9 + 2\sqrt{30}$.

Step 3: Write the full expression

The full expression now looks like this:

$\frac{\sqrt{5} + \sqrt{6} + \sqrt{2}}{9 + 2\sqrt{30}}$

This is the rationalized form of the expression. Since the denominator still contains a square root, you could leave it in this form or further simplify if necessary.

Would you like more details on this process, or do you have any questions?

Related Questions:

1. How do you rationalize a denominator with two square roots?
2. What is the conjugate, and how is it used in rationalization?
3. Can you rationalize the denominator if there are three square roots?
4. Why is rationalizing useful in simplifying expressions?
5. What happens if we multiply the numerator and denominator by something other than the conjugate?

Tip:

Always remember, when dealing with irrational denominators, the key is to use the conjugate to eliminate square roots.