To simplify the expression $\frac{7}{\sqrt{6}} - \sqrt{5}$, follow these steps:

Step 1: Rationalize the first term

To simplify $\frac{7}{\sqrt{6}}$, multiply both the numerator and denominator by $\sqrt{6}$:

$\frac{7}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{7\sqrt{6}}{6}$

Step 2: Combine the terms

Now the expression becomes:

$\frac{7\sqrt{6}}{6} - \sqrt{5}$

This is the simplified form. Since $\frac{7\sqrt{6}}{6}$ and $\sqrt{5}$ are unlike terms, they cannot be further simplified.

Final Answer:

$\frac{7\sqrt{6}}{6} - \sqrt{5}$

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How do you rationalize a denominator with square roots?
2. What are the rules for simplifying expressions involving square roots?
3. Can square roots of different numbers be combined or simplified together?
4. What is the process to simplify the expression $\frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$?
5. How do you simplify expressions like $\frac{a\sqrt{b}}{c\sqrt{d}}$?

Tip: When simplifying expressions with square roots, rationalizing the denominator is a useful technique to eliminate roots from the denominator.