Let's simplify the expression:

$\frac{(5 - 2\sqrt{5})}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}$

First, note that multiplying by $\frac{\sqrt{5}}{\sqrt{5}}$ is equivalent to multiplying by 1, so it doesn't change the value of the expression. However, this multiplication simplifies the expression by rationalizing the denominator.

We can distribute $\frac{\sqrt{5}}{\sqrt{5}}$ over the numerator:

$\frac{(5 - 2\sqrt{5})}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{5\sqrt{5} - 2\sqrt{5} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}}$

Simplify each part:

• $5 \times \sqrt{5} = 5\sqrt{5}$
• $\sqrt{5} \times \sqrt{5} = 5$
• $-2\sqrt{5} \times \sqrt{5} = -2 \times 5 = -10$
• $\sqrt{5} \times \sqrt{5} = 5$

So the expression simplifies to:

$\frac{5\sqrt{5} - 10}{5}$

We can separate the terms in the numerator:

$\frac{5\sqrt{5}}{5} - \frac{10}{5}$

Simplify each term:

• $\frac{5\sqrt{5}}{5} = \sqrt{5}$
• $\frac{10}{5} = 2$

So, the simplified result is:

$\sqrt{5} - 2$

Thus, the final answer is:

$\boxed{\sqrt{5} - 2}$

Would you like more details on any step? Here are five related questions to think about:

1. How would you rationalize the denominator of $\frac{1}{\sqrt{7}}$?
2. What is the result of $\frac{3 + \sqrt{2}}{\sqrt{2}}$?
3. Can you simplify $\frac{\sqrt{12} - \sqrt{3}}{\sqrt{3}}$?
4. How do you simplify $\frac{7 - \sqrt{7}}{\sqrt{7}}$?
5. What is the process for rationalizing complex denominators like $\frac{2}{\sqrt{3} - 1}$?

Tip: When simplifying expressions with square roots, rationalizing the denominator can make the expression easier to work with.