The problem shown involves calculating the following expression:

$\frac{\sqrt{10} - \sqrt{2}}{\sqrt{5} - 1} + \frac{2 - \sqrt{2}}{\sqrt{2} - 1}$

Let's simplify it step by step:

Step 1: Rationalize the denominator of each term.

Rationalizing involves multiplying the numerator and denominator by the conjugate of the denominator.

For the first term: $\frac{\sqrt{10} - \sqrt{2}}{\sqrt{5} - 1}$ Multiply by $\frac{\sqrt{5} + 1}{\sqrt{5} + 1}$.

For the second term: $\frac{2 - \sqrt{2}}{\sqrt{2} - 1}$ Multiply by $\frac{\sqrt{2} + 1}{\sqrt{2} + 1}$.

Let's calculate each part.After rationalizing the denominators, we obtain the following:

1. First term: $\frac{(\sqrt{10} - \sqrt{2})(\sqrt{5} + 1)}{4} = \frac{(\sqrt{10} \cdot \sqrt{5} + \sqrt{10} - \sqrt{2} \cdot \sqrt{5} - \sqrt{2})}{4}$

2. Second term: $\frac{(2 - \sqrt{2})(\sqrt{2} + 1)}{1} = (2\sqrt{2} + 2 - \sqrt{2} - 2)$

The denominator of the first term simplifies to 4, while the second term simplifies directly. Now let's simplify both terms completely and sum them up.The simplified result of the given expression:

$\frac{\sqrt{10} - \sqrt{2}}{\sqrt{5} - 1} + \frac{2 - \sqrt{2}}{\sqrt{2} - 1}$

is:

$2\sqrt{2}$

Would you like a more detailed breakdown of any of the steps or calculations?

Here are 5 related questions to expand on this:

1. How do you rationalize denominators involving square roots?
2. Can you simplify $\frac{\sqrt{a} - \sqrt{b}}{\sqrt{c} - d}$ for arbitrary values of $a$, $b$, $c$, and $d$?
3. What are conjugates, and why are they useful in rationalizing expressions?
4. What happens if we don't rationalize the denominator and try simplifying directly?
5. How does rationalizing complex expressions differ from simpler square-root denominators?

Tip: When simplifying expressions with square roots, always check if the denominator can be rationalized by using conjugates, as it often leads to simpler and more manageable forms.