The expression involves nested radicals and algebraic terms. Let's break it down step by step.

We have:

$\frac{\left(\frac{\sqrt{2} - 1}{\sqrt{2} + 1} + \frac{\sqrt{2} + 1}{\sqrt{2} - 1}\right)}{\left(\frac{\sqrt{3} - 1}{\sqrt{3} + 1} + \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\right)} = ?$

Step 1: Simplifying the Numerator and Denominator

The structure of both the numerator and denominator consists of two terms:

• $\frac{\sqrt{a} - 1}{\sqrt{a} + 1} + \frac{\sqrt{a} + 1}{\sqrt{a} - 1}$.

We simplify these terms using common denominators and algebraic manipulation. Let's calculate the value and proceed step-by-step.

I'll compute the exact value of this expression.The value of the given expression simplifies to:

$\frac{3}{2}$

Would you like further explanation on any of the steps or simplifications?

Here are some related questions to explore:

1. How can you simplify fractions with nested radicals like $\frac{\sqrt{2} - 1}{\sqrt{2} + 1}$?
2. What is the importance of rationalizing the denominator in such expressions?
3. How can the difference of squares help simplify these types of terms?
4. What other methods can be used to solve expressions involving square roots?
5. What are the properties of radicals that help in simplifying complex expressions?

Tip: Rationalizing denominators in complex fractions often reveals simple patterns that help with quick simplifications.