You have the expression:

$\frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} - \sqrt{2}}$

To simplify this, we need to rationalize the denominator. We do this by multiplying both the numerator and denominator by the conjugate of the denominator, which is $\sqrt{3} + \sqrt{2}$.

Let's proceed:

$\frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} - \sqrt{2}} \times \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}} = \frac{(\sqrt{3} + \sqrt{5})(\sqrt{3} + \sqrt{2})}{(\sqrt{3})^2 - (\sqrt{2})^2}$

Now let's simplify the numerator and the denominator:

Denominator:

$(\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1$

Numerator:

We apply the distributive property (FOIL): $(\sqrt{3} + \sqrt{5})(\sqrt{3} + \sqrt{2}) = \sqrt{3} \times \sqrt{3} + \sqrt{3} \times \sqrt{2} + \sqrt{5} \times \sqrt{3} + \sqrt{5} \times \sqrt{2}$ $= 3 + \sqrt{6} + \sqrt{15} + \sqrt{10}$

Thus, the simplified expression is:

$3 + \sqrt{6} + \sqrt{15} + \sqrt{10}$

Therefore, the final simplified expression is:

$3 + \sqrt{6} + \sqrt{15} + \sqrt{10}$

Let me know if you would like any further explanation!

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Here are 5 related questions:

1. How do you rationalize a denominator in a radical expression?
2. What are the conjugates of complex or radical expressions?
3. Can you multiply two binomials containing square roots?
4. How do you simplify expressions involving multiple radicals?
5. How do you handle radical expressions in equations?

Tip: Rationalizing the denominator helps in simplifying expressions by eliminating radicals from the denominator, making them easier to interpret.