The expression you provided is:

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

To simplify this expression, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{5} - \sqrt{3}$. This will help eliminate the square roots in the denominator.

Step-by-step simplification:

1. Multiply both the numerator and denominator by $\sqrt{5} - \sqrt{3}$:

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

2. Expand the numerator and the denominator separately:

   • Numerator: $(\sqrt{3} - \sqrt{5})(\sqrt{5} - \sqrt{3})$ $= \sqrt{3} \cdot \sqrt{5} - \sqrt{3} \cdot \sqrt{3} - \sqrt{5} \cdot \sqrt{5} + \sqrt{5} \cdot \sqrt{3}$ $= \sqrt{15} - 3 - 5 + \sqrt{15}$ $= 2\sqrt{15} - 8$

   • Denominator: $(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})$ $= \sqrt{5}^2 - \sqrt{3}^2$ $= 5 - 3 = 2$

3. Now, the expression simplifies to: $\frac{2\sqrt{15} - 8}{2}$

4. Finally, divide both terms in the numerator by 2: $\sqrt{15} - 4$

Simplified result:

$\sqrt{15} - 4$

Would you like more details on any step?

Here are some related questions to further explore:

1. How do you rationalize complex fractions?
2. What are the properties of conjugates used in simplification?
3. How do you simplify expressions involving square roots?
4. Can you apply similar techniques to more complex radical expressions?
5. What is the geometric interpretation of square roots?

Tip: Always consider multiplying by the conjugate when simplifying fractions with square roots to eliminate radicals in the denominator.