To simplify the expression \(\frac{5}{\sqrt{3} - \sqrt{5}}\), we need to rationalize the denominator. Here are the steps:

1. **Multiply the numerator and the denominator by the conjugate of the denominator**: The conjugate of \(\sqrt{3} - \sqrt{5}\) is \(\sqrt{3} + \sqrt{5}\).

\[ \frac{5}{\sqrt{3} - \sqrt{5}} \cdot \frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}} = \frac{5(\sqrt{3} + \sqrt{5})}{(\sqrt{3} - \sqrt{5})(\sqrt{3} + \sqrt{5})} \]

2. **Simplify the denominator using the difference of squares**: 

\[ (\sqrt{3} - \sqrt{5})(\sqrt{3} + \sqrt{5}) = (\sqrt{3})^2 - (\sqrt{5})^2 = 3 - 5 = -2 \]

3. **Combine the results**:

\[ \frac{5(\sqrt{3} + \sqrt{5})}{-2} = -\frac{5(\sqrt{3} + \sqrt{5})}{2} = -\frac{5\sqrt{3}}{2} - \frac{5\sqrt{5}}{2} \]

So, the simplified form of \(\frac{5}{\sqrt{3} - \sqrt{5}}\) is:

\[ -\frac{5\sqrt{3}}{2} - \frac{5\sqrt{5}}{2} \]

Would you like a detailed explanation of any step? Here are 5 related questions you might find useful:

1. How do you rationalize the denominator of a fraction?
2. What is the difference of squares and how is it used in simplifying expressions?
3. Can you show examples of rationalizing different types of denominators?
4. How do you multiply and simplify expressions involving square roots?
5. What are conjugates and why are they used in rationalizing denominators?

**Tip:** When rationalizing the denominator, always multiply by the conjugate of the denominator to simplify the expression effectively.

Learn how to simplify the expression 5 divided by the square root of 3 minus the square root of 5. Follow a detailed step-by-step solution involving rationalization of denominators and simplification using the difference of squares. Suitable for advanced high school students.

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