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The question asks to rationalize the denominator of the expression \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\).

### Solution:
To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{5} - \sqrt{3}\) is \(\sqrt{5} + \sqrt{3}\).

So, we multiply the fraction by \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}\):

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

### Step 1: Simplify the numerator
\[
(\sqrt{5} + \sqrt{3})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15}
\]

### Step 2: Simplify the denominator
\[
(\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2
\]

### Step 3: Combine the results
\[
\frac{8 + 2\sqrt{15}}{2}
\]

### Step 4: Simplify the fraction
\[
= \frac{8}{2} + \frac{2\sqrt{15}}{2} = 4 + \sqrt{15}
\]

So, the rationalized expression is \(4 + \sqrt{15}\).

Would you like any more details or have any other questions?

Here are five related questions for further practice:
1. Rationalize the denominator of \(\frac{\sqrt{7} + \sqrt{2}}{\sqrt{7} - \sqrt{2}}\).
2. Simplify \(\frac{3\sqrt{2} + 2\sqrt{3}}{2\sqrt{2} - \sqrt{3}}\).
3. Rationalize the denominator and simplify \(\frac{1 + \sqrt{5}}{\sqrt{5} - 1}\).
4. Find the value of \(\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\).
5. Simplify \(\frac{5\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}\).

**Tip:** When rationalizing denominators, always remember to multiply by the conjugate of the denominator, which changes the sign between the two terms.

Rationalize the denominator of \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\).

Learn how to rationalize the denominator of \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}\). This solution covers rationalization, simplifying radicals, and applying the difference of squares theorem. Suitable for students in Grades 9-10.

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