The problem asks us to simplify the expression $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ by rationalizing the denominator.

1. Problem Description

The problem asks us to simplify the expression $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ by rationalizing the denominator.

2. Solution Steps

To rationalize the denominator, we need to 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 have

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

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

Using the formula $(a+b)^2 = a^2 + 2ab + b^2$ and $(a-b)(a+b) = a^2 - b^2$, we have:

$(\sqrt{5} + \sqrt{3})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15}$

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

Therefore,

$\frac{(\sqrt{5} + \sqrt{3})^2}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})} = \frac{8 + 2\sqrt{15}}{2} = \frac{2(4 + \sqrt{15})}{2} = 4 + \sqrt{15}$

3. Final Answer

$4 + \sqrt{15}$