The problem is to simplify the expression $\frac{2\sqrt{2}}{\sqrt{5} - \sqrt{3}}$. This requires rationalizing the denominator.

1. Problem Description

The problem is to simplify the expression $\frac{2\sqrt{2}}{\sqrt{5} - \sqrt{3}}$. This requires rationalizing the denominator.

2. Solution Steps

To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of $\sqrt{5} - \sqrt{3}$ is $\sqrt{5} + \sqrt{3}$.

$$$$

\frac{2\sqrt{2}}{\sqrt{5} - \sqrt{3}} = \frac{2\sqrt{2}}{\sqrt{5} - \sqrt{3}} \cdot \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}

$$$$

Now, multiply the numerators and denominators:

$$$$

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

$$$$

Using the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$, the denominator becomes:

$$$$

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

$$$$

The numerator becomes:

$$$$

2\sqrt{2}(\sqrt{5} + \sqrt{3}) = 2(\sqrt{2}\sqrt{5} + \sqrt{2}\sqrt{3}) = 2(\sqrt{10} + \sqrt{6}) = 2\sqrt{10} + 2\sqrt{6}

$$$$

So we have:

$$$$

\frac{2\sqrt{10} + 2\sqrt{6}}{2}

$$$$

Divide both terms in the numerator by 2:

$$$$

\frac{2\sqrt{10}}{2} + \frac{2\sqrt{6}}{2} = \sqrt{10} + \sqrt{6}

$$$$

3. Final Answer

The simplified expression is $\sqrt{10} + \sqrt{6}$.