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 the numerator and denominator by the conjugate of the denominator. The conjugate of $\sqrt{5}-\sqrt{3}$ is $\sqrt{5}+\sqrt{3}$.

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

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

We can use the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ to simplify the denominator.

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

So, we have

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

Distribute $\sqrt{2}$:

$\sqrt{2}\sqrt{5} + \sqrt{2}\sqrt{3} = \sqrt{2 \cdot 5} + \sqrt{2 \cdot 3} = \sqrt{10} + \sqrt{6}$

3. Final Answer

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