The problem asks us to simplify the expression $\frac{x}{\sqrt{7} - \sqrt{5}}$ by rationalizing the denominator.

1. Problem Description

The problem asks us to simplify the expression $\frac{x}{\sqrt{7} - \sqrt{5}}$ by 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 $\sqrt{7} - \sqrt{5}$ is $\sqrt{7} + \sqrt{5}$.

$\frac{x}{\sqrt{7} - \sqrt{5}} = \frac{x}{\sqrt{7} - \sqrt{5}} \cdot \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} + \sqrt{5}}$

Now we multiply the numerators and denominators:

$\frac{x(\sqrt{7} + \sqrt{5})}{(\sqrt{7} - \sqrt{5})(\sqrt{7} + \sqrt{5})}$

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

$(\sqrt{7} - \sqrt{5})(\sqrt{7} + \sqrt{5}) = (\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2$

So the expression becomes:

$\frac{x(\sqrt{7} + \sqrt{5})}{2}$

3. Final Answer

$\frac{x(\sqrt{7}+\sqrt{5})}{2}$