The problem asks us to simplify the expression $\frac{x}{\sqrt{13} - \sqrt{7}}$ by rationalizing the denominator.

1. Problem Description

The problem asks us to simplify the expression $\frac{x}{\sqrt{13} - \sqrt{7}}$ by rationalizing the denominator.

2. Solution Steps

To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{13} - \sqrt{7}$ is $\sqrt{13} + \sqrt{7}$.

So, we have:

$\frac{x}{\sqrt{13} - \sqrt{7}} = \frac{x}{\sqrt{13} - \sqrt{7}} \cdot \frac{\sqrt{13} + \sqrt{7}}{\sqrt{13} + \sqrt{7}}$

$\frac{x(\sqrt{13} + \sqrt{7})}{(\sqrt{13} - \sqrt{7})(\sqrt{13} + \sqrt{7})}$

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

$(\sqrt{13} - \sqrt{7})(\sqrt{13} + \sqrt{7}) = (\sqrt{13})^2 - (\sqrt{7})^2 = 13 - 7 = 6$

So,

$\frac{x(\sqrt{13} + \sqrt{7})}{6}$

3. Final Answer

The simplified expression is $\frac{x(\sqrt{13} + \sqrt{7})}{6}$.