Simplify the expression $\frac{x}{\sqrt{13}-\sqrt{7}}$ by rationalizing the denominator.

1. Problem Description

Simplify the expression $\frac{x}{\sqrt{13}-\sqrt{7}}$ by rationalizing the denominator.

2. Solution Steps

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

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

Now, we multiply the denominator using the difference of squares formula $(a-b)(a+b) = a^2 - b^2$:

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

So the expression becomes:

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

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

3. Final Answer

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