The problem is to simplify the expression $\frac{3y}{\sqrt{14} + \sqrt{13}}$ by rationalizing the denominator.

1. Problem Description

The problem is to simplify the expression $\frac{3y}{\sqrt{14} + \sqrt{13}}$ by rationalizing the denominator.

2. Solution Steps

To rationalize the denominator, we need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{14} + \sqrt{13}$ is $\sqrt{14} - \sqrt{13}$.

So, we multiply the expression by $\frac{\sqrt{14} - \sqrt{13}}{\sqrt{14} - \sqrt{13}}$:

$\frac{3y}{\sqrt{14} + \sqrt{13}} \cdot \frac{\sqrt{14} - \sqrt{13}}{\sqrt{14} - \sqrt{13}}$

$=\frac{3y(\sqrt{14} - \sqrt{13})}{(\sqrt{14} + \sqrt{13})(\sqrt{14} - \sqrt{13})}$

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

$(\sqrt{14} + \sqrt{13})(\sqrt{14} - \sqrt{13}) = (\sqrt{14})^2 - (\sqrt{13})^2 = 14 - 13 = 1$

Therefore, the expression becomes:

$\frac{3y(\sqrt{14} - \sqrt{13})}{1} = 3y(\sqrt{14} - \sqrt{13})$

$= 3y\sqrt{14} - 3y\sqrt{13}$

3. Final Answer

$3y\sqrt{14} - 3y\sqrt{13}$