The problem asks to simplify the expression $\frac{2y}{\sqrt{11} + \sqrt{10}}$ by rationalizing the denominator.

1. Problem Description

The problem asks to simplify the expression $\frac{2y}{\sqrt{11} + \sqrt{10}}$ by rationalizing the denominator.

2. Solution Steps

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{11} - \sqrt{10}$.

$$\frac{2y}{\sqrt{11} + \sqrt{10}} \cdot \frac{\sqrt{11} - \sqrt{10}}{\sqrt{11} - \sqrt{10}}$$

Multiplying the numerators gives:

$$2y (\sqrt{11} - \sqrt{10})$$

Multiplying the denominators using the difference of squares formula $(a+b)(a-b) = a^2 - b^2$ gives:

$$(\sqrt{11})^2 - (\sqrt{10})^2 = 11 - 10 = 1$$

So the expression becomes:

$$\frac{2y (\sqrt{11} - \sqrt{10})}{1} = 2y (\sqrt{11} - \sqrt{10})$$

$$= 2y\sqrt{11} - 2y\sqrt{10}$$

3. Final Answer

$2y\sqrt{11} - 2y\sqrt{10}$