The problem is to simplify the expression $\frac{1}{\sqrt{2} + 3}$. This involves rationalizing the denominator.

1. Problem Description

The problem is to simplify the expression $\frac{1}{\sqrt{2} + 3}$. This involves rationalizing the denominator.

2. Solution Steps

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{2} - 3$.

$$$$

\frac{1}{\sqrt{2} + 3} = \frac{1}{\sqrt{2} + 3} \cdot \frac{\sqrt{2} - 3}{\sqrt{2} - 3}

$$$$

Multiplying the numerators gives

$$$$

1 \cdot (\sqrt{2} - 3) = \sqrt{2} - 3

$$$$

Multiplying the denominators gives

$$$$

(\sqrt{2} + 3)(\sqrt{2} - 3) = (\sqrt{2})^2 - (3)^2 = 2 - 9 = -7

$$$$

Therefore,

$$$$

\frac{1}{\sqrt{2} + 3} = \frac{\sqrt{2} - 3}{-7} = \frac{3 - \sqrt{2}}{7}

$$$$

3. Final Answer

The simplified expression is $\frac{3 - \sqrt{2}}{7}$.