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

1. Problem Description

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

2. Solution Steps

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{2} + 3$ is $\sqrt{2} - 3$.

Multiply both the numerator and the denominator by $\sqrt{2} - 3$:

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

$= \frac{\sqrt{2} - 3}{(\sqrt{2} + 3)(\sqrt{2} - 3)}$

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

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

So, the expression becomes:

$\frac{\sqrt{2} - 3}{-7}$

We can rewrite this as:

$\frac{3 - \sqrt{2}}{7}$

3. Final Answer

$\frac{3 - \sqrt{2}}{7}$