The given expression is:

$\frac{\sqrt{3}}{\sqrt{2} + 3}$

To simplify this expression, we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is $\sqrt{2} - 3$.

Let's perform the steps:

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

$\frac{\sqrt{3}}{\sqrt{2} + 3} \times \frac{\sqrt{2} - 3}{\sqrt{2} - 3}$

This gives us:

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

2. In the denominator, use the difference of squares formula:

$(a + b)(a - b) = a^2 - b^2$

So:

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

Thus, the expression becomes:

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

3. Now expand the numerator:

$\frac{\sqrt{3} \cdot \sqrt{2} - \sqrt{3} \cdot 3}{-7} = \frac{\sqrt{6} - 3\sqrt{3}}{-7}$

Finally, we can express the result as:

$\frac{-(\sqrt{6} - 3\sqrt{3})}{7}$

Or:

$-\frac{\sqrt{6} - 3\sqrt{3}}{7}$

This is the simplified version of the original expression.

Would you like more details on any step? Here are some related questions:

1. How do you multiply radical expressions?
2. What is the difference of squares formula, and how is it applied?
3. Can you explain rationalizing the denominator in different cases?
4. Why is it important to use the conjugate when simplifying expressions with radicals?
5. How do you simplify expressions involving square roots?

Tip: When simplifying expressions with square roots, always check if you can factor or rationalize the denominator for a cleaner result.