To simplify the expression $\sqrt{9 + \sqrt{3}}$, we start by examining possible simplifications.

Step 1: Assume a Simplified Form

Let's assume that the expression simplifies to a form like:

$\sqrt{9 + \sqrt{3}} = \sqrt{a} + \sqrt{b}$

where $a$ and $b$ are positive real numbers.

Step 2: Square Both Sides

Square both sides to eliminate the square roots:

$9 + \sqrt{3} = (\sqrt{a} + \sqrt{b})^2$

Expanding