Simplify $\sqrt{7 + 4\sqrt{3}}$.

Explanation

1. 01

   Look for $(a + \sqrt{b})^2 = a^2 + b + 2a\sqrt{b}$ that equals $7 + 4\sqrt{3}$.

2. 02

   $2a\sqrt{b} = 4\sqrt{3}$ suggests $a = 2$, $b = 3$; then $a^2 + b = 4 + 3 = 7$. It matches.

3. 03

   So $\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}$.