Find the square root of 7744 using prime factorization.

Divide 7744 by the smallest prime numbers until the quotient is 1.

$$\begin{align*} 7744 \div 2 &= 3872 \\ 3872 \div 2 &= 1936 \\ 1936 \div 2 &= 968 \\ 968 \div 2 &= 484 \\ 484 \div 2 &= 242 \\ 242 \div 2 &= 121 \\ 121 \div 11 &= 11 \\ 11 \div 11 &= 1 \end{align*}$$

So, the prime factorization of 7744 is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 11 \times 11$.

$$\sqrt{7744} = 2 \times 2 \times 2 \times 11$$ $$\sqrt{7744} = 8 \times 11$$ $$\sqrt{7744} = 88$$

The square root of 7744 is $\boxed{88}$.