To find the square root of $2304$ by factorization, let's follow these steps:

1. Prime Factorization of 2304: Let's break down $2304$ into its prime factors:

   • $2304 \div 2 = 1152$
   • $1152 \div 2 = 576$
   • $576 \div 2 = 288$
   • $288 \div 2 = 144$
   • $144 \div 2 = 72$
   • $72 \div 2 = 36$
   • $36 \div 2 = 18$
   • $18 \div 2 = 9$
   • $9 \div 3 = 3$
   • $3 \div 3 = 1$

   So, the prime factorization of $2304$ is: $2304 = 2^8 \times 3^2$

2. Grouping Factors for Square Root: Since we're looking for the square root, we can pair the factors:

   • $(2^4)^2 = 2^8$
   • $(3^1)^2 = 3^2$

3. Taking the Square Root: We take one factor from each pair to get the square root: $\sqrt{2304} = 2^4 \times 3 = 16 \times 3 = 48$

So, the square root of $2304$ is 48.

Would you like further details on any of the steps?

Here are some related questions to expand on this:

1. What is the prime factorization method used for other perfect squares?
2. How does the square root relate to the exponent in factorization?
3. Can we find cube roots or higher roots by similar factorization?
4. What methods exist to estimate square roots of non-perfect squares?
5. How do calculators compute square roots using factorization?

Tip: When dealing with large numbers, breaking them down into prime factors can simplify root calculations significantly!