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The problem asks to find the square roots of the given numbers using the prime factorization method. Let's solve each part by breaking down the numbers into their prime factors.

1. **1444**:
   - Prime factorization of 1444:
     \[
     1444 \div 2 = 722 \quad (2 \, \text{is a factor})
     \]
     \[
     722 \div 2 = 361 \quad (2 \, \text{again})
     \]
     Now, 361 is not divisible by 2. Let's check for 19:
     \[
     361 \div 19 = 19
     \]
     So, the prime factors of 1444 are:
     \[
     1444 = 2^2 \times 19^2
     \]
     Taking the square root:
     \[
     \sqrt{1444} = \sqrt{2^2 \times 19^2} = 2 \times 19 = 38
     \]

2. **784**:
   - Prime factorization of 784:
     \[
     784 \div 2 = 392
     \]
     \[
     392 \div 2 = 196
     \]
     \[
     196 \div 2 = 98
     \]
     \[
     98 \div 2 = 49
     \]
     \[
     49 \div 7 = 7
     \]
     So, the prime factors of 784 are:
     \[
     784 = 2^4 \times 7^2
     \]
     Taking the square root:
     \[
     \sqrt{784} = \sqrt{2^4 \times 7^2} = 2^2 \times 7 = 28
     \]

3. **6.25**:
   - 6.25 is \( \frac{625}{100} \), and 625 is a perfect square:
     \[
     625 = 25^2, \quad 100 = 10^2
     \]
     Taking the square root:
     \[
     \sqrt{6.25} = \frac{\sqrt{625}}{\sqrt{100}} = \frac{25}{10} = 2.5
     \]

4. **0.4225**:
   - 0.4225 is \( \frac{4225}{10000} \), and 4225 is a perfect square:
     \[
     4225 = 65^2, \quad 10000 = 100^2
     \]
     Taking the square root:
     \[
     \sqrt{0.4225} = \frac{\sqrt{4225}}{\sqrt{10000}} = \frac{65}{100} = 0.65
     \]

### Answers:
- \( \sqrt{1444} = 38 \)
- \( \sqrt{784} = 28 \)
- \( \sqrt{6.25} = 2.5 \)
- \( \sqrt{0.4225} = 0.65 \)

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How does prime factorization help in finding square roots?
2. Can you use prime factorization to find cube roots?
3. What is the square root of a negative number, and how does it differ?
4. Can prime numbers have square roots?
5. What happens when a number is not a perfect square? How do you estimate its square root?

**Tip:** Prime factorization is an efficient method for finding square roots when the number is a perfect square.

Find the square roots of the following numbers by the prime factorisation method. (a) 1444, (b) 784, (c) 6.25, (d) 0.4225

Learn how to find the square roots of numbers like 1444, 784, 6.25, and 0.4225 using the prime factorization method. This guide breaks down each number into its prime factors and provides step-by-step solutions for middle school students.

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