It's only work for even number? Like any even number which is not perfect square

No, this method of prime factorization works for both even and odd numbers, regardless of whether they are perfect squares. The process involves breaking down the number into its prime factors, which can then be used to simplify the square root. Let's go through an example with an odd number and a non-perfect square even number to demonstrate this:

### Example 1: An Odd Number

Let's simplify the square root of 45.

1. **Prime Factorization:**
   - 45 ÷ 3 = 15
   - 15 ÷ 3 = 5
   - 5 ÷ 5 = 1

   So, the prime factors of 45 are: 3 × 3 × 5, or \(3^2 \times 5\).

2. **Grouping the Factors:**
   - Group the factors in pairs: (3 × 3), and one 5 remains unpaired.

3. **Simplify the Square Root:**
   - Take one 3 out of the square root (since each pair can be taken out as a single number).
   - So, \(\sqrt{45} = \sqrt{3^2 \times 5} = 3\sqrt{5}\).

Thus, the simplified form of \(\sqrt{45}\) is \(3\sqrt{5}\).

### Example 2: A Non-Perfect Square Even Number

Let's simplify the square root of 50.

1. **Prime Factorization:**
   - 50 ÷ 2 = 25
   - 25 ÷ 5 = 5
   - 5 ÷ 5 = 1

   So, the prime factors of 50 are: 2 × 5 × 5, or \(2 \times 5^2\).

2. **Grouping the Factors:**
   - Group the factors in pairs: (5 × 5), and one 2 remains unpaired.

3. **Simplify the Square Root:**
   - Take one 5 out of the square root.
   - So, \(\sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2}\).

Thus, the simplified form of \(\sqrt{50}\) is \(5\sqrt{2}\).

### Summary

This method works for any number, whether it is even or odd, and whether it is a perfect square or not. The key steps are:

1. Factorize the number into its prime factors.
2. Group the factors in pairs.
3. Take one number out of the square root for each pair.
4. Multiply the numbers taken out, and keep any remaining factors inside the square root.

Would you like more details or have any questions?

Here are five numbered questions you might consider:

1. How do you handle numbers with more than two types of prime factors?
2. What if the number has repeated prime factors more than twice?
3. How can this method be applied to higher-order roots, such as cube roots?
4. What is the significance of perfect squares in this context?
5. Are there efficient algorithms or tools for quickly performing prime factorization?

**Tip:** Understanding the properties of prime numbers and practicing factorization with different types of numbers can improve your skills in simplifying square roots.

Explore how to simplify square roots using prime factorization with detailed examples for odd numbers and non-perfect square even numbers. Learn the process of prime factorization, grouping factors, and simplifying square roots. Suitable for students in Grades 7-9.

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