To factorize any number into its prime factors for simplifying the square root, follow these steps:

1. Find Prime Factors:

   • Start by dividing the number by the smallest prime number (2). If it's divisible, keep dividing by 2 until it's no longer divisible.
   • Move to the next smallest prime number (3) and repeat the process.
   • Continue this process with 5, 7, 11, and so on, until you reach a prime number greater than the square root of the number you are factorizing.

2. Group the Factors:

   • Once you have the prime factors, group them in pairs.

3. Simplify the Square Root:

   • For each pair of prime factors, take one out of the square root.
   • Multiply these outside numbers together to get the simplified form of the square root.
   • The remaining factors, if any, stay inside the square root.

Example

Let's factorize and simplify the square root of 72:

1. Prime Factorization:

   • 72 ÷ 2 = 36
   • 36 ÷ 2 = 18
   • 18 ÷ 2 = 9
   • 9 ÷ 3 = 3
   • 3 ÷ 3 = 1

   So, the prime factors of 72 are: 2 × 2 × 2 × 3 × 3, or $2^3 \times 3^2$.

2. Grouping the Factors:

   • Group the factors in pairs: (2 × 2) and (3 × 3), and one 2 remains unpaired.

3. Simplify the Square Root:

   • Take one 2 and one 3 out of the square root (since each pair can be taken out as a single number).
   • So, $\sqrt{72} = \sqrt{2^3 \times 3^2} = \sqrt{(2^2) \times (3^2) \times 2} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2}$.

Thus, the simplified form of $\sqrt{72}$ is $6\sqrt{2}$.

Would you like more details or have any questions?

Here are five numbered questions you might consider:

1. How do you factorize numbers that are larger or have more complex prime factors?
2. Can you explain why only prime factors are used in this process?
3. What is the significance of grouping factors in pairs for simplifying square roots?
4. How can this method be applied to cube roots or other higher-order roots?
5. Are there any shortcuts or tools for prime factorization?

Tip: Practicing prime factorization with smaller numbers can help build intuition for handling larger numbers more efficiently.