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The question asks to determine which of the following numbers are perfect squares using the prime factorization method:

The options are:
- a) 144
- b) 4050
- c) 2048
- d) 10253

### Prime Factorization Method Explanation:
To check if a number is a perfect square using prime factorization, we:
1. Find the prime factorization of the number.
2. If all the prime factors appear with an even exponent, the number is a perfect square.

#### Step 1: Prime Factorization of 144
144 = \(2^4 \times 3^2\)  
Since all prime factors have even exponents, 144 is a perfect square.

#### Step 2: Prime Factorization of 4050
4050 = \(2 \times 3^2 \times 5^2 \times 9\)  
Since the factor of 2 has an odd exponent (1), 4050 is **not** a perfect square.

#### Step 3: Prime Factorization of 2048
2048 = \(2^{11}\)  
The exponent of 2 is odd (11), so 2048 is **not** a perfect square.

#### Step 4: Prime Factorization of 10253
10253 is a prime number, meaning it cannot be factored further. Since there are no pairs of factors, 10253 is **not** a perfect square.

### Conclusion:
- **144** is a perfect square.
- **4050, 2048, and 10253** are not perfect squares.

Would you like more details on any of these factorizations or the method itself?

Here are some related questions to expand your understanding:
1. How can you determine if a large number is a perfect square without factoring?
2. What is the significance of exponents in determining a perfect square?
3. Can a number be a perfect square if one of its prime factors appears an odd number of times?
4. How do you simplify the prime factorization process for large numbers?
5. How do perfect squares relate to square roots?

**Tip:** Always check the prime factors’ exponents—if they’re all even, the number is a perfect square!

Using prime factorisation method, find which of the following are perfect squares: a) 144, b) 4050, c) 2048, d) 10253. Explain why if they are not.

Learn how to use prime factorization to find if numbers like 144, 4050, 2048, and 10253 are perfect squares. This guide explains the steps to check if a number is a perfect square by examining the exponents of its prime factors. Suitable for students in Grades 6-8.

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