Using the prime factorisation method, find which of the following numbers are perfect squares:
`{:((i) 441, (ii) 576, (iii) 11025, (iv)1176),((v)5625, (vi)9075, (vii) 4225, (viii) 1089):}`

Answer

To determine which of the given numbers are perfect squares using the prime factorization method, we will follow these steps for each number:

### Step-by-Step Solution:

1. **Factor 441:**
- Start dividing by the smallest prime number, 3.
- \( 441 \div 3 = 147 \)
- \( 147 \div 3 = 49 \)
- \( 49 \div 7 = 7 \)
- \( 7 \div 7 = 1 \)
- Prime factorization: \( 441 = 3^2 \times 7^2 \)
- Since all prime factors have even powers, **441 is a perfect square**.

2. **Factor 576:**
- Start dividing by 2.
- \( 576 \div 2 = 288 \)
- \( 288 \div 2 = 144 \)
- \( 144 \div 2 = 72 \)
- \( 72 \div 2 = 36 \)
- \( 36 \div 2 = 18 \)
- \( 18 \div 2 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- Prime factorization: \( 576 = 2^6 \times 3^2 \)
- Since all prime factors have even powers, **576 is a perfect square**.

3. **Factor 11025:**
- Start dividing by 3.
- \( 11025 \div 3 = 3675 \)
- \( 3675 \div 3 = 1225 \)
- \( 1225 \div 5 = 245 \)
- \( 245 \div 5 = 49 \)
- \( 49 \div 7 = 7 \)
- \( 7 \div 7 = 1 \)
- Prime factorization: \( 11025 = 3^2 \times 5^2 \times 7^2 \)
- Since all prime factors have even powers, **11025 is a perfect square**.

4. **Factor 1176:**
- Start dividing by 2.
- \( 1176 \div 2 = 588 \)
- \( 588 \div 2 = 294 \)
- \( 294 \div 2 = 147 \)
- \( 147 \div 3 = 49 \)
- \( 49 \div 7 = 7 \)
- \( 7 \div 7 = 1 \)
- Prime factorization: \( 1176 = 2^3 \times 3^1 \times 7^2 \)
- Since not all prime factors have even powers, **1176 is not a perfect square**.

5. **Factor 5625:**
- Start dividing by 5.
- \( 5625 \div 5 = 1125 \)
- \( 1125 \div 5 = 225 \)
- \( 225 \div 5 = 45 \)
- \( 45 \div 5 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- Prime factorization: \( 5625 = 5^4 \times 3^2 \)
- Since all prime factors have even powers, **5625 is a perfect square**.

6. **Factor 9075:**
- Start dividing by 3.
- \( 9075 \div 3 = 3025 \)
- \( 3025 \div 5 = 605 \)
- \( 605 \div 5 = 121 \)
- \( 121 \div 11 = 11 \)
- \( 11 \div 11 = 1 \)
- Prime factorization: \( 9075 = 3^1 \times 5^2 \times 11^2 \)
- Since not all prime factors have even powers, **9075 is not a perfect square**.

7. **Factor 4225:**
- Start dividing by 5.
- \( 4225 \div 5 = 845 \)
- \( 845 \div 5 = 169 \)
- \( 169 \div 13 = 13 \)
- \( 13 \div 13 = 1 \)
- Prime factorization: \( 4225 = 5^2 \times 13^2 \)
- Since all prime factors have even powers, **4225 is a perfect square**.

8. **Factor 1089:**
- Start dividing by 3.
- \( 1089 \div 3 = 363 \)
- \( 363 \div 3 = 121 \)
- \( 121 \div 11 = 11 \)
- \( 11 \div 11 = 1 \)
- Prime factorization: \( 1089 = 3^2 \times 11^2 \)
- Since all prime factors have even powers, **1089 is a perfect square**.

### Summary of Results:
- Perfect Squares: **441, 576, 11025, 5625, 4225, 1089**
- Not Perfect Squares: **1176, 9075**