Find the square root of the following numbers :
(i) 1024 (ii) 1764 (iii) 900

Answer

To find the square roots of the numbers 1024, 1764, and 900, we can use the method of factorization. Let's solve each part step by step.

### (i) Finding the square root of 1024

1. **Factorization**: Start by dividing 1024 by 2 repeatedly until you reach 1.
- 1024 ÷ 2 = 512
- 512 ÷ 2 = 256
- 256 ÷ 2 = 128
- 128 ÷ 2 = 64
- 64 ÷ 2 = 32
- 32 ÷ 2 = 16
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1

So, the prime factorization of 1024 is \(2^{10}\) (since we divided by 2 a total of 10 times).

2. **Making pairs**: Group the factors into pairs:
- \(2 \times 2\) (5 pairs of 2)

3. **Taking one from each pair**: The square root of 1024 is:
- \(\sqrt{1024} = 2^5 = 32\)

### (ii) Finding the square root of 1764

1. **Factorization**: Start dividing 1764 by 2.
- 1764 ÷ 2 = 882
- 882 ÷ 2 = 441 (now we can’t divide by 2 anymore)

2. **Continue with 441**: Since 441 is not divisible by 2, we check for the next prime number, which is 3.
- 441 ÷ 3 = 147
- 147 ÷ 3 = 49
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1

So, the prime factorization of 1764 is \(2^2 \times 3^2 \times 7^2\).

3. **Making pairs**: Group the factors into pairs:
- \(2 \times 2\), \(3 \times 3\), \(7 \times 7\)

4. **Taking one from each pair**: The square root of 1764 is:
- \(\sqrt{1764} = 2 \times 3 \times 7 = 42\)

### (iii) Finding the square root of 900

1. **Factorization**: Start by breaking down 900.
- 900 can be expressed as \(9 \times 100\).
- \(9 = 3 \times 3\) and \(100 = 10 \times 10\).

2. **Combine the factors**: So, the prime factorization of 900 is:
- \(3^2 \times 10^2\) (or \(3^2 \times 2^2 \times 5^2\)).

3. **Making pairs**: Group the factors into pairs:
- \(3 \times 3\) and \(10 \times 10\).

4. **Taking one from each pair**: The square root of 900 is:
- \(\sqrt{900} = 3 \times 10 = 30\)

### Summary of Results:
- \(\sqrt{1024} = 32\)
- \(\sqrt{1764} = 42\)
- \(\sqrt{900} = 30\)