Write the prime factorisation of the following in the exponential form:
(i) 2600 (ii) 1575 (iii) 8568

Answer

To find the prime factorization of the numbers 2600, 1575, and 8568 in exponential form, we will break down each number step by step.

### (i) Prime Factorization of 2600

1. **Divide by 2**:
- \( 2600 \div 2 = 1300 \)

2. **Divide by 2 again**:
- \( 1300 \div 2 = 650 \)

3. **Divide by 2 again**:
- \( 650 \div 2 = 325 \)

4. **Divide by 5**:
- \( 325 \div 5 = 65 \)

5. **Divide by 5 again**:
- \( 65 \div 5 = 13 \)

6. **Divide by 13**:
- \( 13 \div 13 = 1 \)

Now we can write the prime factors:
- The prime factors are: \( 2, 2, 2, 5, 5, 13 \)

In exponential form:
- \( 2600 = 2^3 \times 5^2 \times 13^1 \)

### (ii) Prime Factorization of 1575

1. **Divide by 5**:
- \( 1575 \div 5 = 315 \)

2. **Divide by 5 again**:
- \( 315 \div 5 = 63 \)

3. **Divide by 3**:
- \( 63 \div 3 = 21 \)

4. **Divide by 3 again**:
- \( 21 \div 3 = 7 \)

5. **Divide by 7**:
- \( 7 \div 7 = 1 \)

Now we can write the prime factors:
- The prime factors are: \( 5, 5, 3, 3, 7 \)

In exponential form:
- \( 1575 = 5^2 \times 3^2 \times 7^1 \)

### (iii) Prime Factorization of 8568

1. **Divide by 2**:
- \( 8568 \div 2 = 4284 \)

2. **Divide by 2 again**:
- \( 4284 \div 2 = 2142 \)

3. **Divide by 2 again**:
- \( 2142 \div 2 = 1071 \)

4. **Divide by 3**:
- \( 1071 \div 3 = 357 \)

5. **Divide by 3 again**:
- \( 357 \div 3 = 119 \)

6. **Divide by 7**:
- \( 119 \div 7 = 17 \)

7. **Divide by 17**:
- \( 17 \div 17 = 1 \)

Now we can write the prime factors:
- The prime factors are: \( 2, 2, 2, 3, 3, 7, 17 \)

In exponential form:
- \( 8568 = 2^3 \times 3^2 \times 7^1 \times 17^1 \)

### Summary of Prime Factorizations
- \( 2600 = 2^3 \times 5^2 \times 13^1 \)
- \( 1575 = 5^2 \times 3^2 \times 7^1 \)
- \( 8568 = 2^3 \times 3^2 \times 7^1 \times 17^1 \)