Express each of the following numbers as the product of powers of their prime factors:
(i) 36
(ii) 675
(iii) 392

Answer

To express the given numbers as the product of powers of their prime factors, we will follow the steps of prime factorization for each number.

### (i) Prime Factorization of 36

1. **Divide by the smallest prime number (2):**
- \( 36 \div 2 = 18 \)
- \( 36 = 2 \times 18 \)

2. **Continue dividing by 2:**
- \( 18 \div 2 = 9 \)
- \( 18 = 2 \times 9 \)
- So, \( 36 = 2 \times 2 \times 9 = 2^2 \times 9 \)

3. **Factor 9 using the next smallest prime (3):**
- \( 9 \div 3 = 3 \)
- \( 9 = 3 \times 3 \)
- So, \( 36 = 2^2 \times 3^2 \)

Thus, the prime factorization of 36 is:
\[ 36 = 2^2 \times 3^2 \]

### (ii) Prime Factorization of 675

1. **Divide by the smallest prime number (3):**
- \( 675 \div 3 = 225 \)
- \( 675 = 3 \times 225 \)

2. **Continue dividing by 3:**
- \( 225 \div 3 = 75 \)
- \( 225 = 3 \times 75 \)
- So, \( 675 = 3 \times 3 \times 75 = 3^2 \times 75 \)

3. **Continue dividing 75 by 3:**
- \( 75 \div 3 = 25 \)
- \( 75 = 3 \times 25 \)
- So, \( 675 = 3^3 \times 25 \)

4. **Factor 25 using the next smallest prime (5):**
- \( 25 \div 5 = 5 \)
- \( 25 = 5 \times 5 \)
- So, \( 675 = 3^3 \times 5^2 \)

Thus, the prime factorization of 675 is:
\[ 675 = 3^3 \times 5^2 \]

### (iii) Prime Factorization of 392

1. **Divide by the smallest prime number (2):**
- \( 392 \div 2 = 196 \)
- \( 392 = 2 \times 196 \)

2. **Continue dividing by 2:**
- \( 196 \div 2 = 98 \)
- \( 196 = 2 \times 98 \)
- So, \( 392 = 2 \times 2 \times 98 = 2^2 \times 98 \)

3. **Continue dividing 98 by 2:**
- \( 98 \div 2 = 49 \)
- \( 98 = 2 \times 49 \)
- So, \( 392 = 2^3 \times 49 \)

4. **Factor 49 using the next smallest prime (7):**
- \( 49 \div 7 = 7 \)
- \( 49 = 7 \times 7 \)
- So, \( 392 = 2^3 \times 7^2 \)

Thus, the prime factorization of 392 is:
\[ 392 = 2^3 \times 7^2 \]

### Final Answers
- (i) \( 36 = 2^2 \times 3^2 \)
- (ii) \( 675 = 3^3 \times 5^2 \)
- (iii) \( 392 = 2^3 \times 7^2 \)