Express each of the following as a product of prime factors:
(i) 96 (ii)84 (iii) 150 (iv) 240 (v) 3072 (vi)324

Answer

To express each of the given numbers as a product of prime factors, we will perform prime factorization step by step.

### Step-by-Step Solution:

**(i) Prime Factorization of 96:**
1. Start with 96. Divide by 2 (the smallest prime number):
- \( 96 \div 2 = 48 \)
2. Divide 48 by 2:
- \( 48 \div 2 = 24 \)
3. Divide 24 by 2:
- \( 24 \div 2 = 12 \)
4. Divide 12 by 2:
- \( 12 \div 2 = 6 \)
5. Divide 6 by 2:
- \( 6 \div 2 = 3 \)
6. Finally, 3 is a prime number.

So, the prime factorization of 96 is:
\[ 96 = 2^5 \times 3^1 \]

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**(ii) Prime Factorization of 84:**
1. Start with 84. Divide by 2:
- \( 84 \div 2 = 42 \)
2. Divide 42 by 2:
- \( 42 \div 2 = 21 \)
3. Divide 21 by 3 (next smallest prime):
- \( 21 \div 3 = 7 \)
4. Finally, 7 is a prime number.

So, the prime factorization of 84 is:
\[ 84 = 2^2 \times 3^1 \times 7^1 \]

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**(iii) Prime Factorization of 150:**
1. Start with 150. Divide by 2:
- \( 150 \div 2 = 75 \)
2. Divide 75 by 3:
- \( 75 \div 3 = 25 \)
3. Divide 25 by 5:
- \( 25 \div 5 = 5 \)
4. Finally, 5 is a prime number.

So, the prime factorization of 150 is:
\[ 150 = 2^1 \times 3^1 \times 5^2 \]

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**(iv) Prime Factorization of 240:**
1. Start with 240. Divide by 2:
- \( 240 \div 2 = 120 \)
2. Divide 120 by 2:
- \( 120 \div 2 = 60 \)
3. Divide 60 by 2:
- \( 60 \div 2 = 30 \)
4. Divide 30 by 2:
- \( 30 \div 2 = 15 \)
5. Divide 15 by 3:
- \( 15 \div 3 = 5 \)
6. Finally, 5 is a prime number.

So, the prime factorization of 240 is:
\[ 240 = 2^4 \times 3^1 \times 5^1 \]

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**(v) Prime Factorization of 3072:**
1. Start with 3072. Divide by 2:
- \( 3072 \div 2 = 1536 \)
2. Divide 1536 by 2:
- \( 1536 \div 2 = 768 \)
3. Divide 768 by 2:
- \( 768 \div 2 = 384 \)
4. Divide 384 by 2:
- \( 384 \div 2 = 192 \)
5. Divide 192 by 2:
- \( 192 \div 2 = 96 \)
6. Divide 96 by 2:
- \( 96 \div 2 = 48 \)
7. Divide 48 by 2:
- \( 48 \div 2 = 24 \)
8. Divide 24 by 2:
- \( 24 \div 2 = 12 \)
9. Divide 12 by 2:
- \( 12 \div 2 = 6 \)
10. Divide 6 by 2:
- \( 6 \div 2 = 3 \)
11. Finally, 3 is a prime number.

So, the prime factorization of 3072 is:
\[ 3072 = 2^{10} \times 3^1 \]

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**(vi) Prime Factorization of 324:**
1. Start with 324. Divide by 2:
- \( 324 \div 2 = 162 \)
2. Divide 162 by 2:
- \( 162 \div 2 = 81 \)
3. Divide 81 by 3:
- \( 81 \div 3 = 27 \)
4. Divide 27 by 3:
- \( 27 \div 3 = 9 \)
5. Divide 9 by 3:
- \( 9 \div 3 = 3 \)
6. Finally, 3 is a prime number.

So, the prime factorization of 324 is:
\[ 324 = 2^2 \times 3^4 \]

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### Summary of Prime Factorizations:
1. \( 96 = 2^5 \times 3^1 \)
2. \( 84 = 2^2 \times 3^1 \times 7^1 \)
3. \( 150 = 2^1 \times 3^1 \times 5^2 \)
4. \( 240 = 2^4 \times 3^1 \times 5^1 \)
5. \( 3072 = 2^{10} \times 3^1 \)
6. \( 324 = 2^2 \times 3^4 \)