Express the following as product of powers of their prime factors.
a. `432`
b. `10125`

Answer

To express the numbers 432 and 10125 as a product of powers of their prime factors, we will follow these steps:

### Step 1: Prime Factorization of 432

1. **Divide by 2** (the smallest prime number):
- 432 ÷ 2 = 216
- 216 ÷ 2 = 108
- 108 ÷ 2 = 54
- 54 ÷ 2 = 27 (27 is not divisible by 2)

2. **Now divide by 3** (the next smallest prime number):
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1 (we stop here)

3. **Count the factors**:
- We divided by 2 four times and by 3 three times.

4. **Express as product of powers**:
- Therefore, \( 432 = 2^4 \times 3^3 \).

### Step 2: Prime Factorization of 10125

1. **Divide by 5** (since the last digit is 5):
- 10125 ÷ 5 = 2025
- 2025 ÷ 5 = 405
- 405 ÷ 5 = 81 (81 is not divisible by 5)

2. **Now divide by 3** (the next smallest prime number):
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1 (we stop here)

3. **Count the factors**:
- We divided by 5 three times and by 3 four times.

4. **Express as product of powers**:
- Therefore, \( 10125 = 5^3 \times 3^4 \).

### Final Answers:
- \( 432 = 2^4 \times 3^3 \)
- \( 10125 = 5^3 \times 3^4 \)

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