Certainly! Let's break down the three numbers into their prime factors and then state their products in the form of prime factorizations.
### (a) Prime Factorization of 72
To find the prime factors of 72, we follow these steps:
1. Divide by 2 (since 72 is even):
[tex]\[ 72 \div 2 = 36 \][/tex]
So, 2 is a factor.
2. Divide by 2 again:
[tex]\[ 36 \div 2 = 18 \][/tex]
So, another factor of 2.
3. Divide by 2 again:
[tex]\[ 18 \div 2 = 9 \][/tex]
Another factor of 2.
4. Now divide by 3 (since 9 is not even but is divisible by 3):
[tex]\[ 9 \div 3 = 3 \][/tex]
Factor of 3.
5. Divide by 3 again (since 3 is divisible by 3):
[tex]\[ 3 \div 3 = 1 \][/tex]
Another factor of 3.
Thus, the prime factorization of 72 is:
[tex]\[ 72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2 \][/tex]
### (b) Prime Factorization of 96
To find the prime factors of 96, we follow these steps:
1. Divide by 2:
[tex]\[ 96 \div 2 = 48 \][/tex]
Factor of 2.
2. Divide by 2 again:
[tex]\[ 48 \div 2 = 24 \][/tex]
Another factor of 2.
3. Divide by 2 again:
[tex]\[ 24 \div 2 = 12 \][/tex]
Another factor of 2.
4. Divide by 2 again:
[tex]\[ 12 \div 2 = 6 \][/tex]
Another factor of 2.
5. Divide by 2 again:
[tex]\[ 6 \div 2 = 3 \][/tex]
Another factor of 2.
6. Now divide by 3 (since 3 is a small prime number):
[tex]\[ 3 \div 3 = 1 \][/tex]
Factor of 3.
Thus, the prime factorization of 96 is:
[tex]\[ 96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 2^5 \times 3 \][/tex]
### (c) Prime Factorization of 125
To find the prime factors of 125, we follow these steps:
1. Divide by 5 (since 125 is divisible by 5):
[tex]\[ 125 \div 5 = 25 \][/tex]
Factor of 5.
2. Divide by 5 again:
[tex]\[ 25 \div 5 = 5 \][/tex]
Another factor of 5.
3. Divide by 5 again:
[tex]\[ 5 \div 5 = 1 \][/tex]
Another factor of 5.
Thus, the prime factorization of 125 is:
[tex]\[ 125 = 5 \times 5 \times 5 = 5^3 \][/tex]
### Final Products Using Prime Factorization
Now we can write the products of the numbers using their prime factorizations:
- [tex]\(72 = 2^3 \times 3^2\)[/tex]
- [tex]\(96 = 2^5 \times 3\)[/tex]
- [tex]\(125 = 5^3\)[/tex]
