To find the prime factorization of 96, we need to break it down into its prime factors, which are the prime numbers that multiply together to give the number.
Here are the detailed steps:
1. Identify the smallest prime number that divides 96. The smallest prime number is 2.
2. Divide 96 by 2:
[tex]\[
96 \div 2 = 48
\][/tex]
3. Continue dividing by 2 until it no longer divides evenly:
[tex]\[
48 \div 2 = 24
\][/tex]
[tex]\[
24 \div 2 = 12
\][/tex]
[tex]\[
12 \div 2 = 6
\][/tex]
[tex]\[
6 \div 2 = 3
\][/tex]
4. After reaching 3, notice that 3 is a prime number and can no longer be divided by 2. Therefore, we stop.
Now we list all the factors:
- From the divisions, 96 can be expressed as [tex]\(2 \times 2 \times 2 \times 2 \times 2 \times 3\)[/tex], which is equivalent to:
[tex]\[
2^5 \times 3
\][/tex]
Given this result, let’s compare with the provided choices:
1. [tex]\(6 \cdot 8 \cdot 2\)[/tex]
- This is not a prime factorization and can be factored further.
2. [tex]\(3 \cdot 2^4\)[/tex]
- This misses an additional division by 2 and hence is incorrect.
3. [tex]\(3 \cdot 2 \cdot 2 \cdot 2 \cdot 2\)[/tex]
- This is actually [tex]\(3 \cdot 2^4\)[/tex], so this misses one 2 factor. Incorrect.
4. [tex]\(3 \cdot 2^5\)[/tex]
- This matches what we found through factorization: [tex]\(2^5 \times 3\)[/tex].
Therefore, the correct expression that represents the prime factorization of 96 is:
[tex]\[
3 \cdot 2^5
\][/tex]
