To determine which expression provides the correct prime factorization of 96, let's examine each given option and compare it with the actual prime factorization result:
1. Prime Factorization of 96:
The prime factorization of 96 indicates which prime numbers multiply together to give 96. By identifying the prime factors and their respective exponents, we can express the number in a unique form.
2. Options Analysis:
- Option A: [tex]\(3^2 \cdot 2^3\)[/tex]:
- Here, [tex]\(3^2 = 9\)[/tex] and [tex]\(2^3 = 8\)[/tex].
- [tex]\(9 \cdot 8 = 72\)[/tex], which is not equal to 96.
- Option B: [tex]\(2^5 \cdot 3\)[/tex]:
- Here, [tex]\(2^5 = 32\)[/tex] and [tex]\(3 = 3\)[/tex].
- [tex]\(32 \cdot 3 = 96\)[/tex], which is equal to 96.
- This option matches the prime factorization of 96.
- Option C: [tex]\(4^2 \cdot 6\)[/tex]:
- Here, [tex]\(4^2 = 16\)[/tex] and [tex]\(6 = 6\)[/tex].
- [tex]\(16 \cdot 6 = 96\)[/tex], which is equal to 96.
- However, this is not a prime factorization as 4 and 6 are not prime numbers.
- Option D: [tex]\(1 \cdot 96\)[/tex]:
- Here, [tex]\(1 \cdot 96 = 96\)[/tex], which is equal to 96.
- However, this is not a prime factorization as neither 1 nor 96 are prime numbers.
3. Conclusion:
The correct expression for the prime factorization of 96 is:
[tex]\(\boxed{2^5 \cdot 3}\)[/tex]
So, the answer is [tex]\(B. 2^5 \cdot 3\)[/tex].
