Which of the following represents the prime factorization of [tex]18[/tex]?

A. [tex]2 \times 9[/tex]
B. [tex]2 \times 3^2[/tex]
C. [tex]2 \times 6[/tex]
D. [tex]2^2 \times 3[/tex]



Answer :

To determine the prime factorization of [tex]\(18\)[/tex], we need to break it down into its prime factors step-by-step. Here's how it can be done:

1. Start with 18:
[tex]\[ 18 \][/tex]

2. Find the smallest prime that divides 18:
The smallest prime number is [tex]\(2\)[/tex]. Check if [tex]\(18\)[/tex] is divisible by [tex]\(2\)[/tex]:
[tex]\[ 18 \div 2 = 9 \][/tex]
So, we have:
[tex]\[ 18 = 2 \times 9 \][/tex]

3. Factorize 9 further:
Now we focus on [tex]\(9\)[/tex]. The smallest prime number that divides [tex]\(9\)[/tex] is [tex]\(3\)[/tex]:
[tex]\[ 9 \div 3 = 3 \][/tex]
And:
[tex]\[ 3 \div 3 = 1 \][/tex]

Thus, [tex]\(9\)[/tex] can be expressed as:
[tex]\[ 9 = 3 \times 3 = 3^2 \][/tex]

4. Combine all the factors:
Substitute back to express [tex]\(18\)[/tex] fully in terms of its prime factors:
[tex]\[ 18 = 2 \times 9 = 2 \times 3 \times 3 = 2 \times 3^2 \][/tex]

Hence, the prime factorization of [tex]\(18\)[/tex] is:
[tex]\[ 2 \times 3^2 \][/tex]

Therefore, among the given options, the correct representation of the prime factorization of [tex]\(18\)[/tex] is:
[tex]\[ 2 \times 3^2 \][/tex]