To determine the correct prime factorization of 144, we need to examine each of the given options:
1. [tex]\(12^2\)[/tex]
2. [tex]\(2^2 \cdot 3^4\)[/tex]
3. [tex]\(2^4 \cdot 3^2\)[/tex]
4. [tex]\(2 \cdot 3\)[/tex]
Let's analyze each option:
### Option 1: [tex]\(12^2\)[/tex]
[tex]\[ 12^2 = 12 \times 12 = 144 \][/tex]
So, this calculation correctly produces the number 144. However, [tex]\(12^2\)[/tex] is not a prime factorization because 12 itself can be further factored into primes.
### Option 2: [tex]\(2^2 \cdot 3^4\)[/tex]
[tex]\[ 2^2 \cdot 3^4 = 4 \cdot 81 = 324 \][/tex]
This does not equal 144, so this option is not correct.
### Option 3: [tex]\(2^4 \cdot 3^2\)[/tex]
[tex]\[ 2^4 \cdot 3^2 = 16 \cdot 9 = 144 \][/tex]
This calculation gives us 144, and it uses prime factors (2 and 3). Therefore, this is a valid prime factorization of 144.
### Option 4: [tex]\(2 \cdot 3\)[/tex]
[tex]\[ 2 \cdot 3 = 6 \][/tex]
This is far from 144, so this cannot be the correct prime factorization.
### Conclusion:
The correct prime factorization of 144 is:
[tex]\[ 2^4 \cdot 3^2 \][/tex]
