To solve this problem, we need to find the prime factorization of the number 72 and then express it in exponential notation. Let’s break down the problem step-by-step:
1. Identify the number:
We need the prime factors of 72.
2. Divide by the smallest prime number:
Begin by dividing 72 by the smallest prime, which is 2:
- [tex]\( 72 \div 2 = 36 \)[/tex]
- [tex]\( 36 \div 2 = 18 \)[/tex]
- [tex]\( 18 \div 2 = 9 \)[/tex]
Now, we have divided by 2 three times, which means [tex]\( 2^3 \)[/tex].
3. Continue with the next smallest prime:
After obtaining 9, we continue the factorization with the next smallest prime, which is 3:
- [tex]\( 9 \div 3 = 3 \)[/tex]
- [tex]\( 3 \div 3 = 1 \)[/tex]
We have divided by 3 twice, which means [tex]\( 3^2 \)[/tex].
4. Combine the prime factors:
The prime factorization of 72 is [tex]\( 2^3 \cdot 3^2 \)[/tex].
5. Match with given options:
Comparing our result with the given options:
- [tex]\( 8 \cdot 9 \)[/tex]:
[tex]\[ 8 = 2^3 \text{ and } 9 = 3^2 \][/tex]
This option is breakdown into prime factors, matching [tex]\(2^3 \cdot 3^2\)[/tex].
- [tex]\( 2 \cdot 6^2 \)[/tex]:
[tex]\[ 6 = 2 \cdot 3 \text{ so } 6^2 = (2 \cdot 3)^2 = 2^2 \cdot 3^2 \][/tex]
This breaks down to [tex]\( 2 \cdot 2^2 \cdot 3^2 = 2^3 \cdot 3^2 \)[/tex].
- [tex]\( 2^3 \cdot 3^2 \)[/tex]:
This directly matches our breakdown of prime factorization [tex]\(2^3 \cdot 3^2\)[/tex].
- [tex]\( 2^2 \cdot 3^2 \)[/tex]:
This does not match our factorization since it is [tex]\( 2^3 \cdot 3^2 \)[/tex], not [tex]\( 2^2 \cdot 3^2 \)[/tex].
Thus, the correct option that shows the prime factorization of 72 using exponential notation is:
[tex]\[
\boxed{2^3 \cdot 3^2}
\][/tex]
