To find the greatest common factor (GCF) of two numbers given their prime factorizations, we compare the powers of each prime factor in both numbers and take the minimum power for each prime.
- The first number has a prime factorization of \(2^3 \cdot 3^2\).
- The second number has a prime factorization of \(2^2 \cdot 3^3\).
Now, let's identify the GCF by comparing the powers of the prime factors:
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For the prime factor \(2\):
- In the first number, the power is \(3\).
- In the second number, the power is \(2\).
- The minimum power is \(\min(3, 2) = 2\).
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For the prime factor \(3\):
- In the first number, the power is \(2\).
- In the second number, the power is \(3\).
- The minimum power is \(\min(2, 3) = 2\).
Putting this together, we find that the GCF is: \[ GCF = 2^2 \cdot 3^2 \]
Now, let's compare this with the provided options:
- \(2^2 \cdot 3^2\)
- \(2^3 \cdot 3^3\)
- \(2^5 \cdot 3^5\)
- \(2 \cdot 3\)
The correct choice that equals the GCF of the two numbers is: \[ \boxed{2^2 \cdot 3^2} \]