To find the least common multiple (LCM) of two numbers given their prime factorizations, you need to take the highest power of each prime factor present in both numbers:
- The first number has the prime factorization \(2^2 \cdot 3^1 \cdot 5^1\).
- The second number has the prime factorization \(2^1 \cdot 3^2 \cdot 7^1\).
Now, let's identify the highest power of each prime factor:
- For the prime \(2\): The highest power is \(2^2\) (from the first number).
- For the prime \(3\): The highest power is \(3^2\) (from the second number).
- For the prime \(5\): The highest power is \(5^1\) (from the first number).
- For the prime \(7\): The highest power is \(7^1\) (from the second number).
So the LCM can be expressed as: \[ LCM = 2^2 \cdot 3^2 \cdot 5^1 \cdot 7^1 \]
Therefore, the correct expression to find the least common multiple of the two numbers is: \[ 2^2 \cdot 3^2 \cdot 5^1 \cdot 7^1 \]
Among the choices you provided, the correct expression representing the LCM is: \[ 2^2 \cdot 3^2 \cdot 5 \cdot 7 \]