Final answer:
To find the LCM of 2A and 7B, we take the highest exponent of each prime factor from 2A and 7B's prime factorizations and multiply them. The LCM is 2⁴ × 3³ × 5⁴ × 11 × 7.
Step-by-step explanation:
The student is asking to find the lowest common multiple (LCM) of 2A and 7B given A = 2³ × 3² × 5³ × 11 and B = 2²× 3³ × 5⁴. To find the LCM, we first express each quantity in terms of prime factorization.
Let's start by finding the prime factorization of 2A and 7B:
- 2A = 2 × (2³ × 3² × 5³ × 11) = 2⁴ × 3² × 5³ × 11
- 7B = 7 × (2² × 3³ × 5⁴) = 2² × 3³ × 5⁴ × 7
To compute the LCM, we take the highest power of each prime that appears in the factorization of either 2A or 7B:
- For the prime 2, the highest power in 2A is 4, so we use 2⁴.
- For the prime 3, the highest power in 7B is 3, so we use 3³.
- For the prime 5, the highest power in 7B is 4, so we use 5⁴.
- Prime numbers 11 and 7 appear in A and B, respectively, and are not in the other number, so we use both 11¹ and 7¹.
The LCM of 2A and 7B is therefore 2⁴ × 3³ × 5⁴ × 11 × 7.