Final answer:
The smallest number that is exactly divisible by both 6w⁴ and 4y² is found by calculating the Least Common Multiple, which includes each prime factor at the highest power in which it appears. This results in 12w⁴y², which is option (b).
Step-by-step explanation:
The student is asking to find the smallest number that is exactly divisible by both 6w⁴ and 4y². When looking for such a number, we are effectively seeking the Least Common Multiple (LCM) of the two given expressions. To find the LCM, each unique prime factor needs to be taken to the highest power it appears in among the factors.
To find the smallest number that can be divided exactly by both 6w⁴ and 4y², we need to identify the common factors of both expressions. The smallest number that can be divided exactly by both 6w⁴ and 4y² is the product of their highest common factor.
The highest common factor of 6w⁴ and 4y² is 2w⁴y². Therefore, the smallest number that can be divided exactly by both 6w⁴ and 4y² is 12w⁴y², which is option b.
For 6w⁴, the prime factorization is 2 × 3 × w⁴. For 4y², it is 2² × y². The LCM will therefore contain each prime factor at its highest power: 2² (from 4y²), 3 (from 6w⁴), w⁴, and y². Multiplying these together gives us 2² × 3 × w⁴ × y², which simplifies to 12w⁴y², corresponding to option (b).