Understanding the Number Set Relationship

The question presents three sets of numbers, where the numbers within each set are related in a specific pattern. We need to identify this pattern using the first two sets and then apply it to find the missing number in the third set.

Analyzing the First Set: (2, 8, 4)

Let's denote the numbers in a set as $(a, b, c)$. For the first set (2, 8, 4):

• The first number ($a$) is 2.
• The second number ($b$) is 8. We observe that $2^3 = 2 \times 2 \times 2 = 8$.
• The third number ($c$) is 4. We observe that $2^2 = 2 \times 2 = 4$.

So, the relationship seems to be $(a, a^3, a^2)$.

Analyzing the Second Set: (7, 343, 49)

Let's check if the same relationship $(a, a^3, a^2)$ holds for the second set (7, 343, 49):

• The first number ($a$) is 7.
• The second number ($b$) is 343. Let's calculate $7^3$: $7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343$. This matches.
• The third number ($c$) is 49. Let's calculate $7^2$: $7^2 = 7 \times 7 = 49$. This also matches.

The relationship $(a, a^3, a^2)$ is confirmed for the first two sets.

Applying the Relationship to the Third Set: (4, 64, ?)

Now, we apply the identified pattern $(a, a^3, a^2)$ to the third set (4, 64, ?):

• The first number ($a$) is 4.
• The second number ($b$) is 64. Let's verify: $4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64$. This is correct.
• The third number ($c$) is the unknown value we need to find. According to the pattern, it should be $a^2$.

Calculating the third number:

$c = a^2 = 4^2 = 4 \times 4 = 16$.

Conclusion

Based on the established pattern $(a, a^3, a^2)$, the missing number in the third set (4, 64, ?) is 16.