In this number analogy, 8 is related to 27 through powers of whole numbers. Following the same idea of consecutive powers, 9 is related to which number?

Introduction / Context:
This question is a number analogy that uses powers of natural numbers. The given pair 8 : 27 links two numbers that are well known perfect cubes. The second pair 9 : ? asks you to extend the pattern using a similar idea with perfect powers and consecutive bases. The challenge is to recognise which family of powers is being used in each pair and then choose the missing number that keeps the analogy logically consistent and mathematically neat.

Given Data / Assumptions:

In the first pair, 8 and 27 are related as perfect cubes of whole numbers.
We know that 8 = 2^3 and 27 = 3^3, so the bases 2 and 3 are consecutive integers.
In the second pair, the starting number is 9, which is a perfect square rather than a cube.
We are asked to pick the related number for 9 from among likely perfect powers.
We assume the test setter wants a clean, exam style pattern that uses simple integer exponents and consecutive bases.

Concept / Approach:
The key concept is to notice that 8 and 27 are cubes of consecutive integers: 2^3 and 3^3. A natural way to continue the pattern is to look for a similar relationship but now in the family of perfect squares. The number 9 is equal to 3^2, so it already fits neatly as a square. If we copy the idea of consecutive bases from the first pair, we should pair 3^2 with 4^2. That gives 9 on the left and 16 on the right, mirroring the way that 2^3 and 3^3 formed a consecutive base pair in the first part of the analogy.

Step-by-Step Solution:
Step 1: Express 8 and 27 as powers. We have 8 = 2^3 and 27 = 3^3. Step 2: Observe that the exponents are both 3 and the bases 2 and 3 are consecutive integers. Step 3: Now look at the second pair. We know 9 = 3^2, which is a perfect square. Step 4: To imitate the idea of consecutive bases, we consider the next base after 3, which is 4, and keep the exponent the same at 2. Step 5: Compute 4^2 = 16, giving the natural partner to 9 as a perfect square with consecutive base. Step 6: Therefore the relationship 8 : 27 :: 9 : 16 maintains the theme of simple perfect powers with consecutive bases.

Verification / Alternative check:
To verify, compare the structure of both pairs. In the first, both numbers are cubes and we move from 2^3 to 3^3. In the second, both numbers are squares and we move from 3^2 to 4^2. In each case, the pattern is “same type of power, consecutive bases”. Other candidate numbers such as 25, 36 and 64 are also perfect squares or cubes, but they do not preserve the specific consecutive base relationship when paired with 9 under the square pattern we have chosen. For example, 25 is 5^2 which skips base 4, and 36 is 6^2 which skips even more. Sixty four is 4^3 and would break the square pattern entirely.

Why Other Options Are Wrong:
25 is equal to 5^2, but that would pair 3^2 and 5^2, which are not consecutive bases and thus do not mirror the neat 2^3 and 3^3 pattern from the first pair. 36 is 6^2, giving 3^2 to 6^2, which again disrupts the idea of simple consecutive bases. Sixty four equals 4^3 and changes both the exponent family and the neat square structure we set for the second pair. None of these alternatives maintain the clear idea of “same exponent, next base” that is seen in both cube and square families.

Common Pitfalls:
A common mistake is to look only for cubes and try to force 9 into a cube pattern, which fails because 9 is not a perfect cube. Another error is to pick a square number at random without checking whether the underlying relationship between bases is preserved. In analogy questions, the pattern must be logically parallel in both halves, not just roughly similar. By carefully writing each number as an exact power and comparing the roles of bases and exponents, you can avoid guesswork and arrive confidently at the intended answer.

Final Answer:
Just as 8 and 27 are 2^3 and 3^3, 9 and its partner are 3^2 and 4^2, so the missing number is 16.