Power analogy — map cubes to cubes by stepping the base. 27 : 125 :: 64 : ?

Introduction / Context:
This number analogy uses perfect cubes. The first number of each pair is a cube, and the second number is also a cube, typically with a shifted base. We must determine the shift and apply it consistently.

Given Data / Assumptions:

• 27 is 3^3, and it maps to 125 which is 5^3.
• 64 is 4^3; we must find its mapped cube.
• Rule should be a consistent step in the base.

Concept / Approach:
Notice 3 (base of 27) increased by 2 to become 5 (base of 125). Apply the same +2 step to the base of the second cube.

Step-by-Step Solution:
Write 27 as 3^3 and 125 as 5^3. The base increased from 3 to 5 (+2). Write 64 as 4^3. Increase the base by 2: 4 + 2 = 6. Compute 6^3 = 216.

Verification / Alternative check:
You can test whether another increment (like +1) would fit, but 3^3 → 4^3 would be 64, not 125. Hence +2 is the unique simple step that matches the first pair.

Why Other Options Are Wrong:
162, 273, 343, 517: these are not 6^3 and therefore break the pattern of “cube mapped to cube via +2 in base.”

Common Pitfalls:
Forgetting to look at exponents and bases; many analogies hinge on recognizing perfect powers rather than doing ad-hoc arithmetic.

Final Answer:
216