In this number analogy, 8 is related to 81 in a specific way. Using the same relationship, 64 is related to which of the following numbers?

Introduction / Context:
This number analogy checks your ability to recognise patterns involving powers of numbers. The pair 8 and 81 is given, and you must apply the same hidden rule to the number 64 to choose the correct answer from the options. Such questions reward familiarity with basic exponents.

Given Data / Assumptions:
1) First pair: 8 and 81.
2) Second pair: 64 and an unknown three digit number chosen from 225, 425, 525, and 625.
3) All options are perfect squares of integers, which hints that the pattern may involve powers and exponent relationships rather than simple addition or subtraction.
4) We assume the same transformation connects both pairs.

Concept / Approach:
Start by expressing 8 and 81 in terms of powers. Notice that 8 can be written as 2^3, and 81 can be written as 3^4. The base and the exponent both increase by 1 when going from 2^3 to 3^4. Now express 64 in a similar power form and see what the rule suggests for its partner. If the rule is consistent, then that partner should appear among the answer choices.

Step-by-Step Solution:
Step 1: Rewrite 8 as 2^3. Step 2: Rewrite 81 as 3^4. Step 3: Observe the pattern: the base in the second number is one more than the base in the first number, and the exponent in the second number is also one more than the exponent in the first number. Step 4: Now express 64 as 4^3, since 4 * 4 * 4 = 64. Step 5: Apply the same rule: increase the base by 1 to get 5, and increase the exponent by 1 to get 4. This gives 5^4. Step 6: Compute 5^4. Since 5 * 5 = 25 and 25 * 25 = 625, we have 5^4 = 625. Step 7: Check the options and confirm that 625 is present, which matches the pattern.

Verification / Alternative check:
We can verify by considering whether any simpler rule like multiplication or addition gives a unique and consistent answer. For example, 8 multiplied by 10 plus 1 gives 81, but if we apply that to 64, we get 641, which is not in the options. Also, 8 squared plus 17 equals 81, but 64 squared plus 17 is enormous and far outside the option range. The power based pattern is simple, symmetric, and matches the structure of the options, so it is much more convincing.

Why Other Options Are Wrong:
225 is 15^2, 425 is not a perfect square, and 525 is also not a perfect square. None of them match 5^4. Only 625 is exactly 5^4, and it naturally extends the pattern from 2^3 to 3^4 to 4^3 to 5^4. Therefore the other options break the pattern and must be rejected.

Common Pitfalls:
A common mistake is to look only for multipliers or differences between the two numbers and to ignore deeper structural relationships involving powers. Another pitfall is to get distracted by the fact that all options except one may be unfamiliar, and then guess based on appearance. Instead, always express small numbers in terms of powers if a pattern with exponents seems plausible.

Final Answer:
Thus, 64 is related to 625 in the same way that 8 is related to 81. The correct answer is 625.