In the following number analogy, 34 is related to 81 through an exponential relationship. Using the same pattern, 23 is related to which number?

Introduction / Context:
This analogy question tests numerical pattern recognition. You are given that 34 is related to 81 and you must identify how 23 will be related to another number using the same rule. The challenge is to notice that the digits in each pair can be interpreted as base and exponent to form a power. Such problems appear frequently in quantitative aptitude and reasoning sections of exams.

Given Data / Assumptions:
- First pair: 34 and 81.
- Second starting number: 23.
- All answer options are positive integers.
- The same mathematical pattern must connect 23 to the correct answer that connects 34 to 81.

Concept / Approach:
Look at 34 and 81. A very strong hint is that 81 is a well known power, namely 3 raised to the power 4. The number 34 consists of the digits 3 and 4. This suggests we interpret 34 not as thirty four but as a compact way of writing 3 as base and 4 as exponent. That is, 34 corresponds to 3^4 which equals 81. Once we see this, we can apply exactly the same logic to 23 by treating it as 2^3.

Step-by-Step Solution:
Step 1: Interpret 34 as 3^4, where 3 is the first digit and 4 is the second digit. Step 2: Compute 3^4 = 3 * 3 * 3 * 3 = 81, which matches the given pair. Step 3: Apply the same idea to 23, treating it as 2^3, where 2 is the base and 3 is the exponent. Step 4: Calculate 2^3 = 2 * 2 * 2 = 8. Step 5: Compare the result 8 with the options and choose option A.

Verification / Alternative check:
Check quickly whether any other simple interpretation of 34 could produce 81. Common operations like 3 + 4, 3 * 4, or 34 + a constant do not yield 81. The only natural and neat explanation is using exponentiation. Similarly, 2^3 is unambiguously equal to 8, and no other choice fits this very specific pattern. This confirms that the relationship is based on base exponent structure and that 8 is correct.

Why Other Options Are Wrong:
Option B 16 might come from 4^2 or squaring, but that does not mirror the 3^4 pattern. Option C 4 would correspond to 2^2, again inconsistent. Option D 12 could arise from addition or multiplication, but it cannot be derived using the same exponent rule used in the first pair. Therefore these answers fail to respect the base exponent idea that underlies the analogy.

Common Pitfalls:
Students sometimes focus only on addition, subtraction or multiplication of digits and completely miss exponentiation as a possibility. Another error is to reverse the base and exponent and try 4^3 instead of 3^4 for the first pair, which gives 64 and does not match 81. Always confirm which interpretation exactly reproduces the given mapping before applying it to the second case.

Final Answer:
Using the pattern base^exponent, 34 corresponds to 3^4 = 81, so 23 corresponds to 2^3 = 8.