In the mixed power series 2, 4, 27, 256, 3125, 46656, which number is the odd man out based on the pattern n^n for integer values of n?

Introduction / Context:
This question asks you to find the odd man out in a sequence: 2, 4, 27, 256, 3125, 46656. The terms look like well known powers of small integers. The key is to see which of these numbers can be written in the special form n^n, where the base and exponent are equal.

Given Data / Assumptions:
The terms in the list are:

• 2
• 4
• 27
• 256
• 3125
• 46656

We assume that most of the numbers are of the form n^n for some integer n ≥ 2, and that exactly one number does not fit this pattern.

Concept / Approach:
The strategy is to rewrite each number as a power of an integer and check whether the exponent equals the base. For example, 27 can be written as 3^3, so it is of the form n^n with n = 3. We go through each number and see which one fails to have such a representation.

Step-by-Step Solution:
Step 1: Consider 4. We have 4 = 2^2, so it is of the form n^n with n = 2.Step 2: Consider 27. We have 27 = 3^3, which is of the form n^n with n = 3.Step 3: Consider 256. We have 256 = 4^4, which is of the form n^n with n = 4.Step 4: Consider 3125. We have 3125 = 5^5, again of the form n^n with n = 5.Step 5: Consider 46656. We have 46656 = 6^6, which fits the n^n pattern with n = 6.Step 6: Finally, consider 2. Powers of integers start with 1^1 = 1 and 2^2 = 4. There is no integer n such that n^n = 2. Therefore 2 does not fit the n^n pattern.

Verification / Alternative check:
We can explicitly list the sequence n^n for n = 1, 2, 3, 4, 5, 6 as 1, 4, 27, 256, 3125, 46656. The numbers 4, 27, 256, 3125 and 46656 all appear in this list, while 2 does not. This confirms that 2 is the only number in the given set that cannot be written as n^n for any integer n, making it the natural odd man out.

Why Other Options Are Wrong:
27, 3125 and 46656 are exactly 3^3, 5^5 and 6^6 respectively, and 4 is 2^2. All of these are perfect n^n powers and are organised nicely in the pattern. Removing any of them as the odd man out would ignore the clear n^n structure and leave 2 still unmatched.

Common Pitfalls:
Some learners might focus only on whether numbers are squares or cubes in general, instead of noticing the more specific requirement that the exponent equals the base. It is important to check whether there is a special form like n^n in play when many terms can be written as powers in a consistent way.

Final Answer:
The only number that is not of the form n^n is 2.