Question

\(m\) is a positive integer less than 4.
 

Column A	Column B
\((m+2)^m\)	\(m^{m+2}\)

Hint:

When a variable in a quantitative comparison is restricted to a small, finite set of integers, the fastest and safest approach is often to plug in every possible value. If the results are inconsistent, the answer is (D).

Answer 1

Step 1: Understanding the Concept:
This question asks us to compare two exponential expressions where the base and exponent are swapped in a way. The variable \(m\) is constrained to be a positive integer less than 4.
Step 2: Key Formula or Approach:
Since the variable \(m\) can only take on a few specific integer values (1, 2, and 3), the most direct method is to test each value and see if the relationship between the columns remains the same.
Step 3: Detailed Explanation:
The possible values for \(m\) are 1, 2, and 3.
Case 1: Let \(m=1\).
Column A: \((1+2)^1 = 3^1 = 3\).
Column B: \(1^{1+2} = 1^3 = 1\).
In this case, Column A is greater than Column B (\(3>1\)).
Case 2: Let \(m=2\).
Column A: \((2+2)^2 = 4^2 = 16\).
Column B: \(2^{2+2} = 2^4 = 16\).
In this case, the two columns are equal.
Case 3: Let \(m=3\).
Column A: \((3+2)^3 = 5^3 = 125\).
Column B: \(3^{3+2} = 3^5 = 243\).
In this case, Column B is greater than Column A (\(243>125\)).
Step 4: Final Answer:
We found a case where A>B, a case where A = B, and a case where B>A. Since the relationship changes depending on the value of \(m\), the relationship cannot be determined from the information given.