Question

\(n\) is a positive integer.
 

Column A	Column B
\(n^{100}\)	\(100^n\)

Hint:

When comparing exponential functions like \(x^a\) and \(a^x\), remember that their relationship is not fixed. Testing small integer values, and sometimes the value where the bases and exponents are equal (like \(n=100\) here), is a good strategy to see if the relationship changes.

Answer 1

Step 1: Understanding the Concept:
This question compares two exponential expressions where the base and the exponent are interchanged. We need to determine if one expression is consistently larger than the other for all positive integers \(n\).
Step 2: Key Formula or Approach:
The best approach is to test several different values for the positive integer \(n\) and observe how the relationship between the two columns changes.
Step 3: Detailed Explanation:
Let's test various values of \(n\), where \(n\) is a positive integer (\(n=1, 2, 3, \dots\)).
Case 1: Let \(n = 1\).
Column A: \(n^{100} = 1^{100} = 1\).
Column B: \(100^n = 100^1 = 100\).
In this case, Column B>Column A.
Case 2: Let \(n = 2\).
Column A: \(n^{100} = 2^{100}\).
Column B: \(100^n = 100^2 = 10000 = 10^4\).
To compare these, we can approximate \(2^{100}\). We know \(2^{10} = 1024 \approx 10^3\).
So, \(2^{100} = (2^{10})^{10} \approx (10^3)^{10} = 10^{30}\).
Clearly, \(10^{30}\) is vastly larger than \(10^4\).
In this case, Column A>Column B.
Case 3: Let \(n = 100\).
Column A: \(n^{100} = 100^{100}\).
Column B: \(100^n = 100^{100}\).
In this case, Column A = Column B.
Step 4: Final Answer:
We have found a case where Column B is greater (\(n=1\)), a case where Column A is greater (\(n=2\)), and a case where they are equal (\(n=100\)). Since the relationship between the two quantities is not constant and depends on the value of \(n\), the relationship cannot be determined from the information given.