Question

25
consider the equation ( a^b = b^a ), where ( a ) and ( b ) are different numbers.

• what is the value of ( a )? enter your answer on the first line.
• what is the value of ( b )? enter your answer on the second line.

Explanation:

Step1: Analyze possible integer solutions

We need to find different integers \(a\) and \(b\) such that \(a^b = b^a\). Let's test small positive integers.

• Try \(a = 2\), \(b = 4\): \(2^4 = 16\), \(4^2 = 16\). So \(2^4 = 4^2\).
• Check if they are different: \(2

eq4\), which satisfies the condition.

Step2: Confirm the values

From the above test, we see that when \(a = 2\), \(b = 4\) (or vice - versa, but we can take this pair as a valid solution), the equation \(a^b=b^a\) holds for different numbers \(a\) and \(b\).

Answer:

For the value of \(a\): \(2\)
For the value of \(b\): \(4\)