Explanation:

To determine the value of \(B^A\) when 8A5146B is divisible by 88, we need to ensure divisibility by both 8 and 11:

- Divisibility by 8: For 8A5146B to be divisible by 8, the last three digits, 46B, must be divisible by 8. Testing possible values of B, we find B = 4 because 464 is divisible by 8.

- Divisibility by 11: The alternating sum of the digits for divisibility by 11 is 8 - A + 5 - 1 + 4 - 6 + B. Substitute B = 4 into the equation: 8 - A + 5 - 1 + 4 - 6 + 4 = 14 - A. To be divisible by 11, 14 - A could be 0, 11, etc. Thus, A = 3.

- Calculate \(B^A\): With B = 4 and A = 3, \(B^A = 4^3 = 64\).

- Options:

- Option 1: 81: Incorrect, \(B^A\) is not 81.

- Option 2: 64: Correct, \(B^A = 64\).

- Option 3: 15: Incorrect, \(B^A\) is not 15.

- Option 4: 12: Incorrect, \(B^A\) is not 12.

Option 2: 64 is the correct choice.