When solving equations like this,
(-1)^m + 2^(m-1) = 9
Take each case on m and solve them. I.e in this case the first case would be, m being positive. and the second case would be m being negative.
Case 1: m is even If m is even, we can write (-1)^m as 1, and the equation becomes: 1 + 2^(m-1) = 9
Subtracting 1 from both sides: 2^(m-1) = 8
Since 2^3 = 8, we have: m - 1 = 3
Adding 1 to both sides: m = 4
Therefore, m = 4 is a solution for the case when m is even.
Case 2: m is odd If m is odd, we can write (-1)^m as -1, and the equation becomes: -1 + 2^(m-1) = 9
Adding 1 to both sides: 2^(m-1) = 10
Since there is no power of 2 that equals 10, there are no solutions for the case when m is odd.
Therefore, the only solution to the equation (-1)^m + 2^(m-1) = 9 is m = 4.