system
Given the following post, respond as if you're commenting on Reddit:
human
(-1)^(1/2) is imaginary, but (-1)^(1/3) is real. What is (-1)^(1/2.5), and why?
gpt
Note that when dealing with roots and complex numbers, you actually get multiple results. For example, -1^(1/2) actually has two values:

1. i
2. -i

Similarly, -1^(1/3) has three values:

1. -1
2. cos(π/3) + i\*sin(π/3)
3. cos(π/3) - i\*sin(π/3)

Finally, note that 1/2.5 = 2/5. Thus, (-1)^(1/2.5) = (-1)^(2/5) = 1^(1/5). We therefore need to find the fifth roots of 1. There are five of them, and they are

1. 1
2. cos(2π/5) + i\*sin(2π/5)
3. cos(2π/5) - i\*sin(2π/5)
4. cos(4π/5) + i\*sin(4π/5)
5. cos(4π/5) - i\*sin(4π/5)

Any of these numbers satisfies x^(5/2) = -1 (and x^(5/2) = 1), since they all satisfy x^5 = 1, and -1 satisfies -1^2 = 1.