How do you multiply negative numbers to the imaginary power?

Question

please, i really need this.

Answer 1

In general raising any number to a complex power is not well-defined. That is, there can be multiple values. With negative numbers it is particularly hard, since the standard branch cut of the complex logarithm excludes the negative numbers. However, this doesn't really matter. Let's say that we just change the branch cut and choose:

log(-a) = log(a) + i*pi for any positive real number a, and log(a) the normal real logarithm. This is a fine definition since it satisfies: e^{log(-a)} = -a.

Then, if a is still positive, for example:

(-a)^i = e^{i*log(-a)} = e^{i*(log(-a)} = e^{i*log(a) - pi}

so

(-a)^i = e^{-pi}*(cos(log(a)) + i sin(log(a)))

Answer 2

Provide an example? Or let us know the problem? The wording is kind of vague.
Like.. a^i * -a^2i? = -a^3i

Answer 3

Very carefully.