Problem with fractional exponents and negative numbers

[Jan_D]

First we have to know how g^b is defined when b is not an integer.

So what would 0.47^3.1 be?

It is defined by means of the well-known functions e^x and ln(x), as follows:

g^b=e^[b*ln(g)]

This seems reasonable because e^[b*ln(g)]=e^[ln(g^b)]=g^b.

But when we use this definition in the case of g being negative we have to calculate the logarithm of a negative number, which does not exist within the realm of real numbers.

Within the realm of complex numbers such a logarithm exists though.

That is the reason why the result is a complex number, written as r+s*i, where r and s are real numbers, and i is the imaginary unit.