Is it possible to raise a negative integer or other negative number to a power that is not an integer ?
I am asking because when plotting some polynomial functions the negative numbers were not plotted and I am curious why.
Why is it possible to raise $-5$ to the power of $7$ but it is not possible to raise $-5$ to the number of $7.5$ and is it also impossible to plot the function $x^{7.5}$ ?
On
The answer is yes, you can.
However, to do so requires working with complex numbers and a definition of the principal $\alpha$th power is
$$ z^\alpha = e^{\alpha\ \text{Log}\ z}$$
where $z, \alpha \in \mathbb C$, and $z \ne 0$
Obviously you will run in to problems when wanting to plot graphs of real functions if the answer is complex valued.
$(-5)^{7.5}$ is not a real number. This is because if $x=(-5)^{7.5}$, that would imply $x^2=(-5)^{15}=-(5^{15}),$ which is not possible for $x$ real, since squares of real numbers cannot be negative.
You can raise a negative number to some fractional powers and get a real number answer, but only if the denominator of the fraction (in its lowest terms) is odd. For example, if you wanted $x=(-5)^{7/3}$, you would need $x^3=(-5)^7=-5^7$. This is possible, because cubes can be negative, and you just get $-(5^{7/3})$.
It does make sense to talk about $(-5)^{7.5}$ as an imaginary number, but since it's not real you can't plot it.