Can a negative number be raised to a fractional power e.g $2.5$?

Question

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 $x^{7.5}$ ?

It will be discontinuous at infinite points in any finite interval. $(-5)^{7.5}$ would mean $(\sqrt{-5})^{15}$ — jonsno, Mar 06 '17 at 13:53
The function $x \mapsto x^a$ is only a polynomial if $a$ is a nonnegative integer. — Travis Willse, Mar 06 '17 at 13:54
@Travis I did not knew that. Can you point me to an explanation of this concept ? — yoyo_fun, Mar 06 '17 at 13:55
There is no explanation --- by definition a polynomial is sums of positive integer powers. Yoyo --- what is $\sqrt{-1}$? — JP McCarthy, Mar 06 '17 at 13:55
Possible duplicate of How do you compute negative numbers to fractional powers? — JP McCarthy, Mar 06 '17 at 13:56
This is just the definition of polynomial. The function $x \mapsto B x^a$ is called a power function. — Travis Willse, Mar 06 '17 at 13:56
So if the a can take values in a continuous interval it is considered a power function as opposed to a polynomial function ? — yoyo_fun, Mar 06 '17 at 13:59

score 1 · Answer 1 · answered Mar 06 '17 at 14:09

$(-5)^{7.5}$ 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.

score 1 · Answer 2 · answered Mar 06 '17 at 14:09

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.

Can a negative number be raised to a fractional power e.g $2.5$?

2 Answers2