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

Question

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

Kyan Hamilton · Accepted Answer

(  −  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&#039;s not real you can&#039;t plot it.

Cristofer Graves · Accepted Answer

The answer is yes, you can.However, to do so requires working with complex numbers and a definition of the principal   αth power is      z    α    =      e          α             Log             z      where   z  ,  α  ∈      C  , and   z  ≠  0Obviously you will run in to problems when wanting to plot graphs of real functions if the answer is complex valued.

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