Re: are rational exponents defined for negative bases?


Arturo Magidin wrote:
> In article <1133117896.005166.266190@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> Kenneth Bull <kenneth.bull@xxxxxxxxx> wrote:
> >Are rational exponents defined for negative bases?
>
> Not generally, at least for real numbers. With complex numbers, you
> can always define them by using the complex natural logarithm and
> complex natural exponential.
>
> >Is this why we get extraneous solutions when dealing with radical
> >equations?
>
> I'm not sure what you mean here... But one reason one often gets
> 'extraneous' solutions is that when you do things like square an
> equation, you are performing a non-reversable operation: that is, the
> function that maps every real number to its square has no inverse. So
> while it is true that if x = y then x^2 = y^2, it is not true that if
> x^2 = y^2 then x=y (as opposed to, for example, the fact that if x=y
> then x+z = y+z, AND if x+z = y+z then x=y).
>

Another reason could be that any of the original "stuff" under any of
the roots end up being negative, which isn't allowed.

.

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