```On Dec 18, 2:13=A0pm, "William James" <> wrote:
> Fleck Jean-Julien wrote:
> > > That is WRONG, you cannot do that.
>
> > Well, I never said that you should do that, I just explained how Ruby
> > interpreted it...
>
> > > That only works for odd roots of negative numbers.
> > > The even root of negative numbers are imaginary.
>
> > > -27**3**-1 =3D> -3 **correct
> > > -27**2**-1 =3D> -5.19615242270663 **WRONG, its 5.196152i
>
> > Sure. That's quite a hint why ** does not accept a negative number
> > with a non integer exponent. To take into account all the special
> > cases, you should first see if your exponent is a rational and in that
> > case, see if the denominator is odd (after all due simplifications of
> > course). In this case (and only this case), you could try to decipher
> > a root for this negative number.
>
> > Cheers,
>
> def root base, n
> =A0 exp =3D 1.0/n
> =A0 return base ** exp if base >=3D 0 or n.even?
> =A0 -( base.abs ** exp )
> end
>
> --

Remember

i =3D (-1)^(1/2)

i^1 =3D  i
i^2 =3D -1
i^3 =3D -i
i^4 =3D  1
Then it repeats, for example: i^5 =3D i*(i^4) =3D i

For negative real value roots:

x =3D (-a)^(1/n) where n is odd integer =3D> x =3D -[a^(1/n)]

But for negative real value roots where n is even:

x =3D (-a)^(1/n) where n is even gives

x =3D |a^(1/n)|*(-1)^(1/n)
x =3D |a^(1/n)|*(i^2)^(1/n)
x =3D |a^(1/n)|*(i)^(2/n)
from e^(i*x) =3D cos(x) + i*sin(x) where x =3D PI/2
x =3D |a^{1/n)|*e^(PI*i/2)^(2/n)
x =3D |a^(1/n)|*e^(PI*i/n)
x =3D |a^(1/n)|*(cos(PI/n) + i*sin(PI/n)) for n even

(-256)^(1/2) =3D |256^(1/2)|*(cos(PI/2) + i*sin(PI/2))
=3D (16)(0 + i) =3D 16i

(-256)^(/4) =3D |256^(1/4)|*(cos(PI/4) + i*sin(PI/4))
=3D (4)*(0.707 + 0.707*i)
=3D 2.828 + i*2.828
=3D 2.828*(1+i)

Check in irb
> require 'complex'
> include Math

x =3D Complex(-256,0)

x**(1/2.0)
=3D> (9.79685083057902e-16+16.0i)

X**(1/4.0)
=3D> (2.82842712474619+2.82842712474619i)

```