#5705 ((-1)^(1/6))^2 simplifies to -1

[David Scherfgen]

By Claude:

With the default domain : real, Maxima reads (-1)^(p/q) with odd q as the real root, so (-1)^(1/3) simplifies to -1. (-1)^(1/6) has no real value and is the principal root exp(%i*%pi/6), which is also how rectform and float evaluate it. But a power or a product of such roots is folded into (-1)^(1/3), which is then read as the real root:

(%i1) ((-1)^(1/6))^2;
(%o1) -1
(%i2) (-1)^(1/6)*(-1)^(1/6);
(%o2) -1
(%i3) ((-8)^(1/6))^2;
(%o3) -2

The correct value of the first two is exp(%i*%pi/3), that is 1/2+sqrt(3)*%i/2:

(%i4) expand(float(rectform((-1)^(1/6)))^2);
(%o4) 0.8660254037844386*%i+0.4999999999999999

As ratsimp multiplies such powers out, it often returns wrong values for expressions with (-1)^(1/6) or (-1)^(1/10). Here the difference should be 0:

(%i5) r : ratsimp(1/(1+(-1)^(1/6))), algebraic : true;
(%o5) -((%i+(-1)^(5/6)+(-1)^(1/6)-1)/2)
(%i6) float(rectform(r - 1/(1+(-1)^(1/6))));
(%o6) 5.551115123125783e-17-0.8660254037844386*%i

With domain : complex, all of these are correct.