#5706 (-2*%i)^(1/3) simplifies to the wrong cube root

[David Scherfgen]

By Claude:

With the default domain : real and radexpand : true, an odd root of a product is split into odd roots of its factors, and a negative factor gets its real root. That is only right when all factors are real. If a factor is complex, the root of the product is the principal one, and splitting gives a different cube root:

(%i1) (-2*%i)^(1/3);
(%o1) -(-1)^(1/6)*2^(1/3)
(%i2) float(rectform(%));
(%o2) -0.6299605249474366*%i-1.0911236359717214
(%i3) float(rectform(exp(log(-2*%i)/3)));
(%o3) 1.0911236359717214-0.6299605249474366*%i

The same happens with other complex factors, such as (-1)^(1/4), a symbol declared imaginary, or the square root of a negative symbol:

(%i4) (-2*(-1)^(1/4))^(1/3);
(%o4) -(-1)^(1/12)*2^(1/3)
(%i5) declare(j, imaginary)$
(%i6) (-2*j)^(1/3);
(%o6) -2^(1/3)*j^(1/3)
(%i7) assume(xn < 0)$
(%i8) (-2*sqrt(xn))^(1/3);
(%o8) -2^(1/3)*xn^(1/6)