Advice: Fractional powers of negative numbers

Maple's mathematics is largely based on complex numbers rather than real numbers. In particular, it uses the "principal branch" of fractional powers: the principal branch of has argument between and , and the argument of is .
So for example:

> evalc((-1)^(1/5));

> evalf(%);

> evalf(argument(%)/Pi);

The surd function provides "elementary" 'th roots, i.e. if is a negative number and is odd, then surd(x,n) is the real (negative) 'th root of .

> surd(-8,3);

In order to have a somewhat nicer syntax, as well as covering rational exponents with numerators other than 1, I have defined an infix-form operator &^ . This is part of the Maple Advisor Database.

> (-1)&^(2/5), (-1) &^ (3/5);

For positive , is the same as . For negative , if is a fraction we have the following cases:

> assume(x<0);

and both odd:

> x &^ (3/5);

even, odd:

> x &^ (4/5);

odd, even: is complex.

> x &^ (3/8);

When is not known to be real, and is a fraction, is written as a surd. When p is given in decimal form, it is converted to a fraction.

> x:= 'x': x &^ (3/8);

> x &^ 0.375;

> plot(x &^ (2/3), x = -1 .. 1);

> plot(x &^ (3/5), x = -1 .. 1);

Maple Advisor Database R. Israel, 1998