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 [Maple Math] has argument between [Maple Math] and [Maple Math] , and the argument of [Maple Math] is [Maple Math] .
So for example:

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

[Maple Math]

> evalf(%);

[Maple Math]

> evalf(argument(%)/Pi);

[Maple Math]

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

> surd(-8,3);

[Maple Math]

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, and must be read in before being used:

> readlib(`&^`);

[Maple Math]

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

[Maple Math]

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

> assume(x<0);

[Maple Math] and [Maple Math] both odd: [Maple Math]

> x &^ (3/5);

[Maple Math]

[Maple Math] even, [Maple Math] odd: [Maple Math]

> x &^ (4/5);

[Maple Math]

[Maple Math] odd, [Maple Math] even: [Maple Math] is complex.

> x &^ (3/8);

[Maple Math]

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

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

[Maple Math]

> x &^ 0.375;

[Maple Math]

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

[Maple Plot]

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

[Maple Plot]

See also: ^ , &^ , surd

Maple Advisor Database R. Israel, 1998