Function: &^ - "elementary" fractional powers

Calling sequence:

x &^ p;


x, p - any algebraic expressions. Numerical values of p should be rational numbers.


Maple's mathematics is largely based on complex numbers rather than real numbers. It uses the "principal branch" of fractional powers. In particular, if [Maple Math] is a negative real number and [Maple Math] is real but not an integer, [Maple Math] is not real. That is not the branch that is generally used in more elementary mathematics, where, for example, the cube root of a negative number would be taken to be negative. The surd function can be used to obtain these "elementary" roots, but it is comparatively clumsy notation and does not cover fractional exponents with numerators other than 1. The &^ operator is intended to address these shortcomings.

[Maple Math] is always a branch of [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:

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

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

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

When [Maple Math] is not known to be real, and [Maple Math] is a fraction, [Maple Math] is written as a surd.

When [Maple Math] is given in decimal form, it is converted to a fraction.

This operator is part of the Maple Advisor Database, and must be read in before use with readlib(`&^`); (note the back quotes).


> readlib(`&^`);

[Maple Math]

> evalc((-1)^(1/3)), surd(-1,3), (-1)&^(1/3);

[Maple Math]

> evalc((-1)^(1/4)),
evalc(surd(-1,4)), evalc((-1)&^(1/4));

[Maple Math]

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

[Maple Math]

> evalc((-1)^(5/8)),evalc((-1)&^(5/8));

[Maple Math]

> assume(n<0): n &^(3/5), n &^ (4/5);

[Maple Math]

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

[Maple Plot]

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

[Maple Plot]

See also: ^ , Fractional powers of negative numbers , surd

Maple Advisor Database R. Israel 1998