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, and must be read in before being used:
> readlib(`&^`):
> (-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);
See also: ^ , &^, surd
Maple Advisor Database R. Israel, 1998