sage math: how to combine or expand exponents in a symbolic expression?

Question

How can I combine or expand the exponents in an expression in sage? In other words, how can I have sage rewrite an expression from (a**b)**c to a**(b*c), or vise versa?

Examples:

sage: var('x y')
(x, y)
sage: assume(x, 'rational')
sage: assume(y, 'rational')
sage: combine_exponents( (x^2)^y )
x^(2*y)
sage: assume(x > 0)
sage: expand_exponents( x^(1/3*y) )
(x^y)^(1/3)

What I have already tried:

sage: b = x^(2*y)
sage: a = (x^2)^y
sage: bool(a == b)
True
sage: a
(x^2)^y
sage: simplify(a)
(x^2)^y
sage: expand(a)
(x^2)^y
sage: b
x^(2*y)
sage: expand(b)
x^(2*y)

Update:

simplify_exp (codelion's answer) works to convert from (a**b)**c to a**(b*c), but not the other way around. Is it possible to get sage to expand exponents as well?

Answer 1

1. Starting from Sage 6.5, to transform a into b, use the method canonicalize_radical.

   sage: a.canonicalize_radical()
   x^(2*y)
   

   Note that the four methods simplify_exp, exp_simplify, simplify_radical, radical_simplify, which had the same effect, are being deprecated in favour of canonicalize_radical. See Sage trac ticket #11912.

2. I don't know if there is a built-in function to transform b into a.

   You could define your own function like this:

   def power_step(expr, step=None):
       a, b = SR.var('a'), SR.var('b')
       if str(expr.operator()) == str((a^b).operator()):
           aa, mm = expr.operands()
           if step is None:
               if str(mm.operator()) == str((a*b).operator()):
                   bb = mm.operands().pop()
                   return (aa^bb)^(mm/bb)
               else:
                   return expr
           return (aa^step)^(mm/step)
       else:
           if step is None: return expr
           else: return (expr^step)^(1/step)
   

   Then you can decompose the powering into steps:

   sage: x, y = var('x y')
   sage: power_step(x^(2*y),y)
   (x^y)^2
   sage: power_step(x^(2*y),2)
   (x^2)^y
   

   Note that if you don't specify the step, it won't always pick the first one that is displayed.

   sage: power_step(2^(x*y))
   (2^y)^x
   sage: power_step(x^(2*y))
   (x^2)^y
   

Answer 2

You can use the simplify_exp() function. So for your example do the following:

sage: a.simplify_exp()
x^(2*y)