Howto simplify exponents

How can I make axiom to do 2^a2^(2a) -> 2^(3*a) ?

axiom
2^a*2^(2*a)

(1)

axiom
simplify %

(2)

But I cannot convince Axiom to do 2^(5a)/2^(4a) -> 2^a

axiom
2^(5*a)/2^(4*a)

(3)

axiom
simplify %

(4)

Unfortunately this seemingly simple transformation does not seem to be easy to perform in Axiom but I have found two possible ways to do this. The first involves the normalize() operation. Unfortunately normalize takes the process one step too far. This can be undone with a simple rule.

axiom
exprule:=rule exp(x*log(n)) == n^x

(5)

axiom
normalize %% 3

(6)

axiom
exprule %

(7)

The second approach uses a single rule to do the whole job.

axiom
fracrule:=rule n^m/n^p == n^(m-p)

(8)

axiom
fracrule %% 3

(9)

How about 2^a*4^a ?

axiom
2^a*4^a

(10)

axiom
simplify %

(11)

Here is one approach. First lets define a function that factors a power and a rule that applies this function.

axiom
powerFac(n,a) == reduce(*,[(t.factor)^(a*t.exponent) for t in factors(n)])

axiom
powerRule := rule n^a == powerFac(n,a)

(12)

Now we can use the rule and simplify the result

axiom
simplify powerRule (2^a*4^a)

Compiling function powerFac with type (PositiveInteger,Variable a)
       -> Expression Integer

Compiling function powerFac with type (PositiveInteger,Polynomial 
      Integer) -> Expression Integer

(13)

Apparently, simplifyExp yields the desired result

axiom
simplifyExp(2^a*2^(2*a))

(14)

• 2^a*4^a -> 2^(3a)

axiom
simplifyExp(2^a*4^a)

(15)

doesn't do it.

axiom
2**(3*a)*2**(4*a)

(16)

axiom
simplifyExp(2**(3*a)*2**(4*a))

(17)