How to simplify exponents

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

(1) -> 2^a*2^(2*a)

(1)

simplify %

(2)

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

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

(3)

simplify %

(4)

Unfortunately this seemingly simple transformation does not seem to be easy to perform in FriCAS 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.

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

(5)

normalize %% 3

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exprule %

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

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

(8)

fracrule %% 3

(9)

How about 2^a*4^a ?

2^a*4^a

(10)

simplify %

(11)

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

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

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

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

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Now we can use the rule and simplify the result

simplify powerRule (2^a*4^a)

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Apparently, simplifyExp yields the desired result

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

(14)

• 2^a*4^a -> 2^(3a)

simplifyExp(2^a*4^a)

(15)

doesn't do it. But normalize plus cleanup works:

normalize(2^a*4^a)

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exprule %

(17)

powerRule %

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2^(3*a)*2^(4*a)

(19)

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

(20)