About rewrited expressions for power, log, exp and so.

Hello,

I try to understand how axiom is sure in Expression domain.
and what suppositions axiom does.

It seems that axiom makes a lot of fuzzy simplifications.
Of corse it's possible to delete such rules in elemntry.spad.

But then how can I test if standard exemples of axiom continue to be right
with a new elemntry.spad ?

    sqrt (u^2)  ---> sqrt (u^2) I agree  sqrt ((-1)^2) = 1
but (u^a)^(1/a) ---> u          not coherent with a=2

    (u^a)^2     ---> u^(2a)     is right
but (u^a)^b     ---> u^(ab)     I prefer (u^a)^b
and (u^2)^a     ---> u^(2a) 

    u^a*u^b     ---> u^a u^b    is right but u^(a+b) is also possible.

The question is the same for asin (sin x), log (exp x), etc.
For sin it's line 486 in elemntry.spad

What rules might apply axiom for expressions ?
Is there a reason that theses rules aren't usual mathematic rules ?
What is the axiom policy ? What is your advice ?

Have a nice day.

Francois