Why is addition and multiplication associative but not exponentiation and beyond? The most similar question I found on math.se was concerning commutativity (Why are addition and multiplication commutative, but not exponentiation?). If this ostensibly simple (but not obvious to me) question has already been discussed, please post the link and I apologize for duplicate. I am wondering too how other rules of arithmetic in basic and other algebras (distributivity, transitivity, additive and multiplicative inverses, etc) apply or don't apply to operators and any rhyme to their properties' applicability or hierarchy.
associativity of addition: (α+β)+γ ≡ (α+β+γ) := α+β+γ ≡ α+(β+γ)
associativity of multiplication: (a×b)×c := (ab)⋅c = abc ≡ a(bc)
non-associativity of exponentiation: A^B^C := (A^B)^C := $(A^B)^C$ = ${A^B}^C$ ≢ $A^{B^C}$ = $A^{(B^C)}$ := A^(B^C)

The real question is why $\times$ would associate; properties like that aren't free.
As with the question of commutativity, that $\times$ inherits the property from $+$, but exponentiation doesn't repeat this, can be explained in terms of $\times$ on $\Bbb R^+$ being isomorphic to $+$ on $\Bbb R$. (Distributivity of $\times$ over $+$ extends this to $\times$ on the whole of $\Bbb R$ being commutative & associative.)
