A property of binary operations: an associative operation does not depend on the order in which operations are performed, so we may dispense with brackets and not have ambiguity. The operation @ is associative if for all x, y and z (in the space on which @ operates), x@(y@z) = (x@y)@z. This is different from being commutative! For instance, if we define the "leftmost" operator x@y=x, we get an operation which is associative, but not commutative.
Addition and multiplication on real numbers are associative (3+(4+5) = 12 = (3+4)+5; 3*(4*5) = 60 = (3*4)*5), but subtraction, division and exponentiation are not (3-(4-5) = 4 ≠ -6 = (3-4)-5; 3/(4/5) = 15/4 ≠ 3/20 = (3/4)/5; 3^(4^5) ≠ 3^20 = (3^4)^5; see below for exact values).
3^(4^5) = 373391848741020043532959754184866588225409776783734007750636931722079\
040617265251229993688938803977220468765065431475158108727054592160858\
581351336982809187314191748594262580938807019951956404285571818041046\
681288797402925517668012340617298396574731619152386723046235125934896\
058590588284654793540505936202376547807442730582144527058988756251452\
817793413352141920744623027518729185432862375737063985485319476416926\
263819972887006907013899256524297198527698749274196276811060702333710\
356481
(3^4)^5 = 3486784401
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