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ASSOCIATIVE AXIOM
It is property of a binary operation ( operation that involves two operands )
ASSOCIATIVITY means within an expression containing two or more occurances in a row of the same associative operator, the order in which the operation are performed does not matter as long as the sequence of the operands is not changed.In associative axiom if we change or re-arrange the parenthesis the result will not change.
Commutative axiom and associative axiom are different. In commutative axiom order in which operands appears can be changed which is not allowed in case of associative axiom
eg: ( 6 + 3 ) + 1 = 6 + ( 3 +1 )
The above expression satisfy the associative axiom, whereas the below expression does not satisfy the rule since the order of operands are changed
( 6 + 3 ) + 1 = 3 + ( 6 + 1 ).
Types of associative axioms
In associative axiom we have 2 types
1) associative axiom for multiplication and
1) Associative axiom for multiplication
To multiply more than two numbers, we must combine them in pairs successively until all the numbers have been multiplied.
Example: To multiply a = 3 * 4 * 5 you may first choose the first 2 terms i,e 3 and 4 and multiply them . Now a = 12 * 5 here we have only 2 terms remaining which are multiplied to get the final result a = 60
The associative property of multiplication is one in which the result does not depend on how we pair the terms. We use small brackets ( ) to indicate the order of multiplication used .
example ( 2 * 3 ) * 4 means first we multiply 2 and 3 then multiply this result with 4.
No matter how terms are grouped in carrying out additions, the sum will always be the same:
The general form is given below
(a + b) + c = a + (b + c). This is called the associative axiom of addition.
eg :( 5 + 6 ) + 4 = 5 + ( 6 + 4 ) = 15
Here you can observe that even if we change or re-arrange the parenthesis the value does not change and it remains the same.

COMMUTATIVE AXIOM:
Algebra is the subdivision of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Simultaneously with geometry, analysis, topology, combinatory, and number theory, algebra is one of the main branches of pure mathematics.
The part of algebra is called elementary algebra is often part of the curriculum in secondary education and introduces the concept of variables representing numbers, such as addition.
Definition of Commutative Property:
The property refers to rules that designate a standard procedure or method to be followed. A proof is a expression of the truth of a statement in mathematics. The property or rules in mathematics are the result from testing the truth or validity of something by experiment or trial to establish a proof.
The commutative property is one of the essential properties of numbers. The word commute means exchange or swap over. Commutative property states that numbers may be added or multiplied in any order.
The Commutative means change the order in which you add or subtract numbers does not change the sum or product.
Commutative Property of Addition describes that changing the order of addends doesnt change the sum. Commutative can be described more formally. If + stands for an operation and A,B are elements from a given set, then + is commutative if, for all such elements.
a + b = b + a
Commutative Property of Multiplication defines that changing the order of factors doesnt change the product. Commutative can be described more formally. If * stands for an operation and A,B are elements from a given set, then * is commutative if, for all such elements.
a * b = b * a
Example problem:meaning of commutative
Example 1:
a =10, b = 12
a + b = b + a a * b = b * a
10 + 12 = 12 + 10 10 * 12 = 12 * 10
22 = 22 120 = 120
Example 2:
a = 18, b = 26
a + b = b + a a * b = b * a
18 + 26 = 26 + 18 18 * 26 = 26 * 18
44 = 44 468 = 468.