Associative and Commutative Property of Addition and Multiplication

In mathematics associative and commutative properties are defined as the laws that are applied to the method of addition or multiplication while solving problems. The associative property is said to be the property that can regenerate a group of numbers that will fetch the same answer regardless of the parenthesis. The commutative property is stated as the property that allows moving of numbers around any place within the equation but still arriving at the same answer. Associative properties include 3 or more than 3 numbers that are considered as one unit. The associative property is related with addition and multiplication. However, the associative property is not applied in case of subtraction or division. In mathematics, commutative property is applied in case of binary operation however the change in the order of the operands does not changes the result of the operation. The detailed description of the specific properties is described as follows:

Associative property:

Associative property comes from the word associate meaning group that refers to the grouping of numbers as well as variables in algebra. With the help of this property one can form any re grouping of numbers while arriving at the same answers.

The following equation shows the associative property of addition:

(a + b) +c = a+ (b+c)

Example:

(2 + 4)+ 5 = 2+(4+5)

The equation of the associative property of multiplication is shown as follows:

(a × b) × c = a × (b × c)

Commutative property:

In mathematics, the commutative property comes from the word commute that means moving around. The rule for commutative property is stated as the ability to move the number or variables in algebra around any position and still getting the same answer in all position.

The equation of commutative property of addition is stated as follows:

a + b = b + a

4 + 2 = 2 + 4

The equation for the commutative property of multiplication is stated as follows:

a × b = b × a

3 × 2 = 2 × 3

While following the associative and commutative property of addition and multiplication, rearranging the numbers makes it easier to solve problems such as:

2 × 16 × 5

= (2 × 5) × 16 = 10 × 16 = 160.

Use of Associative and commutative properties of addition and multiplication in rewriting algebraic expression

In algebra, Commutative property of addition is stated as-

If a and b are real numbers then, a+b=b+a

Commutative property of multiplication is stated as-

If a and b are real numbers then, a⋅b=b⋅a

Like arithmetic, in algebra also if the order of the numbers are changed while adding or multiplying, the end result or the answer will remain the same in all cases.

Example of using commutative property to rewrite the algebraic expressions related to addition:

1. -1+3

Using commutative property to change the order of numbers in addition,

−1+3=3+(−1)

Example of using commutative property to rewrite the algebraic expressions related to multiplication:

1. 2. 4⋅9

Using commutative property to change the order of numbers in multiplication,

4⋅9=9⋅4

In algebra, Associative property of addition is stated as-

If a,b,c are real numbers, then (a+b)+c=a+(b+c)(a+b)+c=a+(b+c)

Associative property of multiplication is stated as,

If a,b and c are real numbers then, (a⋅b)⋅c=a⋅(b⋅c)

Example of using associative property to rewrite the algebraic expressions related to addition:

1. 3+0.6) +0.4

While changing the grouping of the numbers,

(3+0.6) +0.4=3+(0.6+0.4)

While adding, the addition becomes easier when added from the right as 0.6 + 0.4 =1.

Example of using associative property to rewrite the algebraic expressions related to multiplication:

1. ( -4. 2).15

While changing the grouping of numbers we get,

1. ( -4. 2).15

=-4. (2.15)

Examples:

1. Rewrite (−3⋅1)⋅4(-3⋅1)⋅4 using the Associative Property of Multiplication?

Ans: (−3⋅1)⋅4=−3(1⋅4)

1. Using the Associative Property of Multiplication to simplify: 6(3x).

Ans: 6(3x)

While changing the grouping of the numbers: (6.3) x

Multiplying the parenthesis,

=18 x.

1. Use the associative property of multiplication to simplify.

6(−9n)

Ans: -54 n.

1. Rewrite −7+3-7+3 using the Commutative Property of Addition.

Ans: −7+3=3+−7

1. Use the commutative property of addition to write an equivalent expression to 7x+8y.

Ans: 8y+7x

1. Use the associative property of addition to write an equivalent expression to 3x+(5y+6)

Ans: (3x+5y)+6

1. Use the commutative property of multiplication to write an equivalent expression to (3a)(4b)

Ans: (4b)(3a)

1. Use the associative property of multiplication to write an equivalent expression to v(wXx)

Ans: (vw)Xx