Associative property:
Associative law states that the order of grouping the numbers does not matter. This law holds for addition and multiplication but it doesn’t hold for subtraction and division. This can be observed from the following examples.
Addition:
a+ (b+c) = (a+b) + c
Example:
2+ (3+4) = (2+3) + 4
2+7 = 5+4
9 = 9.
So, associative law holds for addition.
Subtraction:
a-(b-c) ≠ (a-b) – c.
Example:
2- (3-4) = (2-3) – 4
2 + 1 = -1-4
3 = -5, which is not true.
So, associative law doesn’t hold for subtraction.
Multiplication:
a x (b x c) = (axb) x c
Solution:
2 x (3×4) = (2×3) x 4
2 x 12 = 6 x 4
24 = 24.
So, associative law holds for multiplication.
Division:
a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c
Example:
8 ÷ (4 ÷ 2) = (8÷4) ÷ 2
8 ÷ 2 = 2 ÷ 2
4 =1, which is not true.
So, associative law doesn’t hold for division.
Distributive property:
This property is used to eliminate the brackets in an expression. The distributive property states that each term inside the bracket should be multiplied with the term outside. This property is very useful while simplifying the expressions and solving the complicated equations.
Distributive property over addition:
a(b+c) = ab + ac
a(b-c) = ab – ac
Here, the terms which are inside the bracket (b and c) are multiplied with the outside term (whih is a)
Example-1:
2(5+3) = 2×5 + 2×3
2 x 8 = 10 + 6
16 = 16.
So, distributive property over addition is proved.
Example-2:
2(5-3) = 2×5 – 2×3
2 x 2 = 10 – 6
4=4.
So, distributive property over subtraction is proved.
Commutative property:
Commutative property states that there is no change in result though the numbers in an expression are interchanged. Commutative property holds for addition and multiplication but not for subtraction and division.
Addition:
a+b = b+a.
Example:
1+2 = 2+1
3=3, which is true.
Subtraction:
a-b ≠ b-a.
Example:
1-2 = 2-1
-1=1, which is not true.
Multiplication:
a x b = b x a
Example:
2 x 3 = 3 x 2
6 = 6, which is true.
Division:
a ÷ b ≠ b ÷ a
Example:
4 ÷ 2 = 2 ÷ 4
2 = ½, which is not true.
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