**Commutative and associative properties :**

The order in which you add or multiply numbers does not change their sum or product.

Commutative property of addition is nothing but the rule which says that, when we are doing addition, it doesn't matter, in which order the numbers are.

We can add a and b or b and a ... and we'll get the same answer.

**More clearly, a + b = b + a**

Commutative property of multiplication is nothing but the rule which says that, when we are doing multiplication, it doesn't matter, in which order the numbers are. We can multiply a and b or b and a ... and we'll get the same answer.

**More clearly, a x b = b x a**

The way you group three or more numbers when adding or multiplying does not change their sum or product.

Associative property of addition is nothing but the rule which says that, when we are doing addition, it doesn't matter, in which order the numbers are.

We can add a, b and c or a, b and c ... and we'll get the same answer.

**More clearly, (a + b) + c = a + (b + c)**

For any numbers a, b, and c,

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

Example :

(2 + 4) + 6 = 2 + (4 + 6)

Associative property of multiplication is nothing but the rule which says that, when we are doing multiplication, it doesn't matter, in which order the numbers are. We can multiply a, b and c or c, b and a ... and we'll get the same answer.

**More clearly,**(ab) c = a (bc)

Example :

(3 ⋅ 5) 4 = 3 (5 ⋅ 4)

**Example 1 :**

Simplify 3(4x + 2) + 2x

**Solution :**

= 3(4x + 2) + 2x

= 3(4x) + 3(2) + 2x

= 12x + 6 + 2x

= 12x + 2x + 6

= 14x + 6

**Example 2 :**

Simplify 7(ac + 2b) + 2ac

**Solution :**

= 7(ac + 2b) + 2ac

= 7(ac) + 7(2b) + 2ac

= 7ac + 14b + 2ac

= 7ac + 2ac + 14b

= 9ac + 14b

**Example 3 : **

Simplify 3(x + 2y) + 4(3x + y)

**Solution :**

= 3(x + 2y) + 4(3x + y)

= 3 (x) + 3 (2y) + 4 (3x) + 4 (y)

= 3x + 6y + 12x + 4y

= 3x + 12x + 6y + 4y

= 15x + 10y

**Example 4 : **

Write an algebraic expression for half the sum of p and 2q increased by three-fourths q. Then simplify.

**Solution :**

half the sum of p ==> (1/2) p

2q increased by three-fourths q ==> 2q + (3/4) q

= (1/2) p + 2q + (3/4) q

= (p/2) + 2q + (3q/4)

= (p/2) + (2q) x (4/4) + (3q/4)

= (p/2) + (8q/4) + (3q/4)

= (p/2) + (8q + 3q)/4

= (p/2) + 11q/4

**Example 5 : **

Simplify 8 ⋅ 1.6 ⋅ 2.5

**Solution :**

= 8 ⋅ 1.6 ⋅ 2.5

= (8 ⋅ 1.6) ⋅ 2.5

= 12.8 (2.5)

= 32

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