Associative property = (a+b) +c= a+(b+c) Distributive property = a×(b+c) =(a×b)+(a×c)

**Associative Property:**

The associative property is a fundamental concept in mathematics that deals with how we group or associate numbers when performing operations. It states that the grouping of numbers does not affect the result of addition or multiplication. In other words, when we add or multiply three or more numbers, we can change the grouping of the numbers without changing the final outcome.

**Addition:**
The associative property of addition can be expressed as (a + b) + c = a + (b + c). This means that when adding three or more numbers, we can regroup them in any way we want, and the sum will remain the same. For example:

(2 + 3) + 4 = 5 + 4 = 9
2 + (3 + 4) = 2 + 7 = 9

Regardless of the grouping, the sum is always 9.

**Multiplication:**
The associative property of multiplication is similar and can be expressed as (a * b) * c = a * (b * c). This means that when multiplying three or more numbers, we can change the grouping without changing the product. For example:

(2 * 3) * 4 = 6 * 4 = 24
2 * (3 * 4) = 2 * 12 = 24

No matter how we group the numbers, the product is always 24.

**Distributive Property:**

The distributive property is another fundamental concept in mathematics that relates multiplication and addition/subtraction. It allows us to simplify expressions by distributing the multiplication across terms inside parentheses.

**Multiplication over Addition:**
The distributive property of multiplication over addition states that a * (b + c) = a * b + a * c. This means that when we have a number multiplied by the sum of two or more terms, we can distribute the multiplication to each term individually. For example:

3 * (2 + 5) = 3 * 2 + 3 * 5 = 6 + 15 = 21

So, instead of performing the multiplication first, we can distribute the 3 to each term inside the parentheses and then perform the addition.

**Multiplication over Subtraction:**
The distributive property also applies to multiplication over subtraction. It states that a * (b - c) = a * b - a * c. This means that when we have a number multiplied by the difference of two terms, we can distribute the multiplication to each term individually. For example:

4 * (8 - 3) = 4 * 8 - 4 * 3 = 32 - 12 = 20

By distributing the 4 to each term inside the parentheses, we can simplify the expression and calculate the result.

In summary, the associative property deals with grouping of numbers in addition or multiplication, while the distributive property allows us to simplify expressions by distributing multiplication over addition or subtraction. These properties are essential in algebraic manipulations and help us solve complex mathematical problems efficiently.