Associative property

In Math, the associative property asserts that the order in which numbers are grouped by brackets (parentheses) has no effect on their sum or product when adding or multiplying them. Addition and multiplication are both affected by the associative property.

When an expression comprises three terms, the associative property states that they can be arranged in any way to solve the equation. The order in which integers are grouped has no bearing on the outcome of their operation. For addition and multiplication, the associative property holds true. The expression’s total and product remain unchanged.

The associative law asserts that the sum or product of any three or more integers is unaffected by the order in which the numbers are grouped by parentheses. It only applies to addition and multiplication. In other words, the outcome is the same if the same numbers are grouped in different ways for addition and multiplication. This can be stated as follows:

a × (b × c) = (a × b) × c and a + (b + c) = (a + b) + c.

Associative property of addition 

The total of three or more integers, according to the associative property of addition, remains the same regardless of how the numbers are arranged. Let’s say we’ve got three numbers: a, b, and c. the following formula will be used to express the associative property of addition in these cases:

Formula: (A + B) + C = A + (B + C)

Examples: 10 + 3 + 7 = 20 

Solution: Using addition’s associative property => (10 + 3) + 7 = 10 + (3 + 7) = 20

If we solve the left-hand side, we get 13 + 7 = 20 If we solve the right-hand side, 

we get 10 + 10 = 20

The total remains the same, despite the fact that the numbers are grouped differently.

Associative property of multiplication

Multiplication Is associative property asserts that the product of 3 or more integers remains the same regardless of how the numbers are arranged. The following formula can be used to express the associative property of multiplication:

Formula: (A × B) × C = A × (B × C)

Example: 10 x 3 x 7 = 210 

Solution: If we solve the left-hand side, we get  => (10 x 3) x 7 = 10 x (3 x 7) = 210

We get 30 x 7 = 210 when we solve the left-hand side of the expression. We get 10 x 21 = 210 when we solve the right-hand side.

Associative property to rewrite the expression

The associative aid in the rewriting of a difficult algebraic equation into a simpler one. When you use the associative property to rephrase an expression, you use parentheses to group a different pair of numbers together. 

Associative Property Verification

Only addition and multiplication are valid for the associative property. We’ll use this fact to compare the results of all of mathematics’ fundamental operations one by one.

For addition: we already know that (K + L) + M = K + (L + M) is the associative law. Consider the phrase 11 + 4 + 6 as an example. When we add the expressions together, we get the number 21.

For multiplication: We know that the associative law for multiplication is (K x L) x M = K x M. ( L x M). Take a fresh example, such as 2 x 13 x 5. The answer is 130 if we multiply the expression.

For Subtraction: Because the associative law of subtraction has no stated expression, let us argue that (K – L) – M = K – (L – M) is the associative law of subtraction.

For division: if the associative property for division holds true. Because the associative law of division does not have a well-written formula, let us assume that ( K ÷ L ) ÷ M = K ÷ ( L ÷ M ) will be the associative law of division.

Conclusion

In this article, we study the associative property of several binary operations, which indicates that rearrangement of parentheses in an expression has no effect on the outcome. Associativity is an acceptable rule of replacement for expressions in logical proof in propositional logic The associative property comes in handy when adding or multiplying many values. By grouping the problems together, we may break them down into smaller pieces. It facilitates and speeds up the addition and multiplication of multiple numbers.