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When you multiply or add any three real numbers, the grouping (or association) of the numbers has no effect on the outcome, according to the associative rules.

a.(b.c) = (a.b).c

As a result of the aforementioned statement, we may deduce that in addition and multiplication, it makes no difference how we group or associate the numbers. The associative law only applies to the addition and multiplication of all real numbers; it does not apply to subtraction or division.

** **

The associative property of multiplication asserts that no matter how the numbers are arranged, the product of three or more integers stays the same. For instance, 3 *(5* 6) Equals (3* 5) *6. The result of both statements remains 90 regardless of how the integers are arranged.

Let us understand the property by taking a look at the proof.

Prove that:1×(2×3) = (1×2)×3

Taking LHS first,

1×(2×3) = 1×6 = 6

Now let us take RHS

(1×2)×3 = 2×3 = 6

Hence, if we compare,

LHS = RHS

Therefore,

1×(2×3) = (1×2)×3. Proved.

Prove that: 4*((-7) *8) = (4*(-7)) *8

Taking the LHS first,

4*((-7) *8)= 4*((-56))=-224

Taking the RHS second;

(4*(-7)) *8 = ((-28) *8) = -224

Hence, we prove that LHS=RHS

Thus, 4*((-7) *8) = (4*(-7)) *8

While associativity is true for conventional arithmetic with real or imaginary numbers, it does not hold true for other applications, such as non-associative algebras.

Furthermore, associative law is not applicable to subtraction and division. This is because the change in the grouping of numbers changes the results.

For example, 3 – (4 – 5) and (3 – 4) – 5 are not equal, since,

3 – (4 – 5) = 3 – (-1) = 3 + 1 = 4

(3 – 4) – 5 = (-1) – 5 = -1 – 5 = -6

Thus, 4 ≠ -6

The definition of associative law or property may be used to determine the formula. The addition or multiplication of three integers is defined as being independent of their grouping or relationship. Or, to put it another way, grouping or combining three integers while adding or multiplying them produces the same result.

Consider the three integers A, B, and C. Then, according to the law;

A+(B+C) = (A+B) +C

A × (B × C) Equals (A × B) × C

The multiplication operation follows the associative law, which states that no matter how numbers are combined, the end product will be the same. If X, Y, and Z are all three digits, then

X*(Y*Z) = (X*Y)*Z = X*Y*Z = X*Y*Z

If you wish to know about the associative law of multiplication and have a good grasp of it, then you should take a look at examples.

**Example 1:**

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

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

**Example 2**

4* (5* (-9)) = 4* (-45) = -180

(4* 5) * (-9) = 20* (-9) = -180

In this section, you will get to see an example of the associative property of multiplication.

3× (-7×9) = 3* -63= -189

(3* -7) *9 = -21* 9= -189

Some of the examples of associative law are:

5* (5* 6) = 5* 30= 150

(5*5) *6= 25* 6= 150

The associative law also works for addition.

(5 + 6) + 8= 11+8=19

5+ (6+8) = 5+ 14= 19** **

The associative property is a mathematical principle that states that the way elements are arranged in a multiplication problem has no effect on the product.

In order to prove associative law of multiplication, we have to** **show that both sides of the equation satisfy each other.

A * (B*C) = (A*B) *C= ABC

For example,** **-2 * (6*8)= ((-2) * 6) *8= -96

The associative property is a mathematical principle that states that the way elements are arranged in a multiplication problem has no effect on the product.

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