Equations are the language of mathematics by which the most complex and fascinating aspects of the real world can be expressed elegantly using symbols and operators. The fundamental point of these equations is the final value we get after solving them. However, as we advance to higher developed mathematics we face harder to solve equations. Mathematics also provides us with certain manipulative principles to help with this problem. We can perform different arithmetic operations using them and solve complex equations with ease. The three main properties that form the backbone of math are:

- Associative Property
- Commutative Property
- Distributive Property

In the following article, we’ll take a closer look at the associative property of multiplication.

**What is the associative property of multiplication?**

Before we answer the question of what is the associative property of multiplication, let us first understand what associative means and some other elementary concepts.

To “associate” means to connect or join with something. In the context of mathematical operations, this means that the way numbers are grouped under a mathematical operation does not change the result. By grouping, we mean how the brackets are placed in the given algebraic expression. In simpler terms, the mathematical operation result remains the same irrespective of how the numbers are grouped.

Now that we’ve learnt a bit about grouping, let us answer the question of what is the associative property of multiplication in detail.

**Associative property of multiplication definition**

As per the associative property of multiplication definition, if three or more terms are multiplied together, we obtain the same end answer irrespective of how the terms are grouped.

Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups and categories) explicitly require their binary operations to be associative.

However, many significant and intriguing mathematical operations are non-associative. A few examples of this include subtraction, division, exponentiation, and vector cross product. We shall see these operations in detail a little later on. A real-world example of this is the addition of floating-point numbers in computer science which is not associative. The selection of how we associate an expression has a significant impact on the rounding error.

Although the associative property of multiplication definition is quite the same as the commutative property, they are not the same. Commutative property addresses whether the order of the terms will have any effect on the result. For example, operations such as function composition and matrix multiplication are associative but usually not commutative.

**The formula for the associative property of multiplication**

The easiest formula for understanding the associative property of multiplication is (a × b) × c = a × (b × c). This expression helps us realize that the placement of brackets is immaterial for the final result. Grouping of terms helps to make smaller components that make the multiplication process easier.

**Generalized Associative Law**

Associative binary operations deliver the same result regardless of how we place the pair of parentheses in the expression. For example, we can write a product of four elements without changing the order of factors in five possible ways:

- ((ab)c)d
- (ab)(cd)
- (a(bc))d
- a((bc)d)
- a(b(cd))

As the product operation is associative, the law states all the above formulas yield the same result. Unless the formula without the parenthesis has a different meaning, it can be considered unnecessary, and the net product can be simply written as abcd.

If we keep increasing the number of elements in the operations, the number of ways to place the parentheses also increases rapidly. However, ultimately they remain unnecessary for disambiguation.

An example where this does not work is the logical biconditional . It is associative, thus A↔ (B↔C) is equivalent to (A↔B) ↔C, but A↔B↔C most commonly means (A↔B and B↔C), which is not equivalent.

**Points to be noted on the associative property of multiplication:**

Here are some important features of the associative property of multiplication:

- The associative property of multiplication is only applicable to expressions containing at least three terms.
- The numbers that are grouped inside the parenthesis are considered to be a single unit.
- There is another associative property for the addition operation. However, Subtraction and Division do not follow the associative property.

**Associative property of multiplication example No. 1**

Let us understand this property with a simple example.

Consider the expression given below.

2 * 4 * 6

We’ll solve this expression by two different grouping methods.

For the first process, we will group the numbers 2 and 4. In the second process, we will group the numbers 4 and 6.

(2 * 4) * 6 | 2 * (4 * 6) |

=(8) * 6 | =2 * (24) |

=8 * 6 | =2 * 24 |

=48 | =48 |

As we can see, we get the same result from both processes.

**Associative property of multiplication example No. 2**

Let us look at another example to verify this property.

Consider the expression given below.

10 * 5 * 7

We’ll solve this expression by two different grouping methods.

For the first process, we will group the numbers 10 and 5. In the second process, we will group the numbers 5 and 7.

(10 * 5) * 7 | 10 * (5 * 7) |

=(50) * 7 | =10 * (35) |

=50 * 7 | =10 * 35 |

=350 | =350 |

As we can see, we get the same result from both processes.

**Associative property of multiplication example No. 3**

This time we will solve an expression that contains four numbers.

Consider the expression given below.

2 * 3 * 7 * 11

We’ll solve this expression by three different grouping methods.

For the first process, we will group the numbers 2 and 3. In the second process, we will group the numbers 3 and 7. At the last one, we’ll group 7 and 11.

(2 * 3) * 7 * 11 | 2 * (3 * 7) * 11 | 2 * 3 * (7 * 11) |

=(6) * 7* 11 | =2 * (21) * 11 | =2 * 3 * (77) |

=6 * 7 * 11 | =2 * 21 * 11 | =2 * 3 * 77 |

=42 * 11 | =42 * 11 | =6 * 77 |

=462 | =462 | =462 |

As we can see, we get the same result from both processes.

**Associative property of multiplication example No. 4**

Rational numbers also follow the associative property of multiplication.

Suppose ab,cd andef are rational, then the associative property of multiplication can be written as:

ab* cd* ef= ab* cd* ef

Consider the expression given below.

12 *34 *56

We shall solve it by two processes as we have done before.

12 * 34*56 | 12 *34*56 |

=38*56 | =12 *1524 |

=38 *56 | =12 *1524 |

=1548 | =1548 |

=516 | =516 |

As we can see, we get the same result from both processes.

**Associative Property for Addition**

As we mentioned before, just like for multiplication, the associative property is also valid under the addition operation.

For example, consider the expression: 2 + 5 + 10

Let us make two groups of numbers 2 and 5, and then another of 5 and 10.

(2 + 5) + 10 | 2 + (5 + 10) |

=(7) + 10 | =2 + (15) |

=7 + 10 | =2 + 15 |

=17 | =17 |

Hence we have proved that the associative property for addition is perfectly valid.

**Associative Property for Subtraction**

As we mentioned before, the associative property does not hold for the subtraction operation. Let us prove it with a problem.

For example, consider the expression: 3 – 2 – 1

Let us make two groups of numbers 3 and 2, and another one of 2 and 1.

(3 – 2) -1 | 3 – (2 – 1) |

=(1) – 1 | =3 – (1) |

=1- 1 | =3 – 1 |

=0 | =2 |

Since, 0≠2,

Hence, we have proved that associative property is not valid for the subtraction operation.

**Associative Property for Division**

As we mentioned before, the associative property does not hold for the division operation. Let us prove it with a problem.

For example, consider the expression: 100 ÷ 10 ÷ 5

Let us make two groups of numbers 100 and 10, and another of 10 and 5.

(100÷10)÷5 | 100÷(10÷5) |

=(10)÷5 | =100÷(2) |

=10÷5 | =100÷2 |

=2 | =50 |

Since, 2≠50,

Hence, we have proved that associative property is not valid for the division operation.

**Associative Property for Vector Cross Product**

As we mentioned before, the associative property does not hold for the vector cross product operation. Let us prove it with a problem.

For example, consider the expression: i×i×j

Let us make two groups of unit vectors i and i, and another one of i and j.

(ii)j | i(ij) |

(0)j | i(k) |

0j | ik |

0 | -j |

Since 0 ≠-j,

Hence, we have proved that associative property is not valid for the vector cross product operation.