A product is the result of carrying out the mathematical operation of multiplication. When you multiply numbers together, you get their product. The other basic arithmetic operations are addition, subtraction and division, and their results are called the sum, the difference and the quotient, respectively. Each operation also has special properties governing how the numbers can be arranged and combined. For multiplication, it's important to be aware of these properties so that you can multiply numbers and combine multiplication with other operations to get the right answer.

#### TL;DR (Too Long; Didn't Read)

The product meaning in math is the result of multiplying two or more numbers together. To get the right product, the following properties are important:

- The order of the numbers doesn't matter.
- Grouping the numbers with brackets has no effect.
- Multiplying two numbers by a multiplier and then adding them is the same as multiplying their sum by the multiplier.
- Multiplying by 1 leaves a number unchanged.

## The Meaning of the Product of a Number

The product of a number and one or more other numbers is the value obtained when the numbers are multiplied together. For example, the product of 2, 5 and 7 is 2 × 5 × 7 = 70. While the product obtained by multiplying specific numbers together is always the same, products are not unique. The product of 6 and 4 is always 24, but so is the product of 2 and 12, or 8 and 3. No matter which numbers you multiply to obtain a product, the multiplication operation has four properties that distinguish it from other basic arithmetic operations, Addition, subtraction and division share some of these properties, but each has a unique combination.

## The Arithmetic Property of Commutation

Commutation means that the terms of an operation can be switched around, and the sequence of the numbers makes no difference to the answer. When you obtain a product by multiplication, the order in which you multiply the numbers does not matter. The same is true of addition. You can multiply 8 × 2 to get 16, and you will get the same answer with 2 × 8. Similarly, 8 + 2 gives 10, the same answer as 2 + 8.

Subtraction and division don't have the property of commutation. If you change the order of the numbers, you'll get a different answer. For example, 8 ÷ 2 equals 4, but 2 ÷ 8 equals 0.25. For subtraction, 8 - 2 equals 6 but 2 - 8 equals -6. Division and subtraction are not commutative operations.

## The Distributive Property

Distribution in math means that multiplying a sum by a multiplier gives the same answer as multiplying the individual numbers of the sum by the multiplier and then adding. For example, 3 × (4 + 2) = 18, and (3 × 4) + (3 × 2) also equals 18. Adding before multiplying gives the same answer as distributing the multiplier over the numbers to be added and then multiplying before adding.

Division and subtraction don't have the distributive property. For example, 3 ÷ (4 - 2) = 1.5, but (3 ÷ 4) - (3 ÷ 2) = -0.75. Subtracting before dividing gives a different answer than dividing before subtracting.

## The Associative Property for Products and Sums

The associative property means that if you are performing an arithmetic operation on more than two numbers, you can associate or put brackets around two of the numbers without affecting the answer. Products and sums have the associative property while differences and quotients do not.

For example, if an arithmetical operation is performed on the numbers 12, 4 and 2, the sum can be calculated as (12 + 4) + 2 = 18 or 12 + (4 + 2) = 18. A product example is (12 × 4) × 2 = 96 or 12 × (4 × 2) = 96. But for quotients, (12 ÷ 4) ÷ 2 = 1.5, while 12 ÷ (4 ÷ 2) = 6, and for differences (12 - 4) - 2 = 6 while 12 - (4 - 2) = 10. Multiplication and addition have the associative property while division and subtraction do not.

## Operational Identities – Difference and Sum vs. Product and Quotient

If you perform an arithmetic operation on a number and an operational identity, the number remains unchanged. All four basic arithmetic operations have identities, but they are not the same. For subtraction and addition, the identity is zero. For multiplication and division, the identity is one.

For example, for a difference, 8 - 0 = 8. The number remains identical. The same is true for a sum, 8 + 0 = 8. For a product, 8 × 1 = 8 and for a quotient, 8 ÷ 1 = 8. Products and sums have the same basic properties except that they have different operational identities. As a result, multiplication and its products have a unique set of properties that you have to know to get the right answers.

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