# What is the Product in Math? - Definition & Overview

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• 0:00 What Is a Product?
• 0:20 How to Find the Product
• 1:35 Properties of Multiplication
• 2:45 Special Products
• 3:20 Lesson Summary
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Lesson Transcript
Instructor
Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

In mathematics, the term 'product' refers to the answer to a multiplication problem. This lesson will define the term in more detail and give some examples of how it is used. Then, there will be a quiz to test your understanding.

## What Is a Product?

When speaking mathematically, the term product means the answer to a multiplication problem.

For example:

5 * 3 = 15

15 is the product

The term product first showed up in England in the 1400s and comes from the Latin word productum, which means 'to produce.'

## How to Find the Product

Multiplication is often referred to as repeated addition because what the multiplication problem is telling you is that you have a certain number of groups of something, all containing a specific number. Confused yet? Here's an example.

You have 3 bags of candy, and each bag contains 5 pieces of candy. How many pieces of candy do you have?

There are two ways to solve this problem. The first is by adding up the pieces of candy:

5 + 5 + 5 = 15

The other way to solve is to use multiplication, because you have 3 groups of candy with 5 pieces of candy in each bag.

3 * 5 = 15

The answer to this multiplication problem is the product, which in this case is 15.

Here's another example. The classroom has 8 rows of chairs and each row has 7 chairs in it. How many chairs are there?

7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 = 56

Or, you can find the product by multiplying:

7 * 8 = 56

## Properties of Multiplication

There are four basic properties of multiplication that are true no matter what is being multiplied together.

1. Commutative property: When two numbers are multiplied together, the product is the same regardless of the order in which they are written.

For example:

5 * 7 = 7 * 5

2. Associative property: When three or more numbers are multiplied together, the product is the same regardless of which two are multiplied first.

For example:

(2 * 4) * 6 = 2 * (4 * 6)

8 * 6 = 2 * 24

48 = 48

3. Multiplicative identity property: The product of any number and 1 is that number.

For example:

3 * 1 = 3

4. Distributive property: The sum of two numbers times a third number is equal to the sum of each addend times the third number.

For example:

2 * (3 + 4) = 2 * 3 + 2 * 4

2 * 7 = 6 + 8

14 = 14

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## Using the Properties of Multiplication to Rewrite Problems

In the video lesson, we learned what a 'product' is and many properties of multiplication. As a quick reminder, the 'product' is the answer to a multiplication problem and the main properties of multiplication of numbers are:

1. Commutative Property : a*b=b*a. This says that it does not matter which order you multiply numbers in.

2. Associative Property : a*(b*c) = (a*b)*c. This says that when multiplying three numbers together, it does not matter which two you multiply together first.

3. Multiplicative Identity Property : a*1 = 1*a = a. This says that any number multiplied by 1 results in the same number you had before. 1 is called a multiplicative identity.

4. Distributive Property of Multiplication over Addition: a*(b+c) = a*b + a*c. This property states that when multiplying a sum by a number, you can multiply each piece of the sum by the number first and then add the results.

## Examples

1. Show that the commutative property is true for the example 5*3 = 3*5 by using repeated addition two different ways.

2. Show that the associative property is true for the example 2*(3*4) = (2*3)*4 by using repeated addition two different ways.

3. Rewrite the multiplication problem 9*52 using the distributive property to make it easier to calculate in your head. (Hint: try to split the 52 up as a sum of two numbers that are easier to multiply by 9)

## Solutions

1. 5*3 using repeated addition is 5+5+5 = 15. 3*5 using repeated addition is 3+3+3+3+3=15. So the commutative property holds since 5*3=15=3*5

2. 2*(3*4) = 2*12 and using repeated addition, this is 2+2+2+2+2+2+2+2+2+2+2+2=24. Then, (2*3)*4 = 6*4 and using repeated addition, this is 6+6+6+6=24. So the associative property holds since 2*(3*4)=24=(2*3)*4

3. There are many correct answers to this problem, but one of the possibilities outlined here is one of the simplest to work with.

Rewrite 9*52 as 9*(50+2) and then use the distributive property to get 9*50+9*2 = 450+18 = 468. Checking the original product, 9*52, in a calculator, we also get 468, but it is much easier to use the distributive property to make it easier to do in our heads. One tip for rewriting problems in this way is to split off multiples of 10, since multiplying by 10 is easy to do in your head.

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