# Pre algebra properties

## Associative, Commutative, and Distributive Properties

### Purplemath

There are three basic properties of numbers, and your textbook will probably have just a little section on these properties, somewhere near the beginning of the course, and then you'll probably never see them again (until the beginning of the next course). My impression is that covering these properties is a holdover from the "New Math" fiasco of the s. While the topic will start to become relevant in matrix algebra and calculus (and become amazingly important in advanced math, a couple years after calculus), they really don't matter a whole lot now.

Why not? Because every math system you've ever worked with has obeyed these properties! You have never dealt with a system where a×b did not in fact equal b×a, for instance, or where (a×bc did not equal a×(b×c). Which is why the properties probably seem somewhat pointless to you. Don't worry about their "relevance" for now; just make sure you can keep the properties straight so you can pass the next test. The lesson below explains how I keep track of the properties.

### Distributive Property

The Distributive Property is easy to remember, if you recall that "multiplication distributes over addition". Formally, they write this property as "a(b + c) = ab + ac". In numbers, this means, for example, that 2(3 + 4) = 2×3 + 2×4. Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or factor something out); any time a computation depends on multiplying through a parentheses (or factoring something out), they want you to say that the computation used the Distributive Property.

• #### Why is the following true? 2(x + y) = 2x + 2y

Since they distributed through the parentheses, this is true by the Distributive Property.

• #### Use the Distributive Property to rearrange: 4x – 8

The Distributive Property either takes something through a parentheses or else factors something out. Since there aren't any parentheses to go into, you must need to factor out of. Then the answer is:

By the Distributive Property, 4x – 8 = 4(x – 2).

"But wait!" I hear you cry; "the Distributive Property says multiplication distributes over addition, not over subtraction! What gives?" You make a good point. This is one of those times when it's best to be flexible. You can either view the contents of the parentheses as the subtraction of a positive number ("x – 2") or else as the addition of a negative number ("x + (–2)"). In the latter case, it's easy to see that the Distributive Property applies, because you're still adding; you're just adding a negative.

The other two properties come in two versions each: one for addition and the other for multiplication. (Yes, the Distributive Property refers to both addition and multiplication, too, but it refers to both of the operations within just the one rule.)

### Associative Property

The word "associative" comes from "associate" or "group"; the Associative Property is the rule that refers to grouping. For addition, the rule is "a + (b + c) = (a + b) + c"; in numbers, this means 2 + (3 + 4) = (2 + 3) + 4. For multiplication, the rule is "a(bc) = (ab)c"; in numbers, this means 2(3×4) = (2×3)4. Any time they refer to the Associative Property, they want you to regroup things; any time a computation depends on things being regrouped, they want you to say that the computation uses the Associative Property.

• #### Rearrange, using the Associative Property: 2(3x)

They want me to regroup things, not simplify things. In other words, they do not want me to say "6x". They want to see me do the following regrouping:

• #### Simplify 2(3x), and justify your steps.

In this case, they do want me to simplify, but I have to say why it's okay to do just exactly what I've always done. Here's how this works:

2(3x) :  original (given) statement

(2×3) x :  by the Associative Property

6x :  simplification of 2×3

• #### Why is it true that 2(3x) = (2×3)x?

Since all they did was regroup things, this is true by the Associative Property.

### Commutative Property

The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is "ab = ba"; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.

• #### Use the Commutative Property to restate "3×4×x" in at least two ways.

They want me to move stuff around, not simplify. In other words, my answer should not be "12x"; the answer instead can be any two of the following:

4 × 3 × x

4 × x × 3

3 × x × 4

x × 3 × 4

x × 4 × 3

• #### Why is it true that 3(4x) = (4x)(3)?

Since all they did was move stuff around (they didn't regroup), this statement is true by the Commutative Property.

### Worked examples

• #### Simplify 3a – 5b + 7a. Justify your steps.

I'm going to do the exact same algebra I've always done, but now I have to give the name of the property that says its okay for me to take each step. The answer looks like this:

3a – 5b + 7a :  original (given) statement

3a + 7a – 5b :  Commutative Property

(3a + 7a) – 5b :  Associative Property

a(3+7) – 5b :  Distributive Property

a(10) – 5b :  simplification (3 + 7 = 10)

10a – 5b :  Commutative Property

The only fiddly part was moving the "– 5b" from the middle of the expression (in the first line of my working above) to the end of the expression (in the second line). If you need help keeping your negatives straight, convert the "– 5b" to "+ (–5b)". Just don't lose that minus sign!

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• #### Simplify 23 + 5x + 7y – x – y – Justify your steps.

I'll do the exact same steps I've always done. The only difference now is that I'll be writing down the reasons for each step.

23 + 5x + 7yxy – 27 :  original (given) statement

23 – 27 + 5xx + 7yy :  Commutative Property

(23 – 27) + (5xx) + (7yy) :  Associative Property

(–4) + (5xx) + (7yy) :  simplification (23 – 27 = –4)

(–4) + x(5 – 1) + y(7 – 1) :  Distributive Property

–4 + x(4) + y(6) :  simplification (5 – 1 = 4, 7 – 1 = 6)

–4 + 4x + 6y :  Commutative Property

• #### Simplify 3(x + 2) – 4x. Justify your steps.

3(x + 2) – 4x :  original (given) statement

3x + 3×2 – 4x :  Distributive Property

3x + 6 – 4x :  simplification (3×2 = 6)

3x – 4x + 6 :  Commutative Property

(3x – 4x) + 6 :  Associative Property

x(3 – 4) + 6 :  Distributive Property

x(–1) + 6 :  simplification (3 – 4 = –1)

x + 6 :  Commutative Property

• #### Why is it true that 3(4 + x) = 3(x + 4)?

All they did was move stuff around.

• #### Why is 3(4x) = (3×4)x?

All they did was regroup.

• #### Why is 12 – 3x = 3(4 – x)?

They factored.

URL: https://www.purplemath.com/modules/numbprop.htm

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## Commutative and Associative Properties

### Learning Objectives

By the end of this section, you will be able to:

• Use the commutative and associative properties
• Evaluate expressions using the commutative and associative properties
• Simplify expressions using the commutative and associative properties

Be Prepared

Before you get started, take this readiness quiz.

Simplify:
If you missed this problem, review Example

Be Prepared

Multiply:
If you missed this problem, review Example

Be Prepared

Find the opposite of
If you missed this problem, review Example

In the next few sections, we will take a look at the properties of real numbers. Many of these properties will describe things you already know, but it will help to give names to the properties and define them formally. This way we’ll be able to refer to them and use them as we solve equations in the next chapter.

### Use the Commutative and Associative Properties

The results are the same.

Notice, the order in which we add does not matter. The same is true when multiplying and

Again, the results are the same! The order in which we multiply does not matter.

These examples illustrate the commutative properties of addition and multiplication.

### Commutative Properties

Commutative Property of Addition: if and are real numbers, then

Commutative Property of Multiplication: if and are real numbers, then

The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.

### Example

Use the commutative properties to rewrite the following expressions:

#### Solution

 ⓐ Use the commutative property of addition to change the order.
 ⓑ Use the commutative property of multiplication to change the order.

Try It

Use the commutative properties to rewrite the following:

Try It

Use the commutative properties to rewrite the following:

What about subtraction? Does order matter when we subtract numbers? Does give the same result as

Since changing the order of the subtraction did not give the same result, we can say that subtraction is not commutative.

Let’s see what happens when we divide two numbers. Is division commutative?

Since changing the order of the division did not give the same result, division is not commutative.

Addition and multiplication are commutative. Subtraction and division are not commutative.

Suppose you were asked to simplify this expression.

Some people would think and then Others might start with and then

Both ways give the same result, as shown in Figure (Remember that parentheses are grouping symbols that indicate which operations should be done first.)

Figure

When adding three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Addition.

The same principle holds true for multiplication as well. Suppose we want to find the value of the following expression:

Changing the grouping of the numbers gives the same result, as shown in Figure

Figure

When multiplying three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Multiplication.

If we multiply three numbers, changing the grouping does not affect the product.

You probably know this, but the terminology may be new to you. These examples illustrate the Associative Properties.

### Associative Properties

Associative Property of Addition: if and are real numbers, then

Associative Property of Multiplication: if and are real numbers, then

### Example

Use the associative properties to rewrite the following:

#### Solution

 ⓐ Change the grouping.

Notice that is so the addition will be easier if we group as shown on the right.

 ⓑ Change the grouping.

Notice that is The multiplication will be easier if we group as shown on the right.

Try It

Use the associative properties to rewrite the following:
ⓐ ⓑ

Try It

Use the associative properties to rewrite the following:
ⓐⓑ

Besides using the associative properties to make calculations easier, we will often use it to simplify expressions with variables.

### Example

Use the Associative Property of Multiplication to simplify:

#### Solution

Change the grouping.
Multiply in the parentheses.

Notice that we can multiply but we could not multiply without having a value for

Try It

Use the Associative Property of Multiplication to simplify the given expression:

Try It

Use the Associative Property of Multiplication to simplify the given expression:

### Evaluate Expressions using the Commutative and Associative Properties

The commutative and associative properties can make it easier to evaluate some algebraic expressions. Since order does not matter when adding or multiplying three or more terms, we can rearrange and re-group terms to make our work easier, as the next several examples illustrate.

### Example

Evaluate each expression when

#### Solution

 ⓑ Substitute for x. Add opposites first.

What was the difference between part ⓐ and part ⓑ? Only the order changed. By the Commutative Property of Addition, But wasn’t part ⓑ much easier?

Try It

Evaluate each expression when ⓐ ⓑ

Try It

Evaluate each expression when ⓐ ⓑ

Let’s do one more, this time with multiplication.

### Example

Evaluate each expression when

#### Solution

What was the difference between part ⓐ and part ⓑ here? Only the grouping changed. By the Associative Property of Multiplication, By carefully choosing how to group the factors, we can make the work easier.

Try It

Evaluate each expression when ⓐⓑ

Try It

Evaluate each expression when ⓐⓑ

### Simplify Expressions Using the Commutative and Associative Properties

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. Notice that in Example part ⓑ was easier to simplify than part ⓐ because the opposites were next to each other and their sum is Likewise, part ⓑ in Example was easier, with the reciprocals grouped together, because their product is In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.

### Example

Simplify:

#### Solution

Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.

Re-order the terms.

Try It

Simplify:

Try It

Simplify:

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is

### Example

Simplify:

#### Solution

Notice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.

Re-order the terms.
Multiply left to right.
Multiply.

Try It

Simplify:

Try It

Simplify:

In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.

### Example

Simplify:

#### Solution

Notice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.

Group the terms with a common denominator.
Simplify the fraction.
Convert to an improper fraction.

Try It

Simplify:

Try It

Simplify:

When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.

### Example

Simplify:

#### Solution

Notice that the sum of the second and third coefficients is a whole number.

Change the grouping.

Many people have good number sense when they deal with money. Think about adding cents and cent. Do you see how this applies to adding

Try It

Simplify:

Try It

Simplify:

No matter what you are doing, it is always a good idea to think ahead. When simplifying an expression, think about what your steps will be. The next example will show you how using the Associative Property of Multiplication can make your work easier if you plan ahead.

### Example

Simplify the expression:

#### Solution

Notice that multiplying is easier than multiplying because it gives a whole number. (Think about having quarters—that makes

Regroup.
Multiply in the brackets first.
Multiply.
Sours: https://openstax.org/books/prealgebra-2e/pages/commutative-and-associative-properties

## Number Properties - Definition with Examples

### Number Properties

The four main number properties are:

### Commutative property:

The commutative property states that the numbers on which we perform the operation can be moved or swapped from their position without making any difference to the answer.

This property holds true for addition and multiplication, but not for subtraction and division.

Example of the commutative property of addition

3 + 5 = 5 + 3 = 8

Hence, the commutative property of addition for any two real numbers a and b is:

a + b = b + a

Example of the commutative property of multiplication

Hence, the commutative property of multiplication for any two real numbers a and b is:

a x b = b x a

We can also say that in commutative property, the numbers can be added or multiplied to each other in any order without changing the answer.

### Associative Property:

The associative property gets its name from the word “Associate” and it refers to the grouping of numbers. This property states that when three or more numbers are added (or multiplied), the sum(or product) is the same regardless of the grouping of the addends (or multiplicands).

For example:

(3 + 4) + 5 = (4 + 5) + 3

( 4 x 7 ) x 5 = ( 4 x 5 ) x 7 =

### Identity Property:

 Additive identity Multiplicative identity Additive identity is a number, which when added to any number, gives the sum as the number itself.This means, the additive identity is “0” as adding 0 to any number, gives the sum as the number itself. Multiplicative identity is a number, which when multiplied by any number, gives the product as the number itself.This means, the multiplicative identity is “1” as multiplying any number by 1, gives the product as the number itself.

### Distributive Property:

Distributive property helps us to simplify the multiplication of a number by a sum or difference. As the name suggests, it distributes the expression.

For example:  a x (b + c)

Using distributive property, we can expand the expression as:

 Example of distributive property using addition Example of distributive property using subtraction

 Fun FactsThe additive identity property is also called the zero property of addition.The commutative property gets its name from the word commutes, meaning move around.

Sours: https://www.splashlearn.com/math-vocabulary/number-sense/number-properties

## 43 Commutative and Associative Properties

### Simplify Expressions Using the Commutative and Associative Properties

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. Notice that in (Figure) part ⓑ was easier to simplify than part ⓐ because the opposites were next to each other and their sum is Likewise, part ⓑ in (Figure) was easier, with the reciprocals grouped together, because their product is In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.

Simplify:

Solution

Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.

Simplify:

Simplify:

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is

Simplify:

Solution

Notice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.

Simplify:

Simplify:

In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.

Simplify:

Solution

Notice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.

Simplify:

Simplify:

When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.

Simplify:

Solution

Notice that the sum of the second and third coefficients is a whole number.

Many people have good number sense when they deal with money. Think about adding cents and cent. Do you see how this applies to adding

Simplify:

Simplify:

No matter what you are doing, it is always a good idea to think ahead. When simplifying an expression, think about what your steps will be. The next example will show you how using the Associative Property of Multiplication can make your work easier if you plan ahead.

Simplify the expression:

Solution

Notice that multiplying is easier than multiplying because it gives a whole number. (Think about having quarters—that makes

Simplify:

Simplify:

When simplifying expressions that contain variables, we can use the commutative and associative properties to re-order or regroup terms, as shown in the next pair of examples.

Simplify:

Solution

 Use the associative property of multiplication to re-group. Multiply in the parentheses.

Simplify:

Simplify:

In The Language of Algebra, we learned to combine like terms by rearranging an expression so the like terms were together. We simplified the expression by rewriting it as and then simplified it to We were using the Commutative Property of Addition.

Simplify:

Solution

Use the Commutative Property of Addition to re-order so that like terms are together.

 Re-order terms. Combine like terms.

Simplify:

Simplify:

The Links to Literacy activity, &#;Each Orange Had 8 Slices&#; will provide you with another view of the topics covered in this section.

#### Practice Makes Perfect

Use the Commutative and Associative Properties

In the following exercises, use the commutative properties to rewrite the given expression.

In the following exercises, use the associative properties to rewrite the given expression.

(21 + 14) + 9 = 21 + (14 + 9)

(−2 + 6) + 7 = −2 + (6 + 7)

(17 + y) + 33 = 17 + (y + 33)

Evaluate Expressions using the Commutative and Associative Properties

In the following exercises, evaluate each expression for the given value.

If evaluate:

If evaluate:

If evaluate:

If evaluate:

If evaluate:

If evaluate:

If evaluate:

If evaluate:

Simplify Expressions Using the Commutative and Associative Properties

In the following exercises, simplify.

Sours: https://opentextbc.ca/prealgebraopenstax/chapter/commutative-and-associative-properties/
Properties of numbers 1 - Arithmetic properties - Pre-Algebra - Khan Academy

## 1 PROPERTIES Pre-Algebra. 2 Vocabulary Equivalent expressions — expressions that have the same value Property — Statement that is true for any # or variable.

### Presentation on theme: "1 PROPERTIES Pre-Algebra. 2 Vocabulary Equivalent expressions — expressions that have the same value Property — Statement that is true for any # or variable."— Presentation transcript:

1 1 PROPERTIES Pre-Algebra

2 2 Vocabulary Equivalent expressions — expressions that have the same value Property — Statement that is true for any # or variable

3 3 Vocabulary Distributive Property—To multiply a sum by a number, multiply each addend of the sum by the number outside the parentheses. Ex: 3(4 + 6) = 3(4) + 3(6)

4 4 Vocabulary Commutative Property—The order in which two numbers are added or multiplied does not change their sum or product. Ex:5 + 4 = 4 + 5 9 x 7 = 7 x 9

5 5 Vocabulary Associative Property—The way in which three numbers are grouped when they are added or multiplied does not change their sum or product. Ex:(2 x 3) x 7 = 2 x (3 x 7) (6 + 4) +8 = 6 + (4 + 8)

6 6 Vocabulary Identity Property—The sum of an addend and zero is the addend. The product of a factor and one is the factor. Additive Identity—Identity is zero. Multiplicative Identity—Identity is one. Ex:5 x 1 = 59 + 0 = 9

7 7 Name the property shown by each statement. a. 6 + (2 + 7) = (6 + 2) +7 b x 10 = 10 x 15 c.4 x 1 = 4 d.4(6 + 8) = 4(6) + 4(8) e.1 x (3 x 4) = (1 x 3) x 4

8 8 Name the property shown by each statement. a. 4(a + 5) = 4(a) + 4(5) b. 7 = 1 x 7 c. 24 + 5 = 5 + 24 d. 7 + 0 = 7 e.(11 x 4) x 8 = 11 x (4 x 8)

9 9 Name the property shown by each statement. m + n = n + m 1. associative 2. commutative 3. identity 4. distributive

10 10 Name the property shown by each statement. 3 + 0 = 3 1. associative 2. commutative 3. identity 4. distributive

11 11 How to use the Distributive Property to write an expression as an equivalent expression… Example: 3(7 + 4) Example: 3(7 + 4) Step 1: Multiply the number outside the parenthesis to each number inside the parenthesis. 3 ∙ 7 + 3 ∙ 4 This is your equivalent expression, now evaluate. Step 2: Add the two products. 21 + 12 = 33

12 12 Use the distribute property to write an equivalent expression then evaluate. (3 + 8)4 (3 + 8)4 (11 + 3)8 (11 + 3)8 7() 7()

13 13 Use the distributive property to write 2(5 + 3) as an equivalent expression then evaluate. 1. 2(8); 16 2. 2(5) + 2(3); 16 3. 2(5) + 3; 13 4. (5 + 3)2; 16

14 14 Class Work

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### Now discussing:

Helen nodded her head, she understood that the sooner she finished, the sooner she would go home to change. I ask for me she gave a signal with her hand and went to the dressing room. "Lord, it would be faster, it. Flows down my legs. " They went into the dressing room.

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