# The Properties of Real Numbers

## Distributive Property


### Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need  that is,  dollars and  quarter. How much money do they need all together? You can think about the dollars separately from the quarters.
They need  times  so  and  times  quarter, so  cents. In total, they need 
If you think about doing the math in this way, you are using the Distributive Property.
Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:
In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression  the order of operations says to work in the parentheses first. But we cannot add  and  since they are not like terms. So we use the Distributive Property, as shown in .
Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in  would look like this:
The distributive property can be used to simplify expressions that look slightly different from  Here are two other forms.
Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.
Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.
In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.
The next example will use the ‘backwards’ form of the Distributive Property, 
When you distribute a negative number, you need to be extra careful to get the signs correct.
In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, 
Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

### Evaluate Expressions Using the Distributive Property

Some students need to be convinced that the Distributive Property always works.
In the examples below, we will practice evaluating some of the expressions from previous examples; in part ⓐ , we will evaluate the form with parentheses, and in part ⓑ we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

### Key Concepts

1. Distributive Property:

### Practice Makes Perfect

Simplify Expressions Using the Distributive Property 
In the following exercises, simplify using the distributive property.
Evaluate Expressions Using the Distributive Property 
In the following exercises, evaluate both expressions for the given value.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.



ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
