**Learning Objectives**

After completing this tutorial, you should be able to:

- Identify and use the addition and multiplication commutative properties.
- Identify and use the addition and multiplication associative properties.
- Identify and use the distributive property.
- Identify and use the addition and multiplication identity properties.
- Identify and use the addition and multiplication inverse properties.

** Introduction**

It is important to be familiar with the properties in this tutorial. They lay the foundation that you need to work with equations, functions, and formulas all of which are covered in later tutorials, as well as, your algebra class. In some cases, it isn't very helpful to rewrite an expression, but in other cases it helps to write an equivalent expression to be able to continue with a problem and solve it. An equivalent expression is one that is written differently, but has the same value. The properties on this page will get you up to speed as to how you can write expressions in equivalent forms.

** Tutorial**

*a + b = b + a and
ab = ba*

The two sides are called equivalent expressions because they look different
but have the same value.

Using the commutative property of addition (where changing the order
of a sum does not change the value of it) we get

**2.5x + 3y = 3y + 2.5x.**

Using the commutative property of multiplication (where changing
the order of a product does not change the value of it), we get

**The Associative Properties of **

**Addition and Multiplication**

*a + (b + c) = (a + b) + c and
a(bc) = (ab)c*

**At this point it is good to remind you that
both the commutative and associative properties do NOT work for subtraction
or division.**

** Example
3:** Use the associative property to write an equivalent
expression to (a + 5b) + 2c.

Using the associative property of addition (where changing the grouping
of a sum does not change the value of it) we get

**( a + 5b) + 2c = a + (5b + 2c).**

Using the associative property of multiplication (where changing the
grouping of a product does not change the value of it) we get

**(1.5 x)y = 1.5(xy)**

*a(b + c) = ab + ac *

or

*(b + c)a = ba + ca*

Remember terms are separated by + and -.

This idea can be extended to more than two terms in the ( ).

** Example
5: **Use the distributive property to write 2(*x* - *y*) without parenthesis.

Basically, when you have a negative sign in front of a ( ), like this
example, you can think of it as taking a -1 times the ( ). What you
end up doing in the end is taking the opposite of every term in the ( ).

3(2

As mentioned above, you can extend the distributive property to as
many terms as are inside the ( ). The basic idea is that you multiply
the outside term times EVERY term on the inside.

*The additive identity is 0*

*a + 0 = 0 + a = a*

In other words, when you add 0 to any number, you end up with that
number as a result.

*Multiplication identity is 1*

*a(1) = 1(a) = a*

And when you multiply any number by 1, you wind up with that number
as your answer.

*For each real number a, there is a unique real number,
denoted -a,*

*a + (-a) = 0.*

In other words, when you add a number to its additive inverse, the
result is 0. Other terms that are synonymous with additive inverse are
negative and opposite.

*For each real number a, except 0, there is a unique
real number such that*

In other words, when you multiply a number by its multiplicative inverse
the result is 1. A more common term used to indicate a multiplicative
inverse is the **reciprocal**. A multiplicative inverse or reciprocal
of a real number *a* (except 0) is found by "flipping" *a *upside
down. The numerator of *a *becomes the
denominator of the reciprocal of *a* and the
denominator of *a* becomes the numerator of the
reciprocal of *a*.

**These two inverses will come in big time handy
when you go to solve equations later on. Keep them in your memory
bank until that time.**

** Example
8:** Write the opposite (or additive inverse) and
reciprocal (or multiplicative inverse) of -3.

When you take the reciprocal, the sign of the original number stays
intact. Remember that you need a number that when you multiply times
the given number you get 1. If you change the sign when you take
the reciprocal, you would get a -1, instead of 1, and that is a no no.

** Practice Problems**

These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. **Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.**
In fact there is no such thing as too much practice.

To get the most out of these, **you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem**. At the link you will find the answer
as well as any steps that went into finding that answer.

Practice Problems 1a - 1b:Use a commutative property to write an equivalent expression.

Practice Problems 2a - 2b:Use an associative property to write an equivalent expression.

2a. (*a* + *b*)
+ 1.5

(answer/discussion
to 2a)

2b. 5(*xy*)

(answer/discussion
to 2b)

Practice Problems 3a - 3b:Use the distributive property to find the product.

Practice Problems 4a - 4b: Write the opposite (additive inverse) and the reciprocal (multiplicative inverse) of each number.

4a. -7

(answer/discussion
to 4a)

4b. 3/5

(answer/discussion
to 4b)

** Need Extra Help on these Topics?**

**http://www.mathleague.com/help/wholenumbers/wholenumbers.htm#commutativeproperty**

This webpage helps with the commutative property.

**http://www.mathleague.com/help/wholenumbers/wholenumbers.htm#associativeproperty**

This webpage helps with the associative property.

**http://www.mathleague.com/help/wholenumbers/wholenumbers.htm#distributiveproperty**

This webpage helps with the distributive property.

**http://home.earthlink.net/~djbach/basic.html#anchor904011**

This webpage goes over the commutative, associative, and distributive
properties.

**Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.**

Last revised on July 24, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.