**Learning Objectives**

After completing this tutorial, you should be able to:

- Use the definition of exponents.
- Simplify exponential expressions involving multiplying like bases, zero as an exponent, dividing like bases, raising a base to two exponents, raising a product to an exponent and raising a quotient to an exponent.

** Introduction**

This tutorial covers the basic definition and some of
the rules of
exponents. The rules it covers are the product rule, quotient
rule,
power rule, power of a product rule and power of a quotient rule as
well
as the definitions for zero and negative exponents. Exponents are
everywhere
in algebra and beyond. Let's see what we can do with exponents.

** Tutorial**

(note there are* n * *x*'s
in the product)

*x = base, n =
exponent*

The exponent tells you how many times a base appears in
a PRODUCT.

** Example
1: **Evaluate .

Note how I included the - when I expanded this
problem out.
If the - is inside the ( ) of an exponent, then it is included as part
of the base.

Hey, this looks a lot like example 2!!!!

It may look alike, but they ARE NOT exactly the same. Can you see the difference between the two?? Hopefully, you noticed that in example 2, there was a ( ) around the - and the 6. In this problem, there is no -. This means the - is NOT part of the base, so it will not get expanded like it did in example 2.

It is interpreted as finding the negative or opposite of 6 squared.

**Specific Illustration**

Note that 2 + 3 = 5, which is the exponent we ended up
with. We
had 2 *x*’s written in a product plus
another
3* x*’s written in the product for a total
of
5 *x*’s in the product. To indicate
that
we put the 5 in the exponent.

**Let's put this idea together into a general rule:**

*in general,*

In other words, **when you
multiply like bases
you add your exponents**.

**The reason is, exponents count how many of your base
you have in
a product, so if you are continuing that product, you are adding on to
the exponents.**

** Example
4: **Use the product rule to simplify the
expression .

Note that the exponent doesn’t become 1, but the whole
expression simplifies
to be the number 1.

** Example
6: **Evaluate .

Be careful on this example. The order of
operations shown in **Tutorial
4: Introduction to Variable Expressions and Equations** says to
evaluate
exponents before doing any multiplication. This means we need to
find *x *raised to the 0 power first and then
multiply it by 3.

**Specific Illustration**

Note how 5 - 2 = 3, the final answer’s exponent. When you multiply you are adding on to your exponent, so it should stand to reason that when you divide like bases you are taking away from your exponent.

**Let's put this idea together into a general rule:**

*in general,*

Keep in mind that you always take the numerator’s
exponent minus your
denominator’s exponent, NOT the other way around.

** Example
8: **Use the quotient rule to simplify the
expression .

**Specific Illustration**

Let’s first start by using the **definition
of exponents** as well as the **law for multiplying like bases**
to help you to understand how we get to the law for raising a base to
two
exponents:

Note how 2 times 3 is 6, which is the exponent of the final answer. We can think of this as 3 groups of 2, which of course would come out to be 6.

*in general,*

In other words, **when you
raise a base to two
exponents, you multiply those exponents together.**

Again, you can think of it as *n* groups of *m* if it helps you to remember.

** Example
10: **Use the power rule for exponents to
simplify
the expression .

**Specific Illustration**

Let’s first start by using the **definition
of exponents** to help you to understand how we get to the law for raising a product
to
an exponent:

Note how both bases of your product ended up being raised by the exponent of 3.

*in general,*

In other words, **when you have
a PRODUCT (not
a sum or difference) raised to an exponent, you can simplify by raising
each base in the product to that exponent.**

** Example
11: **Use the power of a product rule to simplify
the expression .

**Specific Illustration**

Let’s first start by using the **definition
of exponents** to help you to understand how we get to the law for raising a quotient
to an exponent:

Since, division is really multiplication of the reciprocal, it has the same basic idea as when we raised a product to an exponent.

*in general,*

In other words, **when you have
a QUOTIENT (not
a sum or difference) raised to an exponent, you can simplify by raising
each base in the numerator and denominator of the quotient to that
exponent.**

** Example
12: **Use the power of a quotient rule to
simplify
the expression .

***When raising a quotient to an
exponent, raise
each base of the quotient to that exponent**

***Use def. of exponents to
evaluate**

In other words, write it in the most condense form you can.

A lot of times you are having to use more than one rule to get the job done. As long as you are using the rule appropriately, you should be fine.

***When div. like bases you
subtract your exponents**

***When div. like bases you
subtract your exponents**

** 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 - 1e: Simplify.

** Need Extra Help on these Topics?**

**The following are webpages
that can assist
you in the topics that were covered on this page: **

**http://www.purplemath.com/modules/exponent.htm**

This webpage helps you with the rules of exponents.

**http://www.sosmath.com/algebra/logs/log2/log2.html#shortcuts**

This webpage helps you with the definition of exponents.

**http://www.sosmath.com/algebra/logs/log3/log31/log31.html**

This webpage helps with the product rule for exponents.

**http://www.sosmath.com/algebra/logs/log3/log33/log33.html**

This webpage helps with the rule for raising a base to two
exponents.

**http://www.ltcconline.net/greenl/courses/152A/polyExp/intexp.htm**

This webpage goes over the rules of exponents.

**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 August 2, 2011 by Kim Seward.

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