**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, negative exponents, 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 rules of
exponents.
The rules it covers are the product rule, quotient rule, power rule,
products
to powers rule, quotients to powers 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

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

** Example
1: **Evaluate .

View a video of this example

** Example
2: **Evaluate .

**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 1, there was a ( ) around the - and the 3. 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 1. **

**It is interpreted as finding the negative or opposite
of 3 to the
fourth power.**

** Example
3: **Evaluate .

**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,

**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 .

** Example
5: **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 .

** Example
7: **Evaluate .

Be careful on this example. Order of operations
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 -15.

**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: **Find the quotient .

** Example
9: **Find the quotient .

Be careful with negative
exponents. The
temptation is to negate the base, which would not be a correct thing to
do. **Since exponents
are another
way to write multiplication and the negative is in the exponent, to
write
it as a positive exponent we do the multiplicative inverse which is to
take the reciprocal of the base.**

** Example
10:** Simplify .

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

** Example
11: **Simplify .

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

**Base Raised to Two Exponents**

**(Power Rule for Exponents)**

**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 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
12: **Simplify .

** Example
13: **Simplify .

***Use the definition
of neg. exponents to rewrite as the recip. of base**

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

**A Product Raised to an Exponent**

**(Products to Powers Rule for Exponents)**

**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 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
14: **Simplify .

** Example
15: **Simplify .

***Mult. exponents when using power
rule for exponents**

**A Quotient Raised to an Exponent**

**(Quotients to Powers Rule for Exponents)**

**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 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
16: **Simplify .

***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 condensed form you can making sure that all your exponents are positive.

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

** Example
17: ** Simplify the exponential
expression .

***When
div. like bases
you subtract your exponents: -2 - (-20) = 18**

** Example
18: ** Simplify the exponential
expression .

** Example
19: ** Simplify the exponential
expression .

***Rewrite
with a pos.
exp. by taking recip. of base**

Be careful going into the last line. Since *b
*doesn't
have a negative exponent, we DO NOT take the reciprocal of *b*.
The other bases each have a negative exponent, so those bases we have
to
take the reciprocal of.

** 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 - 1f:Simplify the exponential expression.

** Need Extra Help on these Topics?**

This website gives the definition of and some of the basic rules for exponents.

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

This webpage gives the definition of exponents.

**http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/****int_alg_tut24_exppart2.htm**

This website helps you with some of the basic rules for exponents.

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

This webpage gives an overall review of exponents.

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

This webpage goes over the rules of exponents.** **

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

This website helps you with the product rule for exponents.

**http://www.sosmath.com/algebra/logs/log3/log32/log32.html**

This website helps you with the quotient rule for exponents.

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

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

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

Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.

Last revised on Feb. 15, 2008 by Kim Seward.

All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.