Notesale: Turn your study into money

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Preface
Here are my online notes for my Algebra course that I teach here at Lamar University, although I
have to admit that it’s been years since I last taught this course
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Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to
learn Algebra or needing a refresher for algebra
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However, they do assume that you’ve has some
exposure to the basics of algebra at some point prior to this
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Here are a couple of warnings to my students who may be here to get a copy of what happened on
a day that you missed
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Because I wanted to make this a fairly complete set of notes for anyone wanting to learn
algebra I have included some material that I do not usually have time to cover in class
and because this changes from semester to semester it is not noted here
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2
...
Likewise, even
if I do work some of the problems in here I may work fewer problems in class than are
presented here
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Sometimes questions in class will lead down paths that are not covered here
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Sometimes a very good question gets asked in class
that leads to insights that I’ve not included here
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4
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THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!!
Using these notes as a substitute for class is liable to get you in trouble
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College Algebra

© 2007 Paul Dawkins

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College Algebra

Integer Exponents
We will start off this chapter by looking at integer exponents
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We will look at zero and negative exponents in a bit
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If a is any
number and n is a positive integer then,

a n = a14243
× a × a ×L × a
n times

So, for example,

35 = 3 × 3 × 3 × 3 × 3 = 243
We should also use this opportunity to remind ourselves about parenthesis and conventions that
we have in regards to exponentiation and parenthesis
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Consider the following two cases
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When performing exponentiation
remember that it is only the quantity that is immediately to the left of the exponent that gets the
power
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So, in this case we get,

( -2 )

4

= ( -2 )( -2 )( -2 )( -2 ) = 16

In the second case however, the 2 is immediately to the left of the exponent and so it is only the 2
that gets the power
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In this
case we have the following,

-24 = - ( 24 ) = - ( 2 × 2 × 2 × 2 ) = - (16 ) = -16

We put in some extra parenthesis to help illustrate this case
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They are
important and ignoring parenthesis or putting in a set of parenthesis where they don’t belong can
completely change the answer to a problem
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Also, this warning about parenthesis is
not just intended for exponents
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Now, let’s take care of zero exponents and negative integer exponents
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This is important since 00 is not defined
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( -1268 )
© 2007 Paul Dawkins

3

0

=1

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College Algebra

We have the following definition for negative exponents
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Here are a couple of
quick examples for this definition,

5-2 =

1
1
=
2
5
25

( -4 )

-3

=

1

( -4 )

3

=

1
1
=-64
64

Here are some of the main properties of integer exponents
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We will be looking at more complicated examples after the
properties
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a n a m = a n + m

Example : a -9 a 4 = a -9 + 4 = a -5

2
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ì a n-m
an ï
=í 1 , a¹0
a m ï m-n
îa

a4
= a 4-11 = a -7
a11
Example : 4
1
1
a
= 11- 4 = 7 = a -7
11
a
a
a

4
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ç ÷ = n , b ¹ 0
b
èbø
-n

= a ( 7 )( 3) = a 21

-4

= a -4b -4

8

8
æaö a
Example : ç ÷ = 8
èbø b

n

bn
æbö
=ç ÷ = n
a
èaø

3

10

b10
æbö
= ç ÷ = 10
a
èaø

æaö
ç ÷
èbø

7
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1
= an
-n
a

Example :

9
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Example : ç

Example : ( ab )

4

-20

=

1

( ab )

20

1
= a2
-2
a

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College Algebra

(

10
...
ç m ÷ = mk
èb ø b

Notice that there are two possible forms for the third property
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Note as well that many of these properties were given with only two terms/factors but they can be
extended out to as many terms/factors as we need
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( abcd )

n

= a nb n c n d n

We only used four factors here, but hopefully you get the point
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There are several common mistakes that students make with these properties the first time they
see them
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Consider the following case
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Do NOT carry the a down to the denominator with
the b
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( ab )

-2

=

1

( ab )

2

In this case the exponent is on the set of parenthesis and so we can just use property 7 on it and so
both the a and the b move down to the denominator
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1
1 1
1
=
= a5
-5
-5
3a
3a
3
1
Incorrect :
¹ 3a 5
-5
3a

Correct :

In this case the exponent is only on the a and so to use property 8 on this we would have to break
up the fraction as shown and then use property 8 only on the second term
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© 2007 Paul Dawkins

5

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College Algebra

1

( 3a )

-5

= ( 3a )

5

Once again, notice this common mistake comes down to being careful with parenthesis
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We must always be careful with parenthesis
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Let’s take a look at some more complicated examples now
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(
) [Solution]
(b) ( -10z y ) ( z y )
3

(a) 4x -4 y 5

-4 2

2

3

-5

[Solution]

n -2 m
(c)
[Solution]
7 m-4 n -3
5 x -1 y -4
(d)

(3 y )

5 -2

[Solution]

x9

æ z -5 ö
(e) ç -2 -1 ÷
èz x ø

6

æ 24a3b -8 ö
(f) ç
÷
-5
è 6a b ø

[Solution]
-2

[Solution]

Solution
Note that when we say “simplify” in the problem statement we mean that we will need to use all
the properties that we can to get the answer into the required form
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There are many different paths that we can take to get to the final answer for each of these
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All that
this means for you is that as long as you used the properties you can take the path that you find
the easiest
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That is okay
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For instance, we won’t show the actual multiplications
anymore, we will just give the result of the multiplication
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( 4x

-4

y 5 ) = 43 x -12 y15
3

Don’t forget to put the exponent on the constant in this problem
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© 2007 Paul Dawkins

6

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College Algebra

At this point we need to evaluate the first term and eliminate the negative exponent on the second
term
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15
æ 1 ö 15 64 y
-4 5 3
4
x
y
=
64
y
=
(
)
ç 12 ÷
x12
èx ø

We further simplified our answer by combining everything up into a single fraction
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The middle step in this part is usually skipped
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So, from this
point on, that is what we will do without writing in the middle step
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Finally, we will eliminate the negative exponents using the definition of negative
exponents
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First, when using the property 10
on the first term, make sure that you square the “-10” and not just the 10 (i
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don’t forget the
minus sign…)
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The exponent of “-11” is only on the z and so only the z moves to the
denominator
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We will use the definition of negative exponents to move all terms with
negative exponents in them to the denominator
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So, let’s take care of the negative exponents first
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We will use property 1 to combine the m’s in the numerator
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n -2 m
m5 n
=
7 m -4 n -3
7
Again, the 7 will stay in the denominator since there isn’t a negative exponent on it
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math
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edu/terms
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Do not get excited if all the terms move up to the
numerator or if all the terms move down to the denominator
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[Return to Problems]

(d)

5 x -1 y -4

(3 y )

5 -2

x9

This example is similar to the previous one except there is a little more going on with this one
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Any terms in the numerator with negative exponents will get moved to the denominator
and we’ll drop the minus sign in the exponent
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Notice this time, unlike the previous part, there is a term with a set of parenthesis in the
denominator
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Here is the work for this part
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The first step that we’re pretty much
always going to take with these kinds of problems is to first simplify the fraction inside the
parenthesis as much as possible
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6

6

6

æ z -5 ö æ z 2 x1 ö æ x ö
x6
=
=
=
ç -2 -1 ÷ ç 5 ÷ ç 3 ÷
z18
èz x ø è z ø èz ø
In this case we used the second form of property 3 to simplify the z’s since this put a positive
exponent in the denominator
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When
we have exponents of 1 we will drop them
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The main difference is negative on the outer
exponent
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-2

-2

æ 24a 3b -8 ö
æ 4a 3 a 5 ö
æ 4a 8 ö
=
=
ç
÷
ç 8 ÷
ç 9 ÷
-5
è 6a b ø
è bb ø
è b ø

-2

Now at this point we can use property 6 to deal with the exponent on the parenthesis
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math
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This was done only so there would be a consistent final
answer
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In fact,
if you are on a track that will take you into calculus there are a fair number of problems in a
calculus class in which negative exponents are the preferred, if not required, form
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math
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edu/terms

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