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 had 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
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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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http://tutorial
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lamar
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College Algebra

Rational Exponents
Now that we have looked at integer exponents we need to start looking at more complicated
exponents
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That is exponents in
the form

b

m
n

where both m and n are integers
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Once we have this figured out the more general case given above will
actually be pretty easy to deal with
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1

n
is equivalent to
= b= b
a
an

1
n

In other words, when evaluating b we are really asking what number (in this case a) did we
1
n

raise to the n to get b
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Let’s do a couple of evaluations
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(a) 25
(b) 32

1
2

[Solution]

1
5

[Solution]

1
4

(c) 81

[Solution]

(d) ( −8 )

1
3

[Solution]
1

(e) ( −16 ) 4

[Solution]

1
4

(f) −16 [Solution]
Solution
When doing these evaluations we will do actually not do them directly
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In order to evaluate
these we will remember the equivalence given in the definition and use that instead
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1
2

(a) 25
So, here is what we are asking in this problem
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math
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edu/terms
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In this case that is
(hopefully) easy to get
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Therefore,
1

25 2 = 5
[Return to Problems]
1

(b) 32 5
So what we are asking here is what number did we raise to the 5th power to get 32?
1
5

= 2= 32
32
because
25
[Return to Problems]
1
4

(c) 81
What number did we raise to the 4th power to get 81?
1
4

= 3= 81
81
because
34
[Return to Problems]
1

(d) ( −8 ) 3
We need to be a little careful with minus signs here, but other than that it works the same way as
the previous parts
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e
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It is here to make a point
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However, we also know that raising any number
(positive or negative) to an even power will be positive
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Note that this is different from the previous part
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As this part has shown, we can’t always do these evaluations
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Unlike the previous part this one has
an answer
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So, this part is really asking us to
evaluate the following term
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This is 2 and so in
this case the answer is,
1
 1
−16 4 =16 4  =2 ) =
−
−(
−2



[Return to Problems]

As the last two parts of the previous example has once again shown, we really need to be careful
with parenthesis
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Also, don’t be worried if you didn’t know some of these powers off the top of your head
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For instance in the part
b we needed to determine what number raised to the 5 will give 32
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In other
words compute 25 , 35 , 45 until you reach the correct value
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The next thing that we should acknowledge is that all of the properties for exponents that we gave
in the previous section are still valid for all rational exponents
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Now that we know that the properties are still valid we can see how to deal with the more general
rational exponent
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Both
methods involve using property 2 from the previous section
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We will first rewrite the exponent as
follows
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Now we will use the
exponent property shown above
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Also, there are two ways to do it
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2

(a) 8 3 [Solution]
3

(b) 625 4 [Solution]
4

 243  5
(c) 
 [Solution]
 32 
Solution
We can use either form to do the evaluations
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2

(a) 8 3
Let’s use both forms here since neither one is too bad in this case
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2

 1
3
=  8=
8

 
2
3

1

2)
(= 4

= 2 because= 8
83
23

2

Now, let’s take a look at the second form
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Notice however that when we used the
second form we ended up taking the 3rd root of a much larger number which can cause problems
on occasion
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3
4

3

3

6253 ) 4
(=

1


625  625 4 
=
=



=
625 4

1

5)
(=
3

1

625 4 5 because = 625
=
54

125
1

=
( 244140625) 4

125

because 1254 244140625
=

As this part has shown the second form can be quite difficult to use in computations
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[Return to Problems]
4

 243  5
(c) 

 32 

In this case we’ll only use the first form
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© 2007 Paul Dawkins

6

http://tutorial
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lamar
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aspx

College Algebra

4

1


5
4
 243 
 243  5 243


=
 = =

4
1 4
 32 
 5
32 5
 32 


4
5

3)
(=
4
( 2)
4

81
16
[Return to Problems]

We can also do some of the simplification type problems with rational exponents that we saw in
the previous section
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 w−2
(a) 
1

 16v 2

1

4
 [Solution]



 2 −2 
x y 3 
(b)  1
 − −3 
 x 2y 



−

1
7

[Solution]

Solution
(a) For this problem we will first move the exponent into the parenthesis then we will eliminate
the negative exponent as we did in the previous section
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w

1
−2 
4

1
4

16 v

−

1
2

w
1
=
1
1 1
2v 8 2 v 8 w 2

=

1 1 
 
2 4 

[Return to Problems]

(b) In this case we will first simplify the expression inside the parenthesis
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That will happen on
occasion
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−

1

 2 −2  7
y 3 
1
1
 x= =
1
1
5 1
 − −3 
 x 2y 
 5 7  7 x14 y 3


2 3
x y 


[Return to Problems]

© 2007 Paul Dawkins

7

http://tutorial
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lamar
...
aspx

College Algebra

We will leave this section with a warning about a common mistake that students make in regards
to negative exponents and rational exponents
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In other words,

b−n =

1
bn

and NOT
1

b−n ≠ b n
This is a very common mistake when students first learn exponent rules
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math
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