Teaching the Rules of Exponents
Overview
This worksheet should be done by students in stages and then reinforced with
practice from exercises in the text.
For example, you may choose to have students complete Set 1 and 2 and then
have a Share on that section and follow up with individual practice on those
rules.
For the first Mini-Lesson, you will discuss the meaning of an exponent and how
to expand a power. I would also suggest revision of how to cancel fractions,
showing the factors and the canceling explicitly.
Example
9
3 3
1


27 3  3  3 3
Power Rules
Set 1 – Multiplying Powers with the Same Base
Expand the following:
1. 23  2 4 
2.
32  33 
3.
5 4  56 
4.
43  4 2  43 
5.
Can you see what the rule is for multiplying powers that have the same
base?
Write the rule here in words.
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
In symbols:
a m  a n  a __________
Set 2 – Dividing Powers with the Same Base
35
Consider 35  32
This can also be written as 2
3
In the following problem expand the numerator and denominator and then cancel
out the common factors. Finally, write the answer as a power of 3.
35
1. 2 = -------------------------------------- = 3____
3
Do the same with the following problems:
2. 26  23 
3.
26
 --------------------------------------- = 2 ___
3
2
87
 -------------------------------------------- = 8 ___
84
68
4. 3  --------------------------------------------- = 6 ____
6
5.
m5
 ---------------------------------------------- = m ___
3
m
Write the rule for dividing powers with the same base:
________________________________________________________________
________________________________________________________________
In symbols: a m  a n  a __________
Set 3 – Power of a Power
1. Consider (34 )2  34  34
=
= 3___
OR
Using the rule for multiplying powers:
(Expand)
(34 )2  34  34
= 34  4
= 3___
Now do the same for these using one of the above methods:
2.
(42 )3 
3.
(23 ) 4 =
4.
(32 ) 4 =
5.
(d 4 ) 2 =
Write the rule in words for simplifying the power of a power.
________________________________________________________________
________________________________________________________________
In symbols: (a m )n  a ______
Set 4 – Power of a Product
1. Consider the following:
(2  3) 2  (2  3)  (2  3)
= 2  3 2  3
= 2  2  3 3
= 2___  2 ___
Do the same for these:
2. (4  3)3 
3. (5  2)4 
4. (6a)3 
5. (5m) 2 
What is the rule for simplifying the power of a product?
________________________________________________________________
________________________________________________________________
In symbols: (ab)n  a ___  b ____
Set 4 – Power of a Quotient
1. Consider the following:
32  62 
32
              (Expand)
62
3 3
=  (Separate into fractions)
6 6
3
= ( )
______
6
=(
) 2 (Cancel down the fraction)
Now try these:
2. 24  54 
3. 43 123 
4. a 5  b5 
Write a rule for simplifying the power of a quotient:
________________________________________________________________
________________________________________________________________
In symbols:
am
a _____
a  b  m ( )
b
b
m
m
Set 5 – Zero Exponent
Complete the following and find the pattern:
25  32
35  243
105  100 000
2 4  16
34  81
104  10 000
23 
33 
103 
22 
32 
102 
21 
31 
101 
20 
30 
100 
We could do the same for other numbers and the process would be the same.
What can you conclude about a number with an exponent of zero?
________________________________________________________________
________________________________________________________________
In symbols: a0  ____
Another way to show this rule is true is to consider the following:
1. Consider 34  34
Method 1: Use the rule for dividing powers with the same base.
34  34  3
___
34
Method 2: 3  3  4  ------------------------------------ = _________
3
4
4
(1.)
(2.)
Therefore, the answer in (1.) = the answer in (2.)
So 3___  ____
Follow the same procedure for these:
2. 45  45 
3. 52  52 
4. a 3  a 3 
So, once again, any number raised to the power zero is equal to _________