Math 2
Lesson 1-4: Zero & Negative Exponents
Name ______________________________
Date ____________________________
Learning Goal:

I can use exponent rules to simplify expressions with exponents of zero and negative exponents.
The pond at North Chagrin Reservation fills up with lily pads every year. Lily pads reproduce rapidly,
doubling in population every day! Suppose you go the pond today (day zero) and count 32 lily pads. A
week later, you return to find 4096 lily pads!
1.
Fill in the following table to illustrate the growth of the lily pads.
Day
Number of Lily
Pads
Value written as a
power of 2.
-4
four days ago
-3
-2
-1
0
today
1
tomorrow
2
32
64
32  21
32  2 2
3
4
5
6
7
one week from
today
4096
32  2 7
2.
What happened to the lily pad population as time moved forward?
3.
What happened to the lily pad population as time moved backward?
4.
What happened when the exponent was 0?
5.
What happened when the exponent was negative?
OVER 
Page 2
6.
Let’s see what happens when we raise other numbers (bases) to the power of 0.
50  _____
61160  _____
 2 
x 0  _____
Just for fun….. What do you think
0
0
 _____
 0  _____
(0)0 equals? _________
ZERO POWER PROPERTY:
When b is any base and b  0 ,
8.
 89 
 _____
b0  _____
Use your calculator to evaluate the following.
21  _____
22  _____
23  _____
24  _____
23  _____
24  _____
Answer the following using fractions only!
21  _____
22  _____
How are the numbers with the positive exponents related to those of the negative exponents?
9.
What happens when we raise other numbers (bases) to negative exponents. Write your answers as
simplified fractions.
31  _____
10.
43  _____
53  _____
Let’s convert our answers from above to fractions with positive exponents. Fill in the blanks.
31 
1
3
11.
82  _____
82 
1
43 
8
1
5 3 
5
4
One conclusion we can draw about negative exponents is…
NEGATIVE EXPONENT PROPERTY:
When b is any base and b  0 then
b
x
1

1
b
Page 3
12.
Simplify the following using properties discovered in Lesson 1-3. Write your answers as simplified
fractions.
1
2
4
  
 5
13.
______________
4
  
 5
3
4
  
 5
______________
4
______________
4
  
 5
______________
Simplify the following. Write your answers as simplified fractions.
1
4
  
 5
2
______________
4
  
5
3
4
4
  
 5
______________
______________
4
  
5
______________
How are the numbers with the positive exponents related to those of the negative exponents?
14.
Let’s convert our answers from above to fractions with positive exponents. Fill in the blanks.
1
5
4
  
5
4
15.
2
3
5
4
  
5
4
4
5
4
  
5
4
5
4
  
5
4
The other conclusion we can draw about negative exponents is…
NEGATIVE EXPONENTS PROPERTY:
When a and b are not equal to zero,
16.
a
 
b
x

b
a
Simplify the following. Use only positive exponents in your answers.
1
x
 y 
 
2
2
______________
a
  
b
______________
 5x 
 7y  
 
3
______________
 2h 
 j  
 
______________
OVER 
Homework:
Simplify completely. Leave your answers as fractions. No negative exponents!
1.
5 3 
2.
7 2 
3.
6 1 
4.
b 7 
5.
2
  
5
6.
3
  
2
7.
a
 
b
8.
 x 
  
 2y 
9.
19 0 
10.
 7 
11.
y0 
12.
 abcdefg 
1
4
3
8
13.
=
0
0

0

-1
14.
=
=
15. True or False: A positive number raised to a negative power will equal a negative number when
simplified. Explain your answer.
What if a negative exponent appears in the bottom of a fraction?
1
23

2 3 1
1
x5

x 5 1
4
b 4 c 4
b
   4  4
c
b
c
Simply put, if you get a base with a negative exponent in the denominator, move it to the
numerator with a positive exponent.
Simplify completely. Leave your answers as fractions. No negative exponents!
16.
1

y 6
17.
1

8 3
18.
x 7

y 5
19.
23 w6 x 4

y 5 z 2
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