Math 2
Lesson 1-4 Zero and Negative Exponents

Name ___________________________
Date _______________
I can solve problems using properties of exponents learned in previous courses, extending
the properties from integer bases to variable bases.
The pond at North Chagrin Reservation fills up with lily pads every year. Lily pads reproduce
rapidly, doubling in population every day! 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.
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 your exponent was 0?
5.
What happened when your exponent was negative?
6.
NOTES:
In what two ways can we simplify the following expression?
w8
w8
Explain what this fact means about the two different expressions?
7.
Use what you discovered in problem (6) to evaluate the following:
50  _____
61160  _____
x 0  _____
 2 
0
 _____
 89 
8.
ZERO POWER PROPERTY:
When b is any base and b  0 , b0  _____
9.
NOTES:
In what two ways can we simplify the following expression?
0
w5
w8
Explain what this fact means about the two different expressions?
 _____
 0  _____
9.
Evaluate the following.
21  _____
10.
11.
32  22  _____
32  23  _____
32  24  _____
1
32     _____
2
1
32     _____
4
1
32     _____
8
 1
32     _____
 16 
So what can we conclude from the above results? Answer the following using fractions
only!
22  _____
23  _____
24  _____
Let’s see what happens when we raise other numbers (bases) to negative exponents. Write
your answers as simplified fractions.
43  _____
82  _____
53  _____
Let’s convert our answers from above to fractions with positive exponents. Fill in the
blanks.
31 
1
3
14.
24  _____
32  21  _____
31  _____
13.
23  _____
Now let’s explore what happens when we have a negative exponent.
21  _____
12.
22  _____
82 
1
43 
8
1
4
One conclusion we can draw about negative exponents is…
NEGATIVE EXPONENTS PROPERTY:
When b is any base and b  0 ,
b x 
1
b
5 3 
1
5
15.
Simplify the following. Write your answers as simplified fractions.
1
4
  
 5
16.
______________
3
4
4
  
 5
______________
______________
4
  
5
______________
Let’s convert our answers from above to fractions with positive exponents. Fill in the
blanks.
1
2
5
4

 
5
4
17.
2
4
  
5
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,
18.
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  
 
______________
Math 2
Name ___________________________
Lesson 1-4 Zero and Negative Exponents Homework

I can solve problems using properties of exponents learned in previous courses, extending
the properties from integer bases to variable bases.
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  
4
1
3
8
0

0
What if a negative exponent appears in the bottom of a fraction?
1
1
23


1

23 213
1
1
x5

x 5 1
4
b 4 c 4
b


 
c 4 b4
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!
13.)
1

y 6
14.)
1

8 3
15.)
x 7

y 5
16.)
23 w6 x 4

y 5 z 2