Name: ____________________________________
Date: __________________
Zero and Negative Exponents
In our last lesson we learned how to simplify products and quotients of monomials using laws of
exponents with positive integers. But, zero and negative exponents are also possible.
Exercise #1: Recall that
xa
 x a b .
b
x
(a) Using this exponent law, simplify each of the following.
x4

x4
x10

x10
y7

y7
(b) What must each of these quantities equal, assuming none of the variables equals zero?
Exercise #2: Simplify each of the following:
(a) 1250 
(b)  2y  
0
(c) 5x0 
 
(d) 2x 0
3

We can investigate negative exponents in a very similar fashion to the zero exponent. The key is to
define a negative exponent in such a way that our fundamental rules for exponents don’t need to
change.
x2
Exercise #3: Consider the quotient 5 .
x
(a) Write this quotient using the exponent law
from Exercise #1.
(b) Write this quotient in its simplest form
without a negative exponent.
Exercise #4: Rewrite each expression in simplest terms without the use of negative exponents.
(a) 4 2 
(b) x2 
(c) 2 3 
(d) y 10 
Exercise #5: Rewrite each of the following monomials without the use of negative exponents.
(a)
1

x 2
(b)
1

y 5
(c)
1
x 3
(d)
y 5

x 7
NEGATIVE AND ZERO EXPONENTS
If a is any integer and x  0 then
(1) x  a 
1
xa
(2)
1
 xa
a
x
(3) x0  1
x 2 y 5
Exercise #6: Which of the following is equivalent to 5 3 ?
x y
(1)
x3
y8
(3) x3 y8
y8
(2) 3
x
(4)
1
x y
3 8
Exercise #7: Rewrite the following expressions without negative or zero exponents.
(a) 4 2 
(b) 40 
(c) 42 
(g) 3x0 
(h) 2x3 
(i)
6

x 5
(d) 12 
(e) 12 
(j) 2x7 
(k)
a 3

d 2
(f)  1 
0
(l)
r 3t 2

s 4
Exercise #8: Evaluate each of the following expressions using the values a  1 , b  2 and c  3 .
Use the STORE feature on your calculator to aid you.

(a) ab

2  c

1
(b)  abc  
0
 a 2 
(c)  2 15  
b c 
(d) abbcca 
Name: ____________________________________
Date: __________________
Zero and Negative Exponents
Homework
Skills
For problems 1 through 36,
rewrite without zero or
negative exponents.
1. 4 3 
17. 3 
0
2
4. 10
1
20.  
2
5. 4
6. 2
3
4

1
21.  
2

1
22.  
3




8.
1

40
24.
10. 3x 
11. 5x
12.
27.

x5

y 3
a 4
13. 3 
b
0 2
14. 2x y
15. 2
3
38. y 3 for y 


16. 16x 2 y 5  
0
1
2
 50 
 1
2
39. 2 x 4 y 1 for x  2, y 

26. 2 1 
0
4
37. y 3 for y  2
1
23. 16 
25.
36. 2x 1 y 4 
2
1

22
 32 
3x3

y 4
x0 y 3
35.

z2
1
7.
9.
34.
 33 
2. 52 
3. 5 
x2

2 y 3
18. 8x 0 y 3 
19.
0
33.
40.
 x  32
1
3
for x  4
 2 1 
2
41. x  y for x  2, y  2
28.
 2
29.
 2 
30.
2 x 3 y 2

4 x 4 y 1

2 1
31. a3b4 
a 2
32. 4 
b

42.
x y 
4 2
0
4
2
3
7
for x  , y  
2
4
5
3
43. x y x  y for x  , y  
Reasoning
Fill in the missing
each of the following.
44.
47.
55. Evaluate each of the
following products:
(a) 2  2
3
1
3
9
(c) 104 104 
61. 37  34  27
62.
a 
63.
 40  0
65.
1
(a) 2 x
51.
1
4
64
3
1
 3
2x
(b) 2x3 
1

10, 000
1
3
81

1
a6
x 2 y 1 x 5

x 3 y 2 y 3
56. Which of the following
is correct?
 2 1
50.
2 3
64. 23  23  20  2
(d) xa  xa 
2
48. 62 
49. 10

1
1

25
2
3
(b) 52  52 
45. 42 
46.
for
Find the value of x that
makes each statement true.
66. 2 x  24  212
2
x3
Explain why the other
choice is incorrect.
67. 52  5x  59
68.
4 
x 2
 410
True or False
Write the answer to each of
the following as a single
number.
52. [1   5  2  ]3 
1
57.  
2
1
4
58.  
3
1
0
2
1 
1
53.    3  1  
2

8

54. 31  
3

3

59.
69. 3x2  27
2

  2  2 
4
3
70.
4
2
 32  54
1
4
2 x 3 y 2 2 y 2
60.
 3 5
a3 x 2
a x
71. 22 x 6 
1
4

x
1