1. Negative and Zero Exponents
Lesson 5.2.1

2. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
California Standards:
Number Sense 2.1
Understand negative whole-number exponents. Multiply and divide expressions using exponents with a common base.
Algebra and Functions 2.1
Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.
What it means for you:
You’ll learn what zero and negative powers mean, and simplify expressions involving them.
Key words:
base
exponent
power

3. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Up to now you’ve worked with only positive whole-number exponents. These show the number of times a base is multiplied. As you’ve seen, they follow certain rules and patterns. 83
1212
916
3256 84
122 20
100
4–3
2–6
17–2
6–7
4–10
2–256
The effects of negative and zero exponents are trickier to picture. But you can make sense of them because they follow the same rules and patterns as positive exponents.

4. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Any Number Raised to the Power 0 is 1
Any number that has an exponent of 0 is equal to 1.
So, 20 = 1, 30 = 1, 100 = 1, = 1. 1 2
For any number a ¹ 0, a0 = 1
You can show this using the division of powers rule.

5. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
If you start with 1000, and keep dividing by 10, you get this pattern:
1000 = 103
100 = 102
10 = 101
1 = 100
Now divide by 10: 103 ÷ 101 = 10(3 – 1) = 102
Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101
Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100
1000 = 103
100 = 102
10 = 101
1 = 100
Now divide by 10: 103 ÷ 101 = 10(3 – 1) = 102
Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101
Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100
The most important row is the second to last one.
When you divide 10 by 10, you have 101 ÷ 101 = 10(1 – 1) = 100.
You also know that 10 divided by 10 is 1. So you can see that 100 = 1.

6. 61 ÷ 61 = 6(1 – 1) = 60, and 6 divided by 6 is 1. So 60 = 1.
Lesson
5.2.1
Negative and Zero Exponents
This pattern works for any base.
For instance,
61 ÷ 61 = 6(1 – 1) = 60, and 6 divided by 6 is 1. So 60 = 1.
You can use the fact that any number to the power 0 is 1 to simplify expressions.

7. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Example 1
Simplify 34 × Leave your answer in base and exponent form.
Solution
34 × 30 = 34 × 1 = 34
You can use the multiplication of powers rule to show this is right:
34 × 30 = 3(4 + 0) = 34
Add the exponents of the powers
You can see that being multiplied by 30 didn’t change 34.
Solution follows…

8. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Guided Practice
Evaluate the following.
x0 (x ¹ 0)
(7 + 6)0
5. 43 ÷ y2 ÷ y2 (y ¹ 0)
7. 32 × × 20
9. a8 ÷ a0 (a ¹ 0) 1 1 2 1 1 1
32 or 9
24 or 16 a8
Solution follows…

9. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
You Can Justify Negative Exponents in the Same Way
By continuing the pattern of powers shown below you can begin to understand the meaning of negative exponents.
1000 = 103
100 = 102
10 = 101
1 = 100
Now divide by 10: 103 ÷ 101 = 10(3 – 1) = 102
Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101
Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100

10. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Carry on dividing each power of 10 by 10:
100 = 102
10 = 101
1 = 100
Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101
Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100
Now divide by 10: 100 ÷ 101 = 10(0 – 1) = 10–1 1 10
= 10–1
Now divide by 10: 10–1 ÷ 101 = 10(–1 – 1) = 10–2 1
100
= 10–2
Now divide by 10: 10–2 ÷ 101 = 10(–2 – 1) = 10–3 1
1000
= 10–3

11. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Look at the last rows, shown in red, to see the pattern: 1 10
= 10–1
Now divide by 10: 10–1 ÷ 101 = 10(–1 – 1) = 10–2 1
100
= 10–2
Now divide by 10: 10–2 ÷ 101 = 10(–2 – 1) = 10–3 1
1000
= 10–3
One-thousandth, which is , can be rewritten as = 10–3. 1
1000
103
One-hundredth, which is , can be rewritten as = 10–2. 1
100
102
One-tenth, which is , can be rewritten as = 10–1. 1 10
101

12. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
This works with any number, not just with 10.
For example:
60 = 1
60 ÷ 61 = 6–1 and 1 ÷ 6 = , so 6–1 = . 1 6
6–1 ÷ 61 = 6–2 and ÷ 6 = • = = , so 6–2 = . 1 6 62 36
This pattern illustrates the general definition for negative exponents.
For any number a ¹ 0, a–n = 1 an

13. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Example 2
Rewrite 5–3 without a negative exponent.
Solution 1 53
5–3 =
Using the definition of negative exponents = 1
125
Solution follows…

14. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Example 3
Rewrite using a negative exponent. 1 75
Solution 1 75
= 7–5
Using the definition of negative exponents
Solution follows…

15. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Guided Practice
Rewrite each of the following without a negative exponent.
10. 7–3
11. 5–m
12. x–2 (x ¹ 0) 1 73 1 5m 1 x2
Solution follows…

16. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Guided Practice
Rewrite each of the following using a negative exponent.
13.
14.
(q ¹ 0) 1 33
3–3 1 64
6–4 1
q × q × q
q–3
Solution follows…

17. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Independent Practice
Evaluate the expressions in Exercises 1–3.
2. g0 (g ¹ 0)
3. 20 – 30 1 1
Solution follows…

18. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Independent Practice
Write the expressions in Exercises 4–6 without negative exponents.
4. 45–1
5. x–6 (x ¹ 0)
6. y–3 – z–3 (y ¹ 0, z ¹ 0) 1 45 1 x6 1 y3 z3 –
Solution follows…

19. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Independent Practice
Write the expressions in Exercises 7–9 using negative exponents. 7.
(r ¹ 0)
(p + q ¹ 0) 1 82
8–2 1 r6
r–6 1
(p + q)v
(p + q)–v
Solution follows…

20. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Independent Practice
In Exercises 10–12, simplify the expression given.
× 50
11. c5 × c0 (c ¹ 0)
12. f 3 ÷ f 0 (f ¹ 0) 54 c5
f 3
Solution follows…

21. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Independent Practice
13. The number of bacteria in a Petri dish doubles every hour. The numbers of bacteria after each hour are 1, 2, 4, 8, 16, Rewrite these numbers as powers of 2.
20, 21, 22, 23, 24 1 2 4 8
14. Rewrite the numbers 1, , , and as powers of 2.
20, 2–1, 2–2, and 2–3
Solution follows…

22. Negative and Zero Exponents
Lesson
5.2.1
Negative and Zero Exponents
Round Up
So remember — any number (except 0) to the power of 0 is equal to 1. This is useful when you’re simplifying expressions and equations.
Later in this Section, you’ll see how negative powers are used in scientific notation for writing very small numbers efficiently.