Negative and Zero Exponents

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Negative and Zero Exponents
Lesson 5.2.1
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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 Youll learn what zero and
negative powers mean, and simplify expressions
involving them.

• Key words
• base
• exponent
• power

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Lesson 5.2.1
Negative and Zero Exponents
Up to now youve worked with only positive
whole-number exponents. These show the number of
times a base is multiplied. As youve seen, they
follow certain rules and patterns.
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.
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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.
You can show this using the division of powers
rule.
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Lesson 5.2.1
Negative and Zero Exponents
If you start with 1000, and keep dividing by 10,
you get this pattern
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.
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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.
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Lesson 5.2.1
Negative and Zero Exponents
Example 1
Simplify 34 30. 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
Add the exponents of the powers
34 30 3(4 0) 34
You can see that being multiplied by 30 didnt
change 34.
Solution follows
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Lesson 5.2.1
Negative and Zero Exponents
Guided Practice
Evaluate the following. 1. 40 2. x0 (x ¹
0) 3. 110 120 4. (7 6)0 5. 43
43 6. y2 y2 (y ¹ 0) 7. 32 30 8. 24
20 9. a8 a0 (a ¹ 0)
1
1
2
1
1
1
32 or 9
24 or 16
a8
Solution follows
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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.
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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) 101
Now divide by 10 101 101 10(1 1) 102
Now divide by 10 102 101 10(2 1) 103
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Lesson 5.2.1
Negative and Zero Exponents
Look at the last rows, shown in red, to see the
pattern
Now divide by 10 101 101 10(1 1) 102
Now divide by 10 102 101 10(2 1) 103
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Lesson 5.2.1
Negative and Zero Exponents
This works with any number, not just with 10.
For example
60 1
This pattern illustrates the general definition
for negative exponents.
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Lesson 5.2.1
Negative and Zero Exponents
Example 2
Rewrite 53 without a negative exponent.
Solution
53
Using the definition of negative exponents
Solution follows
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Lesson 5.2.1
Negative and Zero Exponents
Example 3
Solution
75
Using the definition of negative exponents
Solution follows
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Lesson 5.2.1
Negative and Zero Exponents
Guided Practice
Rewrite each of the following without a negative
exponent. 10. 73 11. 5m 12. x2 (x ¹
0)
Solution follows
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Lesson 5.2.1
Negative and Zero Exponents
Guided Practice
Rewrite each of the following using a negative
exponent. 13. 14. 15. (q
¹ 0)
33
64
q3
Solution follows
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Lesson 5.2.1
Negative and Zero Exponents
Independent Practice
Evaluate the expressions in Exercises 13. 1.
87020 2. g0 (g ¹ 0) 3. 20 30
1
1
0
Solution follows
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Lesson 5.2.1
Negative and Zero Exponents
Independent Practice
Write the expressions in Exercises 46 without
negative exponents. 4. 451 5. x6 (x ¹
0) 6. y3 z3 (y ¹ 0, z ¹ 0)
Solution follows
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Lesson 5.2.1
Negative and Zero Exponents
Independent Practice
Write the expressions in Exercises 79 using
negative exponents. 7. 8. (r ¹ 0) 9. (p
q ¹ 0)
82
r6
(p q)v
Solution follows
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Lesson 5.2.1
Negative and Zero Exponents
Independent Practice
In Exercises 1012, simplify the expression
given. 10. 54 50 11. c5 c0 (c ¹ 0) 12.
f 3 f 0 (f ¹ 0)
54
c5
f 3
Solution follows
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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
20, 21, 22, and 23
Solution follows
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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 youre
simplifying expressions and equations.
Later in this Section, youll see how negative
powers are used in scientific notation for
writing very small numbers efficiently.