Lesson 3: Properties for Zero and Negative Integer Exponents
Unit: Exponents
Approx. time:
Lesson 3: Properties for Zero and
1 – 2 days
Negative Integer Exponents
A. Focus and Coherence
Students will know…
0
• The proof of x  1 , when x  0 , using the
properties of exponents
• Negative exponents can be expressed using positive
exponents using the rules for multiplying and dividing
exponents with the same base
Students will be able to…

Use the rules that they generated in Lesson 2 (for
multiplying and dividing exponents with the same
base) to generate the property of zero exponents:
1
am
Student prior knowledge:
 The rules for multiplying and dividing exponents with
the same base always work
a) a m  a n  a m  n

b)
Students can use the additive law of exponents and
the multiplicative identity to prove/justify that 30 = 1,
for example:
2
3  30  1 because 320  32 (additive law of exponents)
 
Use the rules that they generated in Lesson 2 (for
multiplying and dividing exponents with the same
base) to generate the property of negative exponents:
a m 
•
9∙
a0  1

CCSS-M Standards: 8.EE.1
Know and apply the properties of integer exponents to
generate equivalent numerical expressions.
B. Evidence of Math Practices
• Student can explain with justification why any number
to the zero power is one because they understand the
rules of additive laws of exponents and multiplicative
m
0
m
m
m
0
identity: a  a  a and a 1  a , so a  1
am
 a mn
n
a
Identify multiple representations for names of 1.
Which math concepts will this lesson lead to?
 Understanding the relationship between square roots
and cube roots and the fractional exponents of ½ and
1/3.
 Propose, justify and communicate solutions involving
properties of exponents.
•
=9
= 1 (multiplicative identity)
Students can use the subtractive law of exponents and
division to prove that 20  1 for example:
any number
2222
24
 1 and
1
 2 4 4  2 0 ,
4
itself
2222
2
So, 2 0  1
•
Students can explain why a negative exponent is
equivalent to its reciprocal because they understand
the rules of additive laws of exponents and they’ve
0
already proven that a  1 . Students will show that
a  m  a m  a 0 , and since a 0  1 , a  m  a m  1 . This
m
m
implies that a and a must be reciprocals.
For example:
3
2 is the unknown, but we know…
23  2 3  20
 
8∙
=1
=
1
1
, which is 3 .
2
8
Essential Question(s):
1) Why is using exponents helpful?
2) How does understanding the exponent rules help you solve real-world problems?
3) Multiplication is to repeated addition, as Exponents are to…. ?
4) How does understanding the additive law of exponents help you develop rules for zero and negative exponents?
Formative Assessments:
Ask the following 3 questions as a Ticket-Out-The-Door closure item, for students to reflect on the day’s learning:
2
0
2
1) In 3  3  3 , where did the exponent “2” in the answer come from?
0
2) Why is 3  1 ? Is this always true?
Grade 8
Exponents
Lesson 3
-2
3) Prove that 3 
1
32
Anticipated Student Preconceptions/Misconceptions:
3 0  0 and other responses
2 -3  8 and other responses
Materials/Resources:
Worksheet/workout paper, notebooks, individual whiteboards and markers
C. Rigor: fluency, deep understanding, application and dual intensity
What are the learning experiences that provide for rigor? What are the learning experiences that provide for evidence of
the Math Practices? (Detailed Lesson Plan)
Warm Up (Day 1)
Use exponent rules to simplify:
1) 73 ∙ 75 =
2) 10013 ∙ 10025 =
3)
4)
610
66
=
6510
6510
=
ANS: 78
ANS: 10038
ANS: 64
ANS: 650
5) What are your thoughts about the answer to #4?
Warm Up (Day 2)
Use exponent rules to simplify:
1) 108 ∙ 105 =
ANS: 1013
2)
3)
4)
610
66
45
45
=
=
235
238
=
ANS: 64
ANS: 1
ANS: 2−3
5) What are your thoughts about the answer to #4?
Lessons
Day 1: Zero Exponents
1) Debrief warm up answers and have a whole-group class discussion about #4.
2) Pose question to the class: What are some guesses of the value of 650 ? Chart student guesses on the
board.
3) Use exponent rules to prove why 650 = 1
For example:
650  652  6502 (additive law of exponents)
There is one unknown that we’re trying to solve for (650 = ?)
x  652  652
x = 1 (multiplicative identity)
4) Show another way: Use exponent rules to prove why 20 = 1:
Grade 8
Exponents
Lesson 3
24
24
= 24−4 = 20 ,
𝑎𝑛𝑦 𝑛𝑢𝑚𝑏𝑒𝑟
𝑖𝑡𝑒𝑠𝑒𝑙𝑓
= 1 and
2∙2∙2∙2
2∙2∙2∙2
=1
So, 20 = 1
5) Have students prove that each of the following equals 1 by using the strategies from #3 and/or #4
a) 7
1
b)  
2
0
0
c) 1,0000
0
6) Have students reason about what a must equal (when a is any non-zero number).
7) Put the rule for zero exponents on the board for class to copy down in notebooks
Zero as an Exponent
For every nonzero number, a
a0  1
Day 2: Negative Exponents:
1) Debrief warm up answers and have a whole-group class discussion about #4.
2) Pose question to class: What are some guesses of the value of 2−3 ? Chart student responses on the board
m
3) Have students use the additive law of exponents to prove that a 
1
am
Example:
2−3 is the unknown but we know…
23 ∙ 2−3= 20
↓ ↓
1
8∙
= 1,
has to be … the multiplicative inverse!
8
3
So 2
1

8
Have students do various examples like the one above until they start to see a pattern.
Example 1:
Example 2:
Example 3:
2
2
1
What is 4 ?
What is 3 ?
What is 9
2
2
0
We know that 3  3  3
(…see above to continue…)
2
So 3 
1
9
1
So 4 
1
4
2
So 9 
1
81
n
4) Have students reason about what a must equal
5) Put the rule for negative exponents on the board for the class to copy down in their notebooks.
Negative Exponents
For every nonzero number, a and integer n,
a n 
Grade 8
1
an
Exponents
Lesson 3
Closure
Use the 3 formative assessment questions as a Ticket-Out-The-Door reflection from today’s work:
1) In 3  3  3 , where did the exponent “2” in the answer come from?
0
2) Why is 3  1 ? Is this always true?
2
0
2
-2
3) Prove that 3 
1
32
Suggested Homework/Independent Practice
m
n
m n
Use the property for multiplying powers with the same base: a  a  a to prove the following:
0
1) 5 = 1
2) (–25)0 = 1
0
3) a  1
1
43
1
 2
7
1
 n
a
3
4) 4 
2
5) 7
n
6) a
Grade 8
Exponents
Lesson 3