Lesson 3: Numbers in Exponential Form Raised to a Power
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Lesson 3: Numbers in Exponential Form Raised to a Power Student Outcomes
Students will know how to take powers of powers. Students will know that when a product is raised to a power, each factor of the product is raised to that power.
Students will write simplified, equivalent numeric, and symbolic expressions using this new knowledge of powers.
Classwork Discussion (10 minutes) Suppose we add
copies of , thereby getting
Now, by the definition of multiplication, adding is then denoted by . So,
, and then add
copies of
is denoted by
copies of the sum. We get
, and adding
A closer examination of the right side of the above equation reveals that we are adding to itself times). Therefore,
to itself
copies of this product
times (i.e., adding
Now, replace repeated addition by repeated multiplication. MP.2 & MP.7
(For example,
What is multiplying
.) copies of
and then multiplying
copies of the product?
Multiplying copies of is , and multiplying copies of the product is . We wish to say this is equal to for some positive integer . By the analogy initiated in Lesson 1, the in should correspond to the exponent in ; therefore, the answer should be .
This is correct because
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
29
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Examples 1–2 Work through Examples 1 and 2 in the same manner as just shown (supplement with additional examples if needed). Example 1
Example 2
In the same way, we have
For any number
and any positive integers
and ,
because
.
Exercises 1–6 (10 minutes) Students complete Exercises 1–4 independently. Check answers, and then have students complete Exercises 5–6. Exercise 1
Exercise 3
Exercise 2
Exercise 4 Let be a number.
Exercise 5 Sarah wrote . Correct her mistake. Write an exponential expression using a base of and that would make her answer correct. Correct way:
and exponents of , ,
, Rewritten Problem:
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
30
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Exercise 6 A number
satisfies
Since
, then
. What equation does the number . Therefore,
satisfy?
would satisfy the equation
.
Discussion (10 minutes) From the point of view of algebra and arithmetic, the most basic question about raising a number to a power has to be the following: How is this operation related to the four arithmetic operations? In other words, for two numbers , and a positive integer , MP.7
1.
How is
related to
and
?
2.
How is
related to
and
3.
How is
related to
and
?
4.
How is
related to
and
?
,
?
The answers to the last two questions turn out to be complicated; students will learn about this in high school under the heading of the binomial theorem. However, they should at least be aware that, in general, , unless
. For example,
.
Allow time for discussion of Question 1. Students can begin by talking in partners or small groups and then share with the class. Some students may want to simply multiply , but remind them to focus on the above stated goal which is to relate to and . Therefore, we want to see copies of and copies of on the right side. Multiplying would take us in a different direction.
Scaffolding: Provide a numeric example for students to work on .
The following computation is a different way of proving the equality.
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
31
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Answer to Question 1: Because in
, the factors
will be repeatedly multiplied
times, resulting in factors of
and
:
because By definition of raising a number to the
th
power
By commutative and associative properties By definition of
For any numbers
and , and positive integer ,
because
.
Exercises 7–13 (10 minutes) Students complete Exercises 17–12 independently. Check answers. Exercise 7
Exercise 10 Let
Exercise 8
be a number.
Exercise 11 Let
and
be numbers.
Exercise 9
Exercise 12
Let , , and be numbers.
Let , , and be numbers.
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
32
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Next, have students work in pairs or small groups on Exercise 13 after you present the problem: Ask students to first explain why we must assume . Students should say that if the denominator were zero then the fraction would be undefined.
The answer to the fourth question is similar to the third: If , numbers, such that and is a positive integer, then
Scaffolding: Have students review problems just completed. Remind students to begin with the definition of a number raised to a power.
are any two
.
Exercise 13 Let
and
be numbers,
, and let
be a positive integer. How is
related to
and
?
Because By definition
By the product formula
By definition
Let the students know that this type of reasoning is required to prove facts in mathematics. They should always supply a reason for each step or at least know the reason the facts are connected. Further, it is important to keep in mind what we already know in order to figure out what we do not know. Students are required to write two proofs for the Problem Set that are extensions of the proofs they have done in class.
Closing (2 minutes) Summarize, or have students summarize the lesson. Students should state that they now know how to take powers of powers.
Exit Ticket (3 minutes)
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
33
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
Name ___________________________________________________
8•1
Date____________________
Lesson 3: Numbers in Exponential Form Raised to a Power Exit Ticket Write each answer as a base raised to a power or as the product of bases raised to powers that is equivalent to the given one. 1.
2.
3.
Let
be numbers.
4.
Let
be numbers and let
be positive integers.
5.
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
34
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Exit Ticket Sample Solutions Write each answer as a base raised to a power or as the product of bases raised to powers that is equivalent to the given one. 1.
2. (associative law)
3.
Let
(because
for all numbers , )
(because
for all numbers , )
(because
for all numbers )
be numbers. (associative law)
4.
Let
be numbers and let
(because
for all numbers , )
(because
for all numbers , )
(because
for all numbers )
be positive integers. (associative law) (because
for all numbers , )
(because
for all numbers , )
(because
for all numbers )
5.
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
35
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Problem Set Sample Solutions 1.
Show (prove) in detail why
. By definition By repeated use of the commutative and associative properties By definition
2.
Show (prove) in detail why
for any numbers
.
The left side of the equation, , means . Using the commutative and associative properties of multiplication, we can write as , which in turn can be written as , which is what the right side of the equation states.
3.
Show (prove) in detail why
for any numbers , , and and for any positive integer .
Beginning with the left side of the equation,
means
associative properties of multiplication, and finally,
. Using the commutative and can be rewritten as
, which is what the right side of the equation states. We can also prove this equality by a
different method, as follows. Beginning with the right side,
means
by the commutative property of multiplication can be rewritten as notation,
can be rewritten as
Lesson 3: Date:
, which . Using exponential
, which is what the left side of the equation states.
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
36
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Lesson 3: Numbers in Exponential Form Raised to a Power Student Outcomes
Students will know how to take powers of powers. Students will know that when a product is raised to a power, each factor of the product is raised to that power.
Students will write simplified, equivalent numeric, and symbolic expressions using this new knowledge of powers.
Classwork Discussion (10 minutes) Suppose we add
copies of , thereby getting
Now, by the definition of multiplication, adding is then denoted by . So,
, and then add
copies of
is denoted by
copies of the sum. We get
, and adding
A closer examination of the right side of the above equation reveals that we are adding to itself times). Therefore,
to itself
copies of this product
times (i.e., adding
Now, replace repeated addition by repeated multiplication. MP.2 & MP.7
(For example,
What is multiplying
.) copies of
and then multiplying
copies of the product?
Multiplying copies of is , and multiplying copies of the product is . We wish to say this is equal to for some positive integer . By the analogy initiated in Lesson 1, the in should correspond to the exponent in ; therefore, the answer should be .
This is correct because
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
29
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Examples 1–2 Work through Examples 1 and 2 in the same manner as just shown (supplement with additional examples if needed). Example 1
Example 2
In the same way, we have
For any number
and any positive integers
and ,
because
.
Exercises 1–6 (10 minutes) Students complete Exercises 1–4 independently. Check answers, and then have students complete Exercises 5–6. Exercise 1
Exercise 3
Exercise 2
Exercise 4 Let be a number.
Exercise 5 Sarah wrote . Correct her mistake. Write an exponential expression using a base of and that would make her answer correct. Correct way:
and exponents of , ,
, Rewritten Problem:
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
30
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Exercise 6 A number
satisfies
Since
, then
. What equation does the number . Therefore,
satisfy?
would satisfy the equation
.
Discussion (10 minutes) From the point of view of algebra and arithmetic, the most basic question about raising a number to a power has to be the following: How is this operation related to the four arithmetic operations? In other words, for two numbers , and a positive integer , MP.7
1.
How is
related to
and
?
2.
How is
related to
and
3.
How is
related to
and
?
4.
How is
related to
and
?
,
?
The answers to the last two questions turn out to be complicated; students will learn about this in high school under the heading of the binomial theorem. However, they should at least be aware that, in general, , unless
. For example,
.
Allow time for discussion of Question 1. Students can begin by talking in partners or small groups and then share with the class. Some students may want to simply multiply , but remind them to focus on the above stated goal which is to relate to and . Therefore, we want to see copies of and copies of on the right side. Multiplying would take us in a different direction.
Scaffolding: Provide a numeric example for students to work on .
The following computation is a different way of proving the equality.
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
31
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Answer to Question 1: Because in
, the factors
will be repeatedly multiplied
times, resulting in factors of
and
:
because By definition of raising a number to the
th
power
By commutative and associative properties By definition of
For any numbers
and , and positive integer ,
because
.
Exercises 7–13 (10 minutes) Students complete Exercises 17–12 independently. Check answers. Exercise 7
Exercise 10 Let
Exercise 8
be a number.
Exercise 11 Let
and
be numbers.
Exercise 9
Exercise 12
Let , , and be numbers.
Let , , and be numbers.
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
32
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Next, have students work in pairs or small groups on Exercise 13 after you present the problem: Ask students to first explain why we must assume . Students should say that if the denominator were zero then the fraction would be undefined.
The answer to the fourth question is similar to the third: If , numbers, such that and is a positive integer, then
Scaffolding: Have students review problems just completed. Remind students to begin with the definition of a number raised to a power.
are any two
.
Exercise 13 Let
and
be numbers,
, and let
be a positive integer. How is
related to
and
?
Because By definition
By the product formula
By definition
Let the students know that this type of reasoning is required to prove facts in mathematics. They should always supply a reason for each step or at least know the reason the facts are connected. Further, it is important to keep in mind what we already know in order to figure out what we do not know. Students are required to write two proofs for the Problem Set that are extensions of the proofs they have done in class.
Closing (2 minutes) Summarize, or have students summarize the lesson. Students should state that they now know how to take powers of powers.
Exit Ticket (3 minutes)
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
33
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
Name ___________________________________________________
8•1
Date____________________
Lesson 3: Numbers in Exponential Form Raised to a Power Exit Ticket Write each answer as a base raised to a power or as the product of bases raised to powers that is equivalent to the given one. 1.
2.
3.
Let
be numbers.
4.
Let
be numbers and let
be positive integers.
5.
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
34
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Exit Ticket Sample Solutions Write each answer as a base raised to a power or as the product of bases raised to powers that is equivalent to the given one. 1.
2. (associative law)
3.
Let
(because
for all numbers , )
(because
for all numbers , )
(because
for all numbers )
be numbers. (associative law)
4.
Let
be numbers and let
(because
for all numbers , )
(because
for all numbers , )
(because
for all numbers )
be positive integers. (associative law) (because
for all numbers , )
(because
for all numbers , )
(because
for all numbers )
5.
Lesson 3: Date:
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
35
Lesson 3
NYS COMMON CORE MATHEMATICS CURRICULUM
8•1
Problem Set Sample Solutions 1.
Show (prove) in detail why
. By definition By repeated use of the commutative and associative properties By definition
2.
Show (prove) in detail why
for any numbers
.
The left side of the equation, , means . Using the commutative and associative properties of multiplication, we can write as , which in turn can be written as , which is what the right side of the equation states.
3.
Show (prove) in detail why
for any numbers , , and and for any positive integer .
Beginning with the left side of the equation,
means
associative properties of multiplication, and finally,
. Using the commutative and can be rewritten as
, which is what the right side of the equation states. We can also prove this equality by a
different method, as follows. Beginning with the right side,
means
by the commutative property of multiplication can be rewritten as notation,
can be rewritten as
Lesson 3: Date:
, which . Using exponential
, which is what the left side of the equation states.
Numbers in Exponential Form Raised to a Power 10/21/14
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
36
