Download Unit 1: Laws of Exponents and Scientific Notation

Unit 1: Laws of
Exponents and Scientific
Notation
Name
Date
Period
Math 8: Mr. Sanford
Lesson 1
8•1
Lesson 1: Powers of 10
Exercise 1
100
10
Exercise 2
5
Write 10 as a product of tens, then find the express the product in standard form (as a single number).
Exercise 3
Write 10 × 100 as a product of tens, then find the product in standard form.
Exercise 4
Write 10 × 100 as a product of tens, then find the product in standard form.
Exercise 5
Write 3 × 102 without using an exponent, then find the product.
Exercise 6
Find the product of each of the following.
a. 3.4 × 103
b.
c.
54.18 × 105
. 035 × 108
Lesson 1:
Powers of 10
1
Lesson 1
8•1
Exercise 7
Write 700 ÷ 102 without using an exponent, then find the quotient.
Exercise 8
Write 7.1 ÷ 102 without using an exponent, then find the quotient.
Exercise 9
Complete the pattern below. Describe what is happening.
0.043; 4.3; 430; __________; __________; __________
Exercise 10
Complete the pattern below. Describe what is happening.
6,300,000; __________; 630; 6.3; _______; ___________
Lesson 1:
Powers of 10
2
Lesson 1
8•1
Problem Set
1. Write the following in exponential form (e.g., 100 = 102).
a. 10,000 =
d. 100 x 100 =
b. 1000
e. 1,000,000 =
=
c. 10 x 10 =
f.
1000 × 1000
=
2. Write the following in standard form (e.g., 5 x 102= 500).
a. 9 x 103
=
e. 4.025 x 103
=
b. 39 x 104
=
f.
40.25 x 104
=
c. 7200 ÷ 102
=
g. 725 ÷ 103
=
d. 7,200,000 ÷ 103 =
h. 7.2 ÷ 102
=
3. Complete the patterns.
a. 0.03
0.3
b. 6,500,000
________
65,000
c. _________
9,430
d. 999
99,900
9,990
Lesson 1:
30
_____________
___________
6.5
_________
94.3
__________
Powers of 10
_____________
_____________
9.43
__________
__________
___________
3
Lesson 2
8•1
Lesson 2: Exponential Notation
𝟗 𝟒
𝟓𝟔 means 𝟓 × 𝟓 × 𝟓 × 𝟓 × 𝟓 × 𝟓 and � � means
𝟕
𝟗
𝟕
×
𝟗
𝟕
×
𝟗
𝟕
𝟗
× .
𝟕
You have seen this kind of notation before, it is called exponential notation. In general, for any number 𝒙 and any
positive integer 𝒏,
(𝒙 ���
𝒙𝒏 = ��
∙ 𝒙 ⋯��
𝒙)
𝒏 𝒕𝒊𝒎𝒆𝒔
𝒏
The number 𝒙 is called 𝒙 raised to the 𝒏-th power, 𝒏 is the exponent of 𝒙 in 𝒙𝒏 and 𝒙 is the base of 𝒙𝒏 .
Exercise 1
Exercise 6
4 ���
��
⋯×
��
4=
7
7
× ⋯× =
��
2 �����
2
Exercise 2
Exercise 7
3.6����
��
⋯ ×���
3.6 = 3.647
(−13)
× ⋯ × (−13) =
�������������
Exercise 3
Exercise 8
(−11.63) × ⋯ × (−11.63) =
�����������������
1
1
�− � × ⋯ × �− � =
���������������
14
14
Exercise 4
Exercise 9
7 𝑡𝑖𝑚𝑒𝑠
21 𝑡𝑖𝑚𝑒𝑠
6 𝑡𝑖𝑚𝑒𝑠
_______ 𝑡𝑖𝑚𝑒𝑠
34 𝑡𝑖𝑚𝑒𝑠
10 𝑡𝑖𝑚𝑒𝑠
15
12 �
��
���
⋯×
���
12 = 12
𝑥 ∙ 𝑥⋯𝑥 =
�����
Exercise 5
Exercise 10
(−5)
× ⋯ × (−5) =
�����������
𝑥 ∙ 𝑥 ⋯ 𝑥 = 𝑥𝑛
�����
_______𝑡𝑖𝑚𝑒𝑠
185 𝑡𝑖𝑚𝑒𝑠
10 𝑡𝑖𝑚𝑒𝑠
_______𝑡𝑖𝑚𝑒𝑠
Lesson 2:
Exponential Notation
4
Lesson 2
8•1
Exercise 11
Will these products be positive or negative? How do you know?
(−1)
× (−1) × ⋯ × (−1) = (−1)12
�����������������
12 𝑡𝑖𝑚𝑒𝑠
(−1)
× (−1) × ⋯ × (−1) = (−1)13
�����������������
13 𝑡𝑖𝑚𝑒𝑠
Exercise 12
Is it necessary to do all of the calculations to determine the sign of the product? Why or why not?
(−5)
× (−5) × ⋯ × (−5) = (−5)95
�����������������
95 𝑡𝑖𝑚𝑒𝑠
(−1.8)
× (−1.8) × ⋯ × (−1.8) = (−1.8)122
���������������������
122 𝑡𝑖𝑚𝑒𝑠
Exercise 13
Fill in the blanks about whether the number is positive or negative.
If 𝑛 is a positive even number, then (−55)𝑛 is __________________________.
If 𝑛 is a positive odd number, then (−72.4)𝑛 is __________________________.
Lesson 2:
Exponential Notation
5
Lesson 2
8•1
Problem Set
1. Use what you know about exponential notation to complete the expressions below.
(−𝟓) × ⋯ × (−𝟓) =
�����������
𝟑. 𝟕��
��
���
⋯×
��𝟑.
��
𝟕 = 𝟑. 𝟕𝟏𝟗
𝟕 ×���
��
⋯×
��
𝟕 = 𝟕𝟒𝟓
𝟔 ×���
��
⋯×
��
𝟔=
𝟒. 𝟑��
��
���
⋯×
��𝟒.
��
𝟑=
(−𝟏. 𝟏) × ⋯ × (−𝟏. 𝟏) =
���������������
𝟐
𝟐
� � × ⋯× � � =
��
𝟑 ���������
𝟑
𝟏𝟏
𝟏𝟏
𝟏𝟏 𝒙
�− � × ⋯ × �− � = �− �
���������������
𝟓
𝟓
𝟓
(−𝟏𝟐) × ⋯ × (−𝟏𝟐) = (−𝟏𝟐)𝟏𝟓
�������������
𝒂 ×���
��
⋯×
��
𝒂=
_____ 𝒕𝒊𝒎𝒆𝒔
𝟏𝟕 𝒕𝒊𝒎𝒆𝒔
𝟒 𝒕𝒊𝒎𝒆𝒔
_____ 𝒕𝒊𝒎𝒆𝒔
𝟏𝟑 𝒕𝒊𝒎𝒆𝒔
𝟗 𝒕𝒊𝒎𝒆𝒔
𝟏𝟗 𝒕𝒊𝒎𝒆𝒔
_____ 𝒕𝒊𝒎𝒆𝒔
𝒎 𝒕𝒊𝒎𝒆𝒔
_____ 𝒕𝒊𝒎𝒆𝒔
2. Write an expression with (−1) as its base that will produce a positive product.
3. Write an expression with (−1) as its base that will produce a negative product.
4. Rewrite each number in exponential notation using 2 as the base.
8=
16 =
32 =
64 =
128 =
256 =
5. Could −2 be used as a base to rewrite 32? 64? Why or why not?
Lesson 2:
Exponential Notation
6
Lesson 3
8•1
Lesson 3: Multiplication of Numbers in Exponential Form
Notes:
In general, if 𝑥 is any number and 𝑚, 𝑛 are positive integers, then
𝑥 𝑚 ∙ 𝑥 𝑛 = 𝑥 𝑚+𝑛
because
𝑥 𝑚 × 𝑥 𝑛 = (𝑥
⋯�𝑥)
⋯�𝑥)
⋯�𝑥)
���
� × (𝑥
���
� = (𝑥
���
� = 𝑥 𝑚+𝑛
𝑚 𝑡𝑖𝑚𝑒𝑠
𝑛 𝑡𝑖𝑚𝑒𝑠
𝑚+𝑛 𝑡𝑖𝑚𝑒𝑠
Exercise 1
Exercise 5
𝟏𝟒𝟐𝟑 × 𝟏𝟒𝟖 =
Let 𝑎 be a number.
Exercise 2
Exercise 6
𝑎23 ∙ 𝑎8 =
(−72)10 × (−72)13 =
Let f be a number.
Exercise 3
Exercise 7
𝑓 10 ∙ 𝑓 13 =
𝟓𝟗𝟒 × 𝟓𝟕𝟖 =
Let 𝑏 be a number.
Exercise 4
Exercise 8
𝒃𝟗𝟒 ∙ 𝒃𝟕𝟖 =
Let 𝑥 be a positive integer. If (−3)9 × (−3) 𝑥 = (−3)14 ,
what is 𝑥?
(−𝟑)𝟗 × (−𝟑)𝟓 =
Lesson 3:
Multiplication of Numbers in Exponential Form
7
Lesson 3
Exercise 9
Exercise 11
65 × 49 × 43 × 614 =
24 × 82 = 24 × 26 =
Exercise 10
Exercise 12
(−4)2 ∙ 175 ∙ (−4)3 ∙ 177 =
37 × 9 =
8•1
Exercise 13
Let 𝑎 and 𝑏 be numbers. Use an array to simplify the expression, 𝑎(𝑎 + 𝑏) =
Exercise 14
Let 𝑎 and 𝑏 be numbers. Use an array to simplify the expression, 𝑏(𝑎 + 𝑏) =
Exercise 15
Let 𝑎 and 𝑏 be numbers. Use an array to simplify the expression, (𝑎 + 𝑏)(𝑎 + 𝑏) =
Lesson 3:
Multiplication of Numbers in Exponential Form
8
Lesson 3
8•1
Problem Set
1. Rewrite the rule for multiplying number with a common base.
2. Multiply each of the following:
a. (−19)5 ∙ (−19)11 =
b. 2.75 × 2.73 =
1 𝑎
5
1 𝑏
5
c. � � ∙ � � =
d. 𝑐 3 ∙ 𝑐 5 =
e. 𝑎10 (−𝑏)5 ∙ 𝑎4 (−𝑏)8 =
f.
1 5
ℎ
1 6
ℎ
� � ∙� � =
g. 6𝑠 6 ∙ 8𝑠 3 ∙ 3𝑠 5 =
h. 2𝑘𝑧 5 ∙ 4𝑘 2 𝑧 6 =
Lesson 3:
Multiplication of Numbers in Exponential Form
9
Lesson 4
8•1
Lesson 4: Division of Numbers in Exponential Form
Notes:
In general, if 𝑥 is nonzero and 𝑚, 𝑛 are positive integers, then
𝑥𝑚
= 𝑥 𝑚−𝑛
𝑥𝑛
𝑖𝑓 𝑚 > 𝑛
Directions: Simplify any expressions below that can be simplified. If they cannot be simplified, explain why not.
Exercise 1
9
8 9
� �
5 =
8 2
� �
5
7
=
76
Exercise 2
(−5)16
(−5)7
Exercise 3
=
Exercise 4
135
=
134
Exercise 5
Lesson 8
27
=
24
35 ∙ 28
=
3 2 ∙ 23
Exercise 7
Lesson 9
323
=
33
(−2)7 ∙ 955
=
(−2)5 ∙ 954
Lesson 4:
Division of numbers in Exponential Form
10
Lesson 4
8•1
Problem Set
1. Rewrite the rule for dividing numbers with a common base.
2. Divide each of the following:
a.
710
b.
1222
c.
� �
d.
𝑦 3𝑧 5
e.
(−3)4
f.
73
=
1210
=
2 15
3
2 9
� �
3
𝑦 2𝑧 4
(−3)3
48 813
810 46
=
=
=
=
Lesson 4:
Division of numbers in Exponential Form
11
Lesson 5
8•1
Lesson 5: Numbers in Exponential Form Raised to a Power
Notes:
For any number 𝑥 and any positive integers 𝑚 and 𝑛,
because
(𝑥 𝑚 )𝑛 = 𝑥 𝑚𝑛
(𝑥 𝑚 )𝑛 = �������
(𝑥 ∙ 𝑥 ⋯ 𝑥)𝑛
𝑚 𝑡𝑖𝑚𝑒𝑠
(𝑥 ���
(𝑥 ���
= ��
∙ 𝑥 ⋯��
𝑥) × ⋯ × ��
∙ 𝑥 ⋯��
𝑥)
=𝑥
Exercise 1
𝑚 𝑡𝑖𝑚𝑒𝑠
𝑚𝑛
𝑚 𝑡𝑖𝑚𝑒𝑠
Exercise 3
(153 )9 =
(3.417 )4 =
Exercise 2
Exercise 4
((−2)5 )8
Let 𝑠 be a number.
(𝑠17 )4 =
=
(𝑛 𝑡𝑖𝑚𝑒𝑠)
Exercise 5
Sarah wrote that (35 )7 = 312 . Correct her mistake. Write an exponential expression using a base of 3 and exponents of
5, 7, and 12 that would make her answer correct.
Lesson 5:
Numbers in Exponential Form Raised to a Power
12
Lesson 5
8•1
Notes:
For any numbers 𝑥 and 𝑦, and positive integer 𝑛,
(𝑥𝑦)𝑛 = 𝑥 𝑛 𝑦 𝑛
because
(𝑥𝑦)𝑛 = (𝑥𝑦)
⋯ (𝑥𝑦)
������
���
𝑛 𝑡𝑖𝑚𝑒𝑠
(𝑦 ���
= (𝑥
∙ 𝑥 ⋯��
𝑥) ∙ ��
∙ 𝑦 ⋯��
𝑦)
�����
𝑛 𝑡𝑖𝑚𝑒𝑠
𝑛 𝑛
=𝑥 𝑦
𝑛 𝑡𝑖𝑚𝑒𝑠
Exercise 6
Exercise 9
(11 × 4)9 =
Let 𝑥 be a number.
(5𝑥)7 =
Exercise 7
Exercise 10
(32
×7
4 )5
Let 𝑥 and 𝑦 be numbers.
(5𝑥𝑦 2 )7 =
=
Exercise 8
Exercise 11
Let 𝑎, 𝑏, and 𝑐 be numbers.
(32 𝑎4 )5 =
Let 𝑎, 𝑏, and 𝑐 be numbers.
(𝑎2 𝑏𝑐 3 )4 =
Exercise 13
Exercise 13
𝑥 𝑛
� � =
𝑦
4
3
� � =
5
Lesson 5:
Numbers in Exponential Form Raised to a Power
13
Lesson 5
8•1
Problem Set
Exponent Laws
1. Multiplying same base: 𝑑 4 ∙ 𝑑 8 = 𝑑 4+8 = 𝑑12
2. Dividing same base:
𝑡6
𝑡4
= 𝑡 6−4 = 𝑡 2
Add exponents
Subtract exponents (top – bottom)
3. Powers of exponents: (𝑎3 )5 = 𝑎3∙5 = 𝑎15
Multiply exponents
4. Powers of products: (𝑎2 𝑏6 )4 = 𝑎2∙4 𝑏6∙4 = 𝑎8 𝑏24
Multiply each exponent
Directions: Simplify each expression using the exponent laws above and state which law you used.
1. (53 )4 =
2. 310 ∙ 35 =
3.
(85 × 62 )6 =
4. 𝑥 4 ∙ 𝑥 4 =
5. (3𝑐 3 )5 =
6.
47
43
=
𝑟 10
7. �
3
� =
𝑟6
𝑥 5 𝑦5
3
8. �𝑦4 ∙ 𝑥 4 � =
Lesson 5:
Numbers in Exponential Form Raised to a Power
14
Lesson 6
8•1
Lesson 6: Negative and Zero Exponents
Notes:
Exercise 1
Write the expanded form of 8,374 using the exponential notation.
Exercise 2
Write the expanded form of 6,985,062 using the exponential notation.
Exercise 3
What is the value of (3 × 10−2 )?
Lesson 6:
Negative and Zero Exponents
15
Lesson 6
8•1
Exercise 4
What is the value of (3 × 10−5 )?
Exercise 5
Write the complete expanded form of the decimal 4.728 in exponential notation.
For Exercises 5–10, write an equivalent expression, in exponential notation, to the one given and simplify as much as
possible.
Exercise 5
Exercise 8
5−3 =
Let 𝑥 be a nonzero number.
Exercise 6
Exercise 9
1
=
89
Let 𝑥 be a nonzero number.
Exercise 7
Exercise 10
3 ∙ 2−4 =
Let 𝑥, 𝑦 be two nonzero numbers.
𝑥 −3 =
1
=
𝑥9
𝑥𝑦 −4 =
Lesson 6:
Negative and Zero Exponents
16
Lesson 6
8•1
Problem Set
Let 𝑥, 𝑦 be numbers (𝑥, 𝑦 ≠ 0). Simplify each of the following expressions of numbers.
1.
𝑦 12
𝑦 12
=
2. 915 ∙
1
915
=
3. (7(123456.789)4 )0 =
4. 22 ∙
5.
1
25
∙ 25 ∙
𝑥 41 𝑦 15
∙
𝑦 15 𝑥 41
1
22
=
=
Write each number below in complete expanded exponential form.
6. 3,482
7. 65.93
8. .3875
9. 1,045,300
Lesson 6:
Negative and Zero Exponents
17