1. 7-Chapter Notes
Algebra 1

2. Multiplication properties of Exponents
7-1 Notes for Algebra 1
Multiplication properties of Exponents

3. 7.1 pg o, 69-84(x3)

4. Monomial (only one term)
A number, a variable, or the product of a number and one or more variables with nonnegative integer exponents. An expression that involves division by a variable is not a monomial.

5. Constant
A monomial that is a real number.

6. Example 1: Identify Monomials
Determine whether each expression is a monomial. Write yes or no. Explain your reasoning. 1.) 17−𝑐 2.) 8 𝑓 2 𝑔 3.) ) 5 𝑡

7. Example 1: Identify Monomials
Determine whether each expression is a monomial. Write yes or no. Explain your reasoning. 1.) 17−𝑐 No, it involves subtraction, so it has more than one term. 2.) 8 𝑓 2 𝑔 Yes, it involves the product of a number and two variables. 3.) 3 4 Yes, the expression is a constant. 4.) 5 𝑡 No, it has a variable in the denominator.

8. Product of Powers 𝑥 3 ∙ 𝑥 6 𝑥 3+6 𝑥 9
To multiply two powers that have the same base, add their exponents.
𝑥 3 ∙ 𝑥 6
𝑥 3+6
𝑥 9

9. Example 2: Product of Powers
Simplify each expression. 1.) 𝑟 4 −12𝑟 7 2.) 6𝑐 𝑑 5 5 𝑐 5 𝑑 2

10. Example 2: Product of Powers
Simplify each expression. 1.) 𝑟 4 −12𝑟 7 2.) 6𝑐 𝑑 5 5 𝑐 5 𝑑 2 −12 𝑟 𝑐 6 𝑑 7

11. Power of a Power
To find the power of a power, multiply the exponents.
𝑥 3 6
𝑥 3∙6
𝑥 18

12. Example 3: Power of a Power
Simplify

13. Example 3: Power of a Power
Simplify

14. Power of a Product 2 𝑥 2 𝑦 5 𝑧 4 3 2 3 ∙ 𝑥 2∙3 ∙ 𝑦 5∙3 ∙ 𝑧 4∙3
To find the power of a product, find the power of each factor and multiply.
2 𝑥 2 𝑦 5 𝑧 4 3
2 3 ∙ 𝑥 2∙3 ∙ 𝑦 5∙3 ∙ 𝑧 4∙3
8𝑥 6 𝑦 15 𝑧 12

15. Example 4: Power of a Product
Express the volume of a cube with side length 5𝑥𝑦𝑧 as a monomial.

16. Example 4: Power of a Product
Express the volume of a cube with side length 5𝑥𝑦𝑧 as a monomial.
125𝑥 3 𝑦 3 𝑧 3

17. Simplify Expressions
To simplify a monomial expression, write an equivalent expression in which:
Each variables base appears exactly once,
There are no powers of powers, and
All fractions are in simplest form.

18. Example 5: Simplify Expressions

19. Example 5: Simplify Expressions

20. Division Property of Exponents
7.2 Notes for Algebra 1
Division Property of Exponents

21. 7.2 pg o, 66-87(x3)

22. Quotient of Powers 15𝑥 7 𝑦 3 45𝑥 3 𝑦 15 45 ∙ 𝑥 7−3 1 ∙ 𝑦 3−1 1
To divide two powers with the same base, subtract the exponents.
15𝑥 7 𝑦 𝑥 3 𝑦
15 45 ∙ 𝑥 7−3 1 ∙ 𝑦 3−1 1
𝑥 4 𝑦 2 3

23. Example 1: Quotient of Powers
Simplify 𝑥 7 𝑦 12 𝑥 6 𝑦 3

24. Example 1: Quotient of Powers
Simplify. 𝑥 7 𝑦 12 𝑥 6 𝑦 3 𝑥𝑦 9

25. Power of a Quotient 2 4 ∙ 𝑦 6∙4 3 4 ∙ 𝑥 3∙4 ∙ 𝑧 2∙4 16 𝑦 24 81𝑥 12 𝑧 8
To find the power of a quotient, find the power of a numerator and the power of the denominator.
2𝑦 6 3𝑥 3 𝑧
2 4 ∙ 𝑦 6∙ ∙ 𝑥 3∙4 ∙ 𝑧 2∙4
16 𝑦 𝑥 12 𝑧 8

26. Example 2: Power of a Quotient
Simplify 𝑐 3 𝑑

27. Example 2: Power of a Quotient
Simplify 𝑐 3 𝑑 𝑐 9 𝑑

28. Zero exponent
Any nonzero number raised to the zero power is equal to 1. 2𝑥 𝑦 3 𝑧 5 0 =1

29. Example 3: Zero Exponent
Simplify. 1.) 12𝑚 8 𝑛 7 8𝑚 50 𝑛 ) 𝑚 0 𝑛 3 𝑛 2

30. Example 3: Zero Exponent
Simplify. 1.) 12𝑚 8 𝑛 7 8𝑚 50 𝑛 ) 𝑚 0 𝑛 3 𝑛 2 𝑛

31. Negative Exponent Property
For any nonzero number 𝑎 and any integer 𝑛, 𝑎 −𝑛 = 1 𝑎 𝑛
10 𝑥 −3 𝑦 𝑥 7 𝑦 −6 𝑧 −2
10 25 ∙ 1 𝑥 7 𝑥 3 ∙ 𝑦 6 𝑦 6 1 ∙ 𝑧 2 1
2𝑦 12 𝑧 2 5𝑥 10

32. Example 4: Negative Exponents
Simplify. 1.) 𝑥 −4 𝑦 9 𝑧 −6 2.) 75𝑝 3 𝑚 −5 15𝑝 5 𝑚 −4 𝑟 −8

33. Example 4: Negative Exponents
Simplify. 1.) 𝑥 −4 𝑦 9 𝑧 −6 𝑦 9 𝑧 6 𝑥 4 2.) 75𝑝 3 𝑚 −5 15𝑝 5 𝑚 −4 𝑟 −8 5𝑟 8 𝑝 2 𝑚

34. Order of Magnitude (used to compare measures to estimate and preform rough calculations)
The quantity of a number rounded to the nearest power of ,000,000,

35. Example 5: Apply Properties of Exponents
Darin has $123,456 in his savings account. Tab has $156 in his savings account. Determine the order of magnitude of Darin’s account and Tab’s account. How many orders of magnitude as great is Darin’s account as Tab’s account?

36. Example 5: Apply Properties of Exponents
Darin has $123,456 in his savings account. Tab has $156 in his savings account. Determine the order of magnitude of Darin’s account and Tab’s account. How many orders of magnitude as great is Darin’s account as Tab’s account? Darin’s: 10 5 , Tab’s: 10 2 , Darin’s account is 3 orders of magnitude as great as Tab’s account.

37. 7-3 Notes for Algebra 1
Rational Exponents

38. 7-3 pg o, (x3)

39. Rational exponents
For any nonnegative real number 𝑏, 𝑏 = 𝑏 17

40. Example 1: Radical and Exponential Forms
Write each expression in radical form, or write each radical in exponential form. 1.) ) 38 3.) 12𝑚 ) 32𝑤

41. Example 1: Radical and Exponential Forms
Write each expression in radical form, or write each radical in exponential form. 1.) ) 38 3.) 12𝑚 ) 32𝑤 𝑚 32𝑤 1 2

42. 𝑛th root
For any real numbers 𝑎 and 𝑏 and any positive integer 𝑛. If 𝑎 𝑛 =𝑏, then 𝑎 is the 𝑛th root of 𝑏.
6 64 2

43. Example 2: 𝑛th roots
Simplify. 1.) ) 6 15,625

44. Example 2: 𝑛th roots
Simplify. 1.) ) 6 15,625 5

45. 𝑏 1 𝑛
For any positive real number 𝑏 and any integer 𝑛>1, 𝑏 1 𝑛 = 𝑛 𝑏 .
4 16
4 2∙2∙2∙2 2

46. Example 3: Evaluate 𝑏 1 𝑛 Expressions
Simplify. 1.) )

47. Example 3: Evaluate 𝑏 1 𝑛 Expressions
Simplify. 1.) )

48. 𝑏 𝑚 𝑛
For any positive real number 𝑏 and any integer 𝑚 and 𝑛>1.
𝑥 3 7
7 𝑥 or 𝑥 3

49. Example 4: Evaluate 𝑏 𝑚 𝑛 Expressions
Simplify. 1.) )

50. Example 4: Evaluate 𝑏 𝑚 𝑛 Expressions
Simplify. 1.) ) , 049

51. Power Property of Equality
For any real number 𝑏>0 and 𝑏≠1, 𝑏 𝑥 = 𝑏 𝑦 if and only if 𝑥=𝑦.
7 𝑥 = 7 13
𝑥=13

52. Example 5: Solve Exponential Equations
Solve each equation. 1.) 9 𝑥 =729 2.) 16 2𝑥−1 =8

53. Example 5: Solve Exponential Equations
Solve each equation. 1.) 9 𝑥 =729 2.) 16 2𝑥−1 =8 9 𝑥 = 𝑥−1 = 𝑥=3 2𝑥−1= 3 4 2𝑥= 7 4 𝑥= 7 8

54. Example 6: solve exponential equations
BIOLOGY: The population 𝑝 of a culture that begins with 40 bacteria and doubles every 8 hours can be modeled by 𝑝= 𝑡 8 , where 𝑡 is time in hours. Find 𝑡 if 𝑝=20,480.

55. Example 6: solve exponential equations
BIOLOGY: The population 𝑝 of a culture that begins with 40 bacteria and doubles every 8 hours can be modeled by 𝑝=40 2 𝑡 8 , where 𝑡 is time in hours. Find 𝑡 if 𝑝=20, ,480=40 2 𝑡 8 512= 2 𝑡 = 2 𝑡 8 9= 𝑡 8 72 ℎ𝑜𝑢𝑟𝑠

56. 7-4 Notes for Algebra 1
Scientific Notation

57. 7.4 pg o, 78-93(x3)

58. Scientific Notation
A way to calculate very large or small numbers easily.

59. Standard form to Scientific Notation
Move the decimal point until it is to the right of the first nonzero digit.
Note the number of places 𝑛 and the direction that you moved the decimal point.
If the decimal point is moved left, write the number as 𝑎× 10 𝑛 If the decimal point is moved right, write the number as 𝑎× 10 −𝑛 .
Remove the unnecessary zeros.

60. Example 1: Standard Form to Scientific Notation
Express each number in scientific notation. 1.) 4,062,000,000,000 2.)

61. Example 1: Standard Form to Scientific Notation
Express each number in scientific notation. 1.) 4,062,000,000,000 2.) × × 10 −7

62. Scientific Notation to Standard Form
In 𝑎× 10 𝑛 , note whether 𝑛>0 or 𝑛<0.
If 𝑛>0, move the decimal point 𝑛 places right If 𝑛<0, move the decimal point −𝑛 places left.
Insert zeros, decimal point and commas as needed for place value.

63. Example 2: Scientific Notation to Standard Form
Express each number in standard form. 1.) 6.49× ) 1.8× 10 −3

64. Example 2: Scientific Notation to Standard Form
Express each number in standard form. 1.) 6.49× ) 1.8× 10 −3 649,

65. Example 3: Multiply with Scientific Notation
Express the result in both Scientific Notation and standard form. Evaluate 5× 10 −6 2.3× 10 12

66. Example 3: Multiply with Scientific Notation
Express the result in both Scientific Notation and standard form. Evaluate 5× 10 −6 2.3× ×1 0 7 and 11,500,000

67. Example 4: Divide with Scientific Notation
Express the result in both scientific notation and standard form. Evaluate 4.5× × 10 10

68. Example 4: Divide with Scientific Notation
Express the result in both scientific notation and standard form. Evaluate 4.5× × × 10 −2 and 0.03

69. Example 5: Use Scientific Notation
WATERCRAFT: Last year Alison’s state registered over 400 thousand watercraft. Boat sales in her state generated more than $15.4 million in state sales taxes that same year.
Express the number of watercraft registered and the state sales tax generated from boat sales last year in Alison’s state in standard form.
Write each number in Scientific Notation.
How many watercraft have been registered in Alison’s state if 12 time the number registered in all? Write your answer in scientific notation and standard form.

70. Example 5: Use Scientific Notation
WATERCRAFT: Last year Alison’s state registered over 400 thousand watercraft. Boat sales in her state generated more than $15.4 million in state sales taxes that same year.
Express the number of watercraft registered and the state sales tax generated from boat sales last year in Alison’s state in standard form.
Watercraft registered: 400, Sales tax generated: $15,400,000
Write each number in Scientific Notation.
Water craft registered: 4× Sales tax generated: 1.54× 10 7
How many watercraft have been registered in Alison’s state if 12 time the number registered in all? Write your answer in scientific notation and standard form ,800, × 10 6

71. Exponential Functions
7-5 Notes for Algebra 1
Exponential Functions

72. 7-5 pg o, 48-66(x3)

73. Exponential Function
A function that can be described by an equation of the form 𝑦=a 𝑏 𝑥 , where 𝑎≠0, 𝑏>0, and 𝑏≠1. Exponential Growth models 𝑦=3 2 𝑥 𝑦= 5 𝑥 Exponential Decay models 𝑦= 2 3 𝑥

74. Example 1: Graph with 𝑎>0 and 𝑏>1
Graph 𝑦= 4 𝑥 . Find the 𝑦-intercept State the domain and range.

75. Example 1: Graph with 𝑎>0 and 𝑏>1
Graph 𝑦= 4 𝑥 . Find the 𝑦-intercept State the domain and range. (0, 1) 𝐷= 𝐴𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑅= 𝐴𝑙𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 7 6 5 4 3 2 1 -4 -3 -2 -1

76. Example 2: Graph with 𝑎>0 and 0<𝑏<1
Graph 𝑦= 1 4 𝑥 . Find the 𝑦-intercept State the domain and range.

77. Example 2: Graph with 𝑎>0 and 0<𝑏<1
Graph 𝑦= 1 4 𝑥 . Find the 𝑦-intercept State the domain and range. 0, 1 𝐷= 𝐴𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑅= 𝐴𝑙𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 7 6 5 4 3 2 1 -4 -3 -2 -1

78. Example 3: Use Exponential Function to Solve Problem
DEPRECIATION: Some people say that the value of a new car decreases as soon as it is driven off the lot.
The function 𝑉=25,000∙ 0.82 𝑡 models the depreciation in the value of a new car that originally cost $25,000. 𝑉 represents the value of the car and 𝑡 represents the time in years from the time of purchase.
Graph the function. What values of 𝑉 and 𝑡 are meaningful in the context of the problem?
What is the cars value after five years?

79. Example 3: Use Exponential Function to Solve Problem
25000
20000
15000
10000
5000 1 2 3 4 5 6 7 8 9
Graph the function What values of 𝑉 and 𝑡 are meaningful in the context of the problem?
What is the cars value after five years?
about $9270

80. Example 4: Identify Exponential Behavior
Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not. 𝑥 10 20 30 𝑦 25
62.5
156.25

81. Example 4: Identify Exponential Behavior
Determine whether the set of data shown below displays exponential behavior. Write yes or no. Explain why or why not. The domain values are at regular intervals, and the range values have a common factor of 2.5, so the set is probably exponential. Also, the graph shows rapidly increasing values of 𝑦 and 𝑥 increases. 𝑥 10 20 30 𝑦 25
62.5
156.25

82. 7-6 Notes for Algebra 1
Growth and Decay

83. 7-6 pg

84. Equation for Exponential Growth
𝑦=a 1+𝑟 𝑡 𝑦– is the final amount 𝑎– is the initial amount 𝑟—is the rate of change expressed as a decimal (𝑟>0) 𝑡—is time

85. Example 1: Exponential Growth
POPULATION: In 2008 the town of Flat Creek had a population of about 280,000 and a growth rate of 0.85% per year.
Write an equation to represent the population of Flat Creek since
According to the equation what will be the population of Flat Creek in the year 2025?

86. Example 1: Exponential Growth
POPULATION: In 2008 the town of Flat Creek had a population of about 280,000 and a growth rate of 0.85% per year.
Write an equation to represent the population of Flat Creek since
𝑦=280, 𝑡
According to the equation what will be the population of Flat Creek in the year 2025?
about 304,731

87. Compound Interest
Interest earned or paid on both the initial investment and previously earned interest. (applied exponential growth)

88. Equation for Compound Interest
𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 𝐴– is the current amount 𝑃– is the (Principle) initial amount 𝑟—is the annual rate of interest expressed as a decimal (𝑟>0) 𝑛—is the number of times the interest is compounded each year 𝑡—is time

89. Example 2: Compound Interest
COLLEGE When Jean was born, her grandparents invested $1000 in a fixed rate savings account at a rate of 7% compounded annually. Jean will receive the money when she turns 18 to help with her college expenses. What amount of money will Jean receive from the investment.

90. Example 2: Compound Interest
COLLEGE When Jean was born, her grandparents invested $1000 in a fixed rate savings account at a rate of 7% compounded annually. Jean will receive the money when she turns 18 to help with her college expenses. What amount of money will Jean receive from the investment. She will receive about $3380

91. Equation for Exponential Decay
𝑦=a 1−𝑟 𝑡 𝑦– is the final amount 𝑎– is the initial amount 𝑟—is the rate of change expressed as a decimal (0<𝑟<1) 𝑡—is time

92. Example 3: Exponential Decay
CHARITY During an economic recession, a charitable organization found that its donations dropped by 1.1% per year. Before the recession, its donations were$390,000.
Write an equation to represent the charity’s donations since the beginning of the recession
𝐴=390, 𝑡
Estimate the amount of the donations 5 years after the start of the recession.
about $369,017

93. Geometric Sequences as Exponential Functions
7-7 Notes for Algebra 1
Geometric Sequences as Exponential Functions

94. 7-7 pg , 45-69(x3)

95. Geometric Sequence
The first term is a nonzero and each term after the first is found by multiplying the previous term by a nonzero constant 𝑟 called the common ratio.

96. Example 1: Identify Geometric Sequences
Determine whether each sequence is arithmetic, geometric, or neither. Explain. 1.) 0, 8, 16, 24, 32, … 2.) 64, 48, 36, 27, …

97. Example 1: Identify Geometric Sequences
Determine whether each sequence is arithmetic, geometric, or neither. Explain. 1.) 0, 8, 16, 24, 32, … Arithmetic; the common difference is 8 2.) 64, 48, 36, 27, … Geometric; the common ratio is 3 4

98. Example 2: Find Terms of Geometric Sequences
Find the next three terms in each geometric sequence. 1.) 1, −8, 64, −512, … 2.) 40, 20, 10, 5, …

99. Example 2: Find Terms of Geometric Sequences
Find the next three terms in each geometric sequence. 1.) 1, −8, 64, −512, 4096, −32,768, 262,144 2.) 40, 20, 10, 5, 2.5, 1.25, , 5 4 , 5 8

100. 𝑛th term of a Geometric Sequence
The 𝑛th term 𝑎 𝑛 of a geometric sequence with first term 𝑎 1 and common ratio 𝑟 is given by the following formula where 𝑛 is any positive integer and 𝑎 1 , 𝑟≠0. 𝑎 𝑛 = 𝑎 1 𝑟 𝑛−1

101. Example 3: Find the 𝑛th term of a Geometric Sequence
1.) Write an equation for the nth term of the geometric sequence 1, −2, 4, −8, … 2.) Find the 12th term of this sequence.

102. Example 3: Find the 𝑛th term of a Geometric Sequence
1.) Write an equation for the nth term of the geometric sequence 1, −2, 4, −8, … 𝑎 𝑛 =1 −2 𝑛−1 2.) Find the 12th term of this sequence. −2048

103. Example 4: Graph a Geometric Sequence
ART A 50-pound ice sculpture is melting at a rate in which 80% of its weight remains each hour Draw graph to represent how many pounds of the sculpture is left at each hour.

104. Example 4: Graph a Geometric Sequence
ART A 50-pound ice sculpture is melting at a rate in which 80% of its weight remains each hour Draw graph to represent how many pounds of the sculpture is left at each hour. 60 50 40 30 20 10 1 2 3 4 5 6 7
0, 50
1, 40
2, 32
4, 20.48
3, 25.6 6, 5, 7,

105. 7-8 Notes for Algebra 1
Recursive Formulas

106. 7-8 pg

107. Recursive Formula
Allows you to find the nth term of a sequence by performing operations to one or more of the preceding terms.

108. Example 1: Use a Recursive Formula
Find the first five terms of the sequence in which 𝑎 1 =−8 and 𝑎 𝑛 =−2 𝑎 𝑛−1 +5, if 𝑛≥2.

109. Example 1: Use a Recursive Formula
Find the first five terms of the sequence in which 𝑎 1 =−8 and 𝑎 𝑛 =−2 𝑎 𝑛−1 +5, if 𝑛≥2. 𝑎 2 =21 𝑎 2 =−2 −8 +5 𝑎 2 =−37 𝑎 2 =− 𝑎 2 =79 𝑎 2 =−2 −37 +5 𝑎 2 =−153 𝑎 2 =−

110. Writing Recursive Formulas
Determine if the sequence is arithmetic or geometric by finding a common difference or common ratio.
Write a recursive formula Arithmetic Sequence 𝑎 𝑛 = 𝑎 𝑛−1 +𝑑 Geometric Sequence 𝑎 𝑛 = 𝑟∙𝑎 𝑛−1
State the first term and domain for 𝑛

111. Example 2: Write Recursive Formulas
Write a recursive formula for each sequence. 1.) 23, 29, 35, 41, … 2.) 7, −21, 63, −189, …

112. Example 2: Write Recursive Formulas
Write a recursive formula for each sequence. 1.) 23, 29, 35, 41, … 𝑎 1 =23; 𝑎 𝑛 = 𝑎 𝑛−1 +6; 𝑛≥2 2.) 7, −21, 63, −189, … 𝑎 1 =7; 𝑎 𝑛 = −3∙𝑎 𝑛−1 ; 𝑛≥2

113. Example 3: Write Recursive and Explicit Formulas
CARS The price of a car depreciates at the end of each year. a.) Write a recursive formula for the sequence b.) Write an explicit formula for the sequence
Year
Price ($) 1
12,000 2
7200 3
4320 4
2592

114. Example 3: Write Recursive and Explicit Formulas
CARS The price of a car depreciates at the end of each year. a.) Write a recursive formula for the sequence 𝑎 1 =12,000; 𝑎 𝑛 =0.6 𝑎 𝑛−1 b.) Write an explicit formula for the sequence 𝑎 𝑛 =12, 𝑛−1
Year
Price ($) 1
12,000 2
7200 3
4320 4
2592

115. Example 4: Translate between Recursive and Explicit Formulas
1.) Write a recursive formula for 𝑎 𝑛 =2𝑛−4. 2.) Write an explicit formula for 𝑎 1 =84; 𝑎 𝑛 =1.5 𝑎 𝑛−1 ; 𝑛≥2

116. Example 4: Translate between Recursive and Explicit Formulas
1.) Write a recursive formula for 𝑎 𝑛 =2𝑛−4. 𝑎 1 =−2; 𝑎 𝑛 = 𝑎 𝑛−1 +2; 𝑛≥2 2.) Write an explicit formula for 𝑎 1 =84; 𝑎 𝑛 =1.5 𝑎 𝑛−1 ; 𝑛≥2 𝑎 𝑛 = 𝑛−1 ; 𝑛≥2