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ALGEBRA 1

Teacher’s Name:

Unit 4 Chapter 7

This book belongs to:

Updated FALL 2016

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Algebra 1 Section 7.1 Day 1 Notes: Multiplication Properties of Exponents What is a monomial?

Example 1: Determine whether each expression is a monomial. Explain your reasoning.

a) 17 − 𝑐 b) 8f 2g c) 34

d) 5t

Power: Product of Powers Property:

Expression Expression as repeated multiplication Number of Factors Simplified expression

4 57 7⋅

5x x⋅

()7 ⋅ ()2

Example 2: Simplify:

a) (r4)(-12r7) b) (6cd5)(5c5d2)

34 =

∴ am•an =

4

Power of a Power Property:

Expression Expanded Expression Expression as repeated multiplication Number of Factors

Simplified expression

( )235

( )33a

[(♥)2]5

Example 3: Simplify: a) (23)5 b) [(42)2]3 Power of a Product Property:

Expression Expanded Expression Group Like Terms (using the Commutative Property)

Simplified expression

( )24z

(−3𝑥)4

−(3𝑥)4

(2♥∆)3

Example 4:

a) Simplify (3a3b)4 b) Find the volume of a cube with side length 5xyz.

∴

∴

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Algebra 1 Section 7.1 Day 2 Notes: Multiplication Properties of Exponents Warm-up 1.) Is 2

𝑥 a monomial? Why or why not? 2.) Simplify. (2𝑥𝑦2)(3𝑥3𝑦5)

3.) Simplify. (𝑥3)2 4.) Simplify. (2𝑥2𝑦)3 ____________________________________________________________________________________________________________ To simplify a monomial expression, be sure that: - each variable base appears _______________. - there are no ________________________. - all fractions are ______________. Example 5: Simplify: a) [(8g3h4)2]2(2gh5)4 b) [(2c2d3)2]3(3c5d2)3 Example 6: Express the area of each figure as a monomial. a) b)

Example 7: Express the volume of the solid as a monomial.

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Algebra 1 Section 7.1 Worksheet Determine whether each expression is a monomial. Write yes or no. Explain your reasoning.

1. 21𝑎2

7𝑏 2.

𝑏3𝑐2 𝟐

Simplify each expression.

3. (–5𝑥2y)(3𝑥4) 4. (2a𝑏2𝑓2)(4𝑎3𝑏2𝑓2) 5. (3a𝑑4)(–2𝑎2) 6. (4𝑔3h)(–2𝑔5) 7. (–15x𝑦4)�− 1

3 𝑥𝑦3� 8. (−𝑥𝑦)3(xz)

9. (−18𝑚2𝑛)2 �– 1

6𝑚𝑛2� 10. (0.2𝑎2𝑏3)2

11. � 23𝑝�

2 12. � 1

4𝑎𝑑3�

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13. (0.4𝑘3)3 14. [(42)2]2 GEOMETRY Express the area of each figure as a monomial. 15. 16.

GEOMETRY Express the volume of each solid as a monomial. 17. 18. COUNTING A panel of four light switches can be set in 24 ways. A panel of five light switches can set in twice this many ways.

In how many ways can five light switches be set?

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Algebra 1 Section 7.2 Notes: Division Properties of Exponents Warm-up Determine whether the following is a monomial. Explain your reasoning. 1.) −𝟓𝒙𝟐 2.) 𝒙𝟑 − 𝒚𝟑 What is the product? 3.) (3ab4)(–a4b2) 4.) (–3x2y3z2)(–17x3z4) ___________________________________________________________________________________________________________ Quotient of Powers:

Expression Expression as repeated multiplication Simplified Expression Simplified Expression as Power

7

2

22

𝑥5

𝑥

Example 1: Simplify: a) b) Power of a Quotient:

Expression Expanded Expression Product of Fractions Simplified Expression 3

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�𝑥2�

4

Example 2: Simplify: a) b)

7 12

6 3

x yx y

3 9

2

a bab

33 245

c d

24 6

5

34m np q

∴

∴

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Zero Exponent Property:

Expression Simplified using Quotient Rule Answer

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𝑥7

𝑥7

Example 3: Simplify: a) b) Negative Exponent Property: Use your calculator to write each expression using positive exponents only. Write each with a base of 2.

a) 12−

b) 22−

c) 32−

Use your calculator to simplify each negative exponent.

a) 11

2−

b) 21

2−

c) 31

2−

Example 4: Simplify: a) b)

Example 5: A lab technician draws a sample of blood. A cubic millimeter of the blood contains 223 white blood cells and 225 red blood cells. What is the ratio of white blood cells to red blood cells?

08 7

5 10

128

m nm n

0 3

2

m nn

4 9

6

x yz

−

−

3 5

5 4 8

7515

p qp q r

−

− −

∴

∴

∴

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Algebra 1 Section 7.2 Worksheet Simplify each expression. Assume that no denominator equals zero.

1. 88

84 2. 𝑎

4𝑏6

𝑎𝑏3 3. 5𝑐

2𝑑3

−4𝑐2𝑑

4. 8𝑦7𝑧6

4𝑦6𝑧5 5. � 6𝑤5

7𝑝6𝑟3�2 6. −4𝑥

2

24𝑥5

7. 𝑥3(𝑦−5)( 𝑥−8) 8. 12−2 9. �37�−2

10. −15𝑤0𝑢−1

5𝑢3 11. 8𝑐3𝑑2𝑓4

4𝑐−1𝑑2𝑓−3 12. �𝑥

−3𝑦5

4−3�0

13. 6𝑓−2𝑔3ℎ5

54𝑓−2𝑔−5ℎ3 14. 𝑚

−2𝑛−5

(𝑚4𝑛3)−1 15. �𝑗

−1𝑘3�−4

𝑗3𝑘3

16. �𝑞−1𝑟3

𝑞𝑟3�−5

17. �7𝑐−3𝑑3

𝑐5𝑑ℎ−4�−1

18. The Moon is approximately 254 kilometers away from Earth on average. The Olympus Mons volcano on Mars stands 25

kilometers high. How many Olympus Mons volcanoes, stacked on top of one another, would fit between the surface of the Earth and the Moon?

19. COUNTING The number of three-letter “words” that can be formed with the English alphabet is 263. The number of five-letter

“words” that can be formed is 265. How many times more five-letter “words” can be formed than three-letter “words”? 20. E-MAIL Spam (also known as junk e-mail) consists of identical messages sent to thousands of e-mail users. People often obtain

anti-spam software to filter out the junk e-mail messages they receive. Suppose Yvonne’s anti-spam software filtered out 102 e-mails, and she received 104 e-mails last year. What fraction of her e-mails were filtered out? Write your answer as a monomial.

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Algebra 1 7.1 – 7.2 Review Simplify the expression.

1. 53 4 4 • 2. ( )432 3. ( )224xy 4. (-2a)4 5. 3 4 1b b b−• • 6. 0a Evaluate the expression. Write answer as a fraction in simplest form.

7. 42− 8. 2

41 −

9. (-2)0 24

1−⋅ 10. (3-3)-1

Rewrite the expression with positive exponents in simplest form. 11. 34 yx − 12.

22 yx21

−− 13. y3 2− 14. (2x2y3)-2

Simplify the expression.

15. 4

2

2

3xyy4x

16.

323

3xyyx2

−

17. 2 3 2 3[( 2 ) ]xy z−−

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Simplify the expression. 18. 1 3 2 3 2 2 4 2(3 ) (4 )x y z x y z− − − −

19. Determine if the expression is a monomial. 20. 3

4𝑥 21. 3

−1

𝑥−3 22. 𝑦

2

4𝑥3 23. 3𝑥

4

5𝑦−2

Express the area of the figure as a monomial. 24. A square with side lengths of x3. 25. A circle with a radius of x4.

x3 x4

4𝑎2𝑏−3

(2𝑎−3𝑏5)4

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Algebra 1 Section 7.3 Notes Day 1: Rational Exponents Warm-up

1. Simplify (2𝑥2)2𝑦3

24𝑥−2𝑦 2. Simplify �2

3𝑥𝑦3

8𝑥2𝑦�2

___________________________________________________________________________________________________________ Rational Exponents – exponents that are fractions or decimals that terminate or repeat. Use your calculator to evaluate each expression.

a) 1612 b) 9

12 c) 64

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What do you notice? What is another way to write each of these? Example 1: Write each expression in radical form. a) b) c) Example 2: Write each radical in exponential form. a) b) nth Root: Use a calculator to determine the following.

a) √83 b) √164 So what does the nth root mean? Example 3: Simplify: a) b)

1281

12(6 )a

1225b

38 5x

4 256 5 32

∴𝑏12 =

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Example 4: Combine the two ideas from the previous page to simplify these problems: a) b) More Rational Exponents:

How do you think you can rewrite 823 in radical form?

Example 5: Simplify: a) b)

131331

142401

2532

5281

∴𝑏𝑚𝑛 =

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Algebra 1 – 7.3 Exponent Chart 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81 23 = 8 33 = 27 43 = 64 53 = 125 63 = 216 73 = 343 83 = 512 93 = 729 24 = 16 34 = 81 44 = 256 54 = 625 64 = 1296 74 = 2401 84 = 4096 94 = 6561 25 = 32 35 = 243 45 = 1024 55 = 3125 65 = 7776 75 = 16807 26 = 64 36 = 729 46 = 4096 56 = 15625 27 = 128 37 = 2187 47 = 16384 28 = 256 38 = 6561 29 = 512 210 = 1024 211 = 2048 212 = 4096 213 = 8192 214 = 16384

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Algebra 1 Section 7.3 Notes Day 2: Rational Exponents Warm-up Rewrite in radical form or exponential form and then simplify. 1. √𝟏𝟔𝟒

2. 𝟐𝟕𝟒𝟑

Simplify using exponent rules.

3. �𝒙𝟏𝟐�

𝟐

4. (𝒚𝟒)𝟏𝟒

____________________________________________________________________________________________________________ Power Property of Equality: Determine the value of x in each of the following:

a) 5𝑥 = 53 b) 4𝑥 = 16 c) 4𝑥 = 8 Example 6: Simplify: a) b) Real-World Example: a) The velocity v in feet per second of a freely falling object that has fallen h feet can be represented

by v = 8ℎ12 . Find the distance that an object has fallen if its velocity is 96 feet per second.

b) The profit P of a company in thousands of dollars can be modeled by P = 12.75�𝑐2 5

, where c is the number of customers in hundreds. If the profit of the company is $51,000, how many customers do they have?

9 729x = 2 116 8x− =

∴𝑏𝑥 =

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Algebra 1 Section 7.3 Worksheet Write each expression in radical form, or write each radical in exponential form.

1. √13 2. √37 3. √17𝑥

4. (7𝑎𝑏)12 5. 21𝑧

12 6. 13(𝑎𝑏)

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Simplify.

7. √10245 8. 51213 9. � 32

1024�15

Solve each equation.

10. 3𝑥 = 729 11. 4𝑥 = 4096 12. 5𝑥 = 15,625

13. 6𝑥 + 3 = 7776 14. 3𝑥 − 3 = 2187 15. 43𝑥 + 4 = 16,384

16. WATER The flow of water F in cubic feet per second over a wier, a small overflow dam, can be represented by

F = 1.26𝐻32, where H is the height of the water in meters above the crest of the wier. Find the height of the water if the flow of the

water is 10.08 cubic feet per second.

17. ELECTRICITY The relationship of the current, the power, and the resistance in an appliance can be modeled by I𝑅12 = �p ,

where I is the current in amperes, P is the power in watts, and R is the resistance in ohms. Find the power that an appliance is using if the current is 2.5 amps and the resistance is 16 ohms.

18. BIOLOGY The relationship between the mass m in kilograms of an organism and its metabolism P in Calories per day can be

represented by P = 73.3�𝑚3 4

. Find the mass of an organism that has a metabolism of 586.4 Calories per day.

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Algebra 1 Section 7.5 Day 1 Notes: Exponential Functions Warm-up Simplify. Solve the equation.

1. (𝟐𝟕)𝟓𝟑 2. 𝟓𝟐𝒙−𝟓 = 𝟏𝟐𝟓

___________________________________________________________________________________________________________ Exponential Function: Input – the values ________________________. (_______________) Output – the __________________ from substituting in x. (______________) Example 1:

Graph y = 4x. Find the y-intercept and state the domain and range.

Example 2: Graph y = 3x + 1. Find the y-intercept and state the domain and range.

What does adding 1 to the function do?

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Example 3: Graph . Find the y-intercept and state the domain and range.

How does a fraction impact the graph?

Example 4:

Graph 𝑦 = 3 �12�𝑥. Find the y-intercept and state the domain and range.

Graphs of Exponential Functions:

Summary of Graphs of Exponential Functions Exponential Growth Exponential Decay Domain Range

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x

y =

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Algebra 1 Section 7.5 Day 1 Homework 10. 12. 14. 16. 18.

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Algebra 1 Section 7.5 Day 2 Notes: Exponential Functions Warm-up The graph of y = 4x is shown. State the y-intercept. ___________________________________________________________________________________________________________ Example 5: Some people say that the value of a new car decreases as soon as it is driven off the dealer’s lot. The function V = 25,000 ● 0.82t

models the depreciation of 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 the car was purchased. Graph the function.

(Graph on Calculator.)

a. Which variable resembles the input values? Which variable resembles the output values?

b. What values of V and t are meaningful in the function? What is the domain? What is the range?

c. What is the value of the car after five years?

Example 6: The value of Royce Company’s computer equipment is decreasing in value according to the following function. y = 4000(0.87)𝑥 In the equation, x is the number of years that have elapsed since the equipment was purchased and y is

the value in dollars. What was the value 5 years after it was purchased? Round your answer to the nearest dollar.

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Example 7: Determine whether the set of data displays exponential behavior or linear behavior. Explain why for each. a) b) c) Example 8: Is the graph given a linear or an exponential function? a. b. Think about it: Which function would increase more in the long run? Exponential or linear?

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Algebra 1 Section 7.5 Worksheet Graph each function. Find the y-intercept and state the domain and range. 1. y = � 1

10�𝑥 2. y = 3𝑥

3. y = �1

6�𝑥 4. y = 2𝑥

Determine whether the set of data shown below displays exponential behavior or linear behavior. Explain why for each.

7. BIOLOGY Suppose a certain cell reproduces itself in four hours. If a lab researcher begins with 50 cells, how many cells will there

be after one day, two days, and three days? (Hint: Use the exponential function y = 50(2𝑥).)

5. x 2 5 8 11

y 480 120 30 7.5

6. x 21 18 15 12

y 30 23 16 9

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8. LEARNING Ms. Klemperer told her English class that each week students tend to forget one sixth of the vocabulary words they learned the previous week. Suppose a student learns 60 words. The number of words remembered can be described by the function W(x) = 60 �5

6�𝑥, where x is the number of weeks that pass.

a. Which variable resembles the input values? Which variable resembles the output values?

b. What is the domain? What is the range?

c. How many words will the student remember after 3 weeks? 9. Is the graph given a linear or an exponential function? a. b.

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Algebra 1 Section 7.6 Notes: Growth and Decay Warm-up 1. Approximate 𝟓(𝟑𝟏.𝟖) to the nearest hundredth. 2. What is the y-intercept of 𝒚 = 𝟒𝒙+𝟐? ____________________________________________________________________________________________________________

Example 1: a. 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.

b. Estimate the amount of the donations 5 years after the start of the recession.

Example 2: a. 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 2008.

b. What will be the population of Flat Creek in the year 2018?

Equations for Exponential Growth or Decay y=a(1 + r)t y=a(1 – r)t

y = r = a = t =

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Example 3: When Lucy was 10 years old, her father invested $2500 in a fixed rate savings account at a rate of 8% compounded semiannually. When Lucy turns 18, the money will help to buy her a car. What amount of money will Lucy receive from the investment?

Example 4: Determine what type of function it is and complete the problem.

a. When Jing May was born, her grandparents invested $1000 in a fixed rate savings account at a rate of 7% compounded annually. The money will go to Jing May when she turns 18 to help with her college expenses. What amount of money will Jing May receive from the investment?

b. A charitable organization found that the value of its clothing donations dropped by 2.5% per year. Before this downturn in donations, the organization received clothing valued at $24,000. Write an equation to represent the value of the charity’s clothing donations since the beginning of the downturn. Estimate the value of the clothing donations 3 years after the start of the downturn.

c. In 2008, Scioto School District had a student population of about 4500 students, and a growth rate of about 0.15% per year. Write an equation to represent the student population of the Scioto School District since the year 2008. According to the equation, what will be the student population of the Scioto School District in the year 2014?

Equation for Compound Interest A = P(1 + 𝒓

𝒏)nt

A = r = t = P = n =

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Algebra 1 Section 7.6 Worksheet 1. COMMUNICATIONS Sports radio stations numbered 220 in 1996. The number of sports radio stations has since increased by

approximately 14.3% per year. a. Write an equation for the number of sports radio stations for t years after 1996. b. If the trend continues, predict the number of sports radio stations in 2015.

2. INVESTMENTS Determine the amount of an investment if $500 is invested at an interest rate of 4.25% compounded quarterly for 12 years.

3. INVESTMENTS Determine the amount of an investment if $300 is invested at an interest rate of 6.75% compounded semiannually for 20 years.

4. HOUSING The Greens bought a condominium for $110,000 in 2010. If its value appreciates at an average rate of 6% per year, what will the value be in 2015?

5. DEFORESTATION During the 1990s, the forested area of Guatemala decreased at an average rate of 1.7%. a. If the forested area in Guatemala in 1990 was about 34,400 square kilometers, write an equation for the forested area for t years

after 1990. b. If this trend continues, predict the forested area in 2015.

6. BUSINESS A piece of machinery valued at $25,000 depreciates at a steady rate of 10% yearly. What will the value of the piece of machinery be after 7 years?

7. TRANSPORTATION A new car costs $18,000. It is expected to depreciate at an average rate of 12% per year. Find the value of the car in 8 years.

8. POPULATION The population of Osaka, Japan, declined at an average annual rate of 0.05% for the five years between 1995 and 2000. If the population of Osaka was 11,013,000 in 2000 and it continues to decline at the same rate, predict the population in 2050.

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9. POPULATION The population of New York City increased from 8,008,278 in 2000 to 8,168,388 in 2005. The annual rate of population increase for the period was about 0.4%.

a. Write an equation for the population t years after 2000. b. Use the equation to predict the population of New York City in 2015. 10. SAVINGS The Fresh and Green Company has a savings plan for its employees. If an employee makes an initial contribution of

$1000, the company pays 8% interest compounded quarterly. a. If an employee participating in the plan withdraws the balance of the account after 5 years, how much will be in the account? b. If an employee participating in the plan withdraws the balance of the account after 35 years, how much will be in the account? 11. HOUSING Mr. and Mrs. Boyce bought a house for $96,000 in 1995. The real estate broker indicated that houses in their area

were appreciating at an average annual rate of 7%. If the appreciation remained steady at this rate, what was the value of the Boyce’s home in 2009?

12. MANUFACTURING Zeller Industries bought a piece of weaving equipment for $60,000. It is expected to depreciate at an

average rate of 10% per year. a. Write an equation for the value of the piece of equipment after t years. b. Find the value of the piece of equipment after 6 years. 13. FINANCES Kyle saved $500 from a summer job. He plans to spend 10% of his savings each week on various forms of

entertainment. At this rate, how much will Kyle have left after 15 weeks? 14. TRANSPORTATION Tiffany’s mother bought a car for $9000 five years ago. She wants to sell it to Tiffany based on a 15%

annual rate of depreciation. At this rate, how much will Tiffany pay for the car?

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Algebra 1 7.3, 7.5, 7.6 Review Write each expression in radical form, or write each radical in exponential form.

1. 2 ab 2. 1282 3.

1235y

Evaluate:

4. 3 216 5. 153125 6.

56729

Solve: 7. 8 4096x = 8. 34 32x− =

9. a. Graph 12

x

y =

. b. Find: y-intercept: ______________

Domain: ________________ Range: _________________ 10. Determine whether the set of data displays exponential behavior or linear behavior. 11. The value of Royce Company’s computer equipment is decreasing in value according to the following function. y = 4000(0.87)𝑥 In the equation, x is the number of years that have elapsed since the equipment was purchased and y is the value in dollars. What was the value 5 years after it was purchased?

x 2 5 8 11

y 480 120 30 7.5

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12. Hans opens a savings account by depositing $1200 in an account that earns 3 percent interest compounded weekly. How much will his investment be worth in 10 years? Assume that there are exactly 52 weeks in a year and round your answer to the nearest cent. 13. In 2007 the U.S. Census Bureau estimated the population of the United States estimated at 301 million. The annual rate of growth is about 0.89%. At this rate, what is the expected population at the time of the 2020 census? Round your answer to the nearest million.