student file a - greeleyschools...4๐ฅ๐ฅโ1 โ3 5. apply the properties of exponents to verify...
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10 9 8 7 6 5 4 3 2 1
Eureka Mathโข
Algebra II, Module 3
Student File_AContains copy-ready classwork and homework
Published by the non-profit Great Minds.
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A Story of Functionsยฎ
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Lesson 1: Integer Exponents
M3 Lesson 1 ALGEBRA II
Lesson 1: Integer Exponents
Classwork
Opening Exercise
Can you fold a piece of notebook paper in half 10 times?
How thick will the folded paper be?
Will the area of the paper on the top of the folded stack be larger or smaller than a postage stamp?
Exploratory Challenge
a. What are the dimensions of your paper?
b. How thick is one sheet of paper? Explain how you decided on your answer.
c. Describe how you folded the paper.
A STORY OF FUNCTIONS
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Lesson 1: Integer Exponents
M3 Lesson 1 ALGEBRA II
d. Record data in the following table based on the size and thickness of your paper.
Number of Folds 0 1 2 3 4 5 6 7 8 9 10
Thickness of the Stack (in.)
Area of the Top of the Stack (sq. in.)
e. Were you able to fold a piece of notebook paper in half 10 times? Why or why not?
f. Create a formula that approximates the height of the stack after ๐๐ folds.
g. Create a formula that will give you the approximate area of the top after ๐๐ folds.
h. Answer the original questions from the Opening Exercise. How do the actual answers compare to your original predictions?
A STORY OF FUNCTIONS
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Lesson 1: Integer Exponents
M3 Lesson 1 ALGEBRA II
Example 1: Using the Properties of Exponents to Rewrite Expressions
The table below displays the thickness and area of a folded square sheet of gold foil. In 2001, Britney Gallivan, a California high school junior, successfully folded a 100-square-inch sheet of gold foil in half 12 times to earn extra credit in her mathematics class.
Rewrite each of the table entries as a multiple of a power of 2.
Number of Folds
Thickness of the Stack (millionths of a meter)
Thickness Using a Power of 2
Area of the Top (square inches)
Area Using a Power of 2
0 0.28 0.28 โ 20 100 100 โ 20
1 0.56 0.28 โ 21 50 100 โ 2โ1
2 1.12 25
3 2.24 12.5
4 4.48 6.25
5 8.96 3.125
6 17.92 1.5625
Example 2: Applying the Properties of Exponents to Rewrite Expressions
Rewrite each expression in the form ๐๐๐ฅ๐ฅ๐๐, where ๐๐ is a real number, ๐๐ is an integer, and ๐ฅ๐ฅ is a nonzero real number.
a. (5๐ฅ๐ฅ5) โ (โ3๐ฅ๐ฅ2)
b. 3๐ฅ๐ฅ5
(2๐ฅ๐ฅ)4
c. 3
(๐ฅ๐ฅ2)โ3
A STORY OF FUNCTIONS
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Lesson 1: Integer Exponents
M3 Lesson 1 ALGEBRA II
d. ๐ฅ๐ฅโ3๐ฅ๐ฅ4
๐ฅ๐ฅ8
Exercises 1โ5
Rewrite each expression in the form ๐๐๐ฅ๐ฅ๐๐, where ๐๐ is a real number and ๐๐ is an integer. Assume ๐ฅ๐ฅ โ 0.
1. 2๐ฅ๐ฅ5 โ ๐ฅ๐ฅ10
2. 13๐ฅ๐ฅ8
3. 6๐ฅ๐ฅโ5
๐ฅ๐ฅโ3
4. ๏ฟฝ 3๐ฅ๐ฅโ22
๏ฟฝโ3
5. ๏ฟฝ๐ฅ๐ฅ2๏ฟฝ๐๐ โ ๐ฅ๐ฅ3
A STORY OF FUNCTIONS
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Lesson 1: Integer Exponents
M3 Lesson 1 ALGEBRA II
Problem Set 1. Suppose your class tried to fold an unrolled roll of toilet paper. It was originally 4 in. wide and 30 ft. long.
Toilet paper is approximately 0.002 in. thick. a. Complete each table, and represent the area and thickness using powers of 2.
Number of Folds
๐๐
Thickness After ๐๐ Folds
(๐ข๐ข๐ข๐ข.)
Number of Folds
๐๐
Area on Top After ๐๐ Folds
(๐ข๐ข๐ข๐ข๐๐) 0 0 1 1 2 2 3 3 4 4 5 5 6 6
b. Create an algebraic function that describes the area in square inches after ๐๐ folds. c. Create an algebraic function that describes the thickness in inches after ๐๐ folds.
Lesson Summary
The Properties of Exponents
For real numbers ๐ฅ๐ฅ and ๐ฆ๐ฆ with ๐ฅ๐ฅ โ 0, ๐ฆ๐ฆ โ 0, and all integers ๐๐ and ๐๐, the following properties hold.
1. ๐ฅ๐ฅ๐๐ โ ๐ฅ๐ฅ๐๐ = ๐ฅ๐ฅ๐๐+๐๐ 2. (๐ฅ๐ฅ๐๐)๐๐ = ๐ฅ๐ฅ๐๐๐๐ 3. (๐ฅ๐ฅ๐ฆ๐ฆ)๐๐ = ๐ฅ๐ฅ๐๐๐ฆ๐ฆ๐๐ 4. 1๐ฅ๐ฅ๐๐ = ๐ฅ๐ฅ
โ๐๐
5. ๐ฅ๐ฅ๐๐
๐ฅ๐ฅ๐๐= ๐ฅ๐ฅ๐๐โ๐๐
6. ๏ฟฝ๐ฅ๐ฅ๐ฆ๐ฆ๏ฟฝ๐๐
= ๐ฅ๐ฅ๐๐
๐ฆ๐ฆ๐๐
7. ๐ฅ๐ฅ0 = 1
A STORY OF FUNCTIONS
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Lesson 1: Integer Exponents
M3 Lesson 1 ALGEBRA II
2. In the Exit Ticket, we saw the formulas below. The first formula determines the minimum width, ๐๐, of a square piece of paper of thickness ๐๐ needed to fold it in half ๐๐ times, alternating horizontal and vertical folds. The second formula determines the minimum length, ๐ฟ๐ฟ, of a long rectangular piece of paper of thickness ๐๐ needed to fold it in half ๐๐ times, always folding perpendicular to the long side.
๐๐ = ๐๐ โ ๐๐ โ 23๏ฟฝ๐๐โ1๏ฟฝ
2 ๐ฟ๐ฟ =๐๐๐๐6
(2๐๐ + 4)(2๐๐ โ 1)
Use the appropriate formula to verify why it is possible to fold a 10 inch by 10 inch sheet of gold foil in half 13 times. Use 0.28 millionth of a meter for the thickness of gold foil.
3. Use the formula from Problem 2 to determine if you can fold an unrolled roll of toilet paper in half more than 10 times. Assume that the thickness of a sheet of toilet paper is approximately 0.002 in. and that one roll is 102 ft. long.
4. Apply the properties of exponents to rewrite each expression in the form ๐๐๐ฅ๐ฅ๐๐, where ๐๐ is an integer and ๐ฅ๐ฅ โ 0.
a. (2๐ฅ๐ฅ3)(3๐ฅ๐ฅ5)(6๐ฅ๐ฅ)2
b. 3๐ฅ๐ฅ4
(โ6๐ฅ๐ฅ)โ2
c. ๐ฅ๐ฅโ3๐ฅ๐ฅ5
3๐ฅ๐ฅ4
d. 5(๐ฅ๐ฅ3)โ3(2๐ฅ๐ฅ)โ4
e. ๏ฟฝ ๐ฅ๐ฅ2
4๐ฅ๐ฅโ1๏ฟฝโ3
5. Apply the properties of exponents to verify that each statement is an identity.
a. 2๐๐+1
3๐๐= 2 ๏ฟฝ2
3๏ฟฝ๐๐
for integer values of ๐๐
b. 3๐๐+1 โ 3๐๐ = 2 โ 3๐๐ for integer values of ๐๐
c. 1
(3๐๐)2โ 4
๐๐
3=13๏ฟฝ23๏ฟฝ2๐๐
for integer values of ๐๐
6. Jonah was trying to rewrite expressions using the properties of exponents and properties of algebra for nonzero values of ๐ฅ๐ฅ. In each problem, he made a mistake. Explain where he made a mistake in each part, and provide a correct solution.
Jonahโs Incorrect Work
a. (3๐ฅ๐ฅ2)โ3 = โ9๐ฅ๐ฅโ6
b. 2
3๐ฅ๐ฅโ5= 6๐ฅ๐ฅ5
c. 2๐ฅ๐ฅโ๐ฅ๐ฅ3
3๐ฅ๐ฅ=23โ ๐ฅ๐ฅ3
A STORY OF FUNCTIONS
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Lesson 1: Integer Exponents
M3 Lesson 1 ALGEBRA II
7. If ๐ฅ๐ฅ = 5๐๐4 and ๐๐ = 2๐๐3, express ๐ฅ๐ฅ in terms of ๐๐.
8. If ๐๐ = 2๐๐3 and ๐๐ = โ 12 ๐๐โ2, express ๐๐ in terms of ๐๐.
9. If ๐ฅ๐ฅ = 3๐ฆ๐ฆ4 and ๐ฆ๐ฆ = ๐ ๐ 2๐ฅ๐ฅ3, show that ๐ ๐ = 54๐ฆ๐ฆ13.
10. Do the following tasks without a calculator.
a. Express 83 as a power of 2. b. Divide 415 by 210.
11. Use powers of 2 to perform each calculation without a calculator or other technology.
a. 27โ 25
16
b. 512000320
12. Write the first five terms of each of the following recursively defined sequences:
a. ๐๐๐๐+1 = 2๐๐๐๐, ๐๐1 = 3 b. ๐๐๐๐+1 = (๐๐๐๐)2, ๐๐1 = 3 c. ๐๐๐๐+1 = 2(๐๐๐๐)2, ๐๐1 = ๐ฅ๐ฅ, where ๐ฅ๐ฅ is a real number Write each term in the form ๐๐๐ฅ๐ฅ๐๐. d. ๐๐๐๐+1 = 2(๐๐๐๐)โ1, ๐๐1 = ๐ฆ๐ฆ, (๐ฆ๐ฆ โ 0) Write each term in the form ๐๐๐ฅ๐ฅ๐๐.
13. In Module 1, you established the identity (1 โ ๐๐)(1 + ๐๐ + ๐๐2 + โฏ+ ๐๐๐๐โ1) = 1 โ ๐๐๐๐, where ๐๐ is a real number and ๐๐ is a positive integer.
Use this identity to respond to parts (a)โ(g) below.
a. Rewrite the given identity to isolate the sum 1 + ๐๐ + ๐๐2 + โฏ+ ๐๐๐๐โ1 for ๐๐ โ 1. b. Find an explicit formula for 1 + 2 + 22 + 23 + โฏ+ 210. c. Find an explicit formula for 1 + ๐๐ + ๐๐2 + ๐๐3 + โฏ+ ๐๐10 in terms of powers of ๐๐. d. Jerry simplified the sum 1 + ๐๐ + ๐๐2 + ๐๐3 + ๐๐4 + ๐๐5 by writing 1 + ๐๐15. What did he do wrong? e. Find an explicit formula for 1 + 2๐๐ + (2๐๐)2 + (2๐๐)3 + โฏ+ (2๐๐)12 in terms of powers of ๐๐. f. Find an explicit formula for 3 + 3(2๐๐) + 3(2๐๐)2 + 3(2๐๐)3 + โฏ+ 3(2๐๐)12 in terms of powers of ๐๐. Hint: Use
part (e).
g. Find an explicit formula for ๐๐ + ๐๐(1 + ๐๐) + ๐๐(1 + ๐๐)2 + ๐๐(1 + ๐๐)3 + โฏ+ ๐๐(1 + ๐๐)๐๐โ1 in terms of powers of (1 + ๐๐).
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Lesson 2: Base 10 and Scientific Notation
M3 Lesson 2 ALGEBRA II
Lesson 2: Base 10 and Scientific Notation
Classwork
Opening Exercise
In the last lesson, you worked with the thickness of a sheet of gold foil (a very small number) and some very large numbers that gave the size of a piece of paper that actually could be folded in half more than 13 times.
a. Convert 0.28 millionth of a meter to centimeters, and express your answer as a decimal number.
b. The length of a piece of square notebook paper that can be folded in half 13 times is 3,294.2 in. Use this number to calculate the area of a square piece of paper that can be folded in half 14 times. Round your answer to the nearest million.
c. Match the equivalent expressions without using a calculator.
3.5 ร 105 โ6 โ6 ร 100 0.6 3.5 ร 10โ6 3,500,000 350,000 6 ร 10โ1 0.0000035 3.5 ร 106
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Lesson 2: Base 10 and Scientific Notation
M3 Lesson 2 ALGEBRA II
A positive, finite decimal ๐ ๐ is said to be written in scientific notation if it is expressed as a product ๐๐ ร 10๐๐, where ๐๐ is a finite decimal number so that 1 โค ๐๐ < 10, and ๐๐ is an integer.
The integer ๐๐ is called the order of magnitude of the decimal ๐๐ ร 10๐๐.
Example 1
Write each number as a product of a decimal number between 1 and 10 and a power of 10. a. 234,000
b. 0.0035
c. 532,100,000
d. 0.0000000012
e. 3.331
Exercises 1โ6
For Exercises 1โ6, write each number in scientific notation.
1. 532,000,000
2. 0.0000000000000000123 (16 zeros after the decimal place)
3. 8,900,000,000,000,000 (14 zeros after the 9)
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Lesson 2: Base 10 and Scientific Notation
M3 Lesson 2 ALGEBRA II
4. 0.00003382
5. 34,000,000,000,000,000,000,000,000 (24 zeros after the 4)
6. 0.0000000000000000000004 (21 zeros after the decimal place)
Exercises 7โ8
7. Use the fact that the average distance between the sun and Earth is 151,268,468 km, the average distance between the sun and Jupiter is 780,179,470 km, and the average distance between the sun and Pluto is 5,908,039,124 km to approximate the following distances. Express your answers in scientific notation (๐๐ ร 10๐๐), where ๐๐ is rounded to the nearest tenth. a. Distance from the sun to Earth:
b. Distance from the sun to Jupiter:
c. Distance from the sun to Pluto:
d. How much farther Jupiter is from the sun than Earth is from the sun:
e. How much farther Pluto is from the sun than Jupiter is from the sun:
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Lesson 2: Base 10 and Scientific Notation
M3 Lesson 2 ALGEBRA II
8. Order the numbers in Exercise 7 from smallest to largest. Explain how writing the numbers in scientific notation helps you to quickly compare and order them.
Example 2: Arithmetic Operations with Numbers Written Using Scientific Notation
a. (2.4 ร 1020) + (4.5 ร 1021)
b. (7 ร 10โ9)(5 ร 105)
c. 1.2 ร 1015
3 ร 107
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Lesson 2: Base 10 and Scientific Notation
M3 Lesson 2 ALGEBRA II
Exercises 9โ10
9. Perform the following calculations without rewriting the numbers in decimal form. a. (1.42 ร 1015) โ (2 ร 1013)
b. (1.42 ร 1015)(2.4 ร 1013)
c. 1.42 ร 10โ5
2 ร 1013
10. Estimate how many times farther Jupiter is from the sun than Earth is from the sun. Estimate how many times farther Pluto is from the sun than Earth is from the sun.
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Lesson 2: Base 10 and Scientific Notation
M3 Lesson 2 ALGEBRA II
Problem Set 1. Write the following numbers used in these statements in scientific notation. (Note: Some of these numbers have
been rounded.)
a. The density of helium is 0.0001785 gram per cubic centimeter. b. The boiling point of gold is 5,200ยฐF. c. The speed of light is 186,000 miles per second. d. One second is 0.000278 hour. e. The acceleration due to gravity on the sun is 900 ft/s2. f. One cubic inch is 0.0000214 cubic yard. g. Earthโs population in 2012 was 7,046,000,000 people. h. Earthโs distance from the sun is 93,000,000 miles. i. Earthโs radius is 4,000 miles. j. The diameter of a water molecule is 0.000000028 cm.
2. Write the following numbers in decimal form. (Note: Some of these numbers have been rounded.) a. A light year is 9.46 ร 1015 m. b. Avogadroโs number is 6.02 ร 1023 molโ1.
c. The universal gravitational constant is 6.674 ร 10โ11 N ๏ฟฝmkg๏ฟฝ2
.
d. Earthโs age is 4.54 ร 109 years. e. Earthโs mass is 5.97 ร 1024 kg. f. A foot is 1.9 ร 10โ4 mile. g. The population of China in 2014 was 1.354 ร 109 people. h. The density of oxygen is 1.429 ร 10โ4 gram per liter. i. The width of a pixel on a smartphone is 7.8 ร 10โ2 mm. j. The wavelength of light used in optic fibers is 1.55 ร 10โ6 m.
3. State the necessary value of ๐๐ that will make each statement true. a. 0.000โ027 = 2.7 ร 10๐๐ b. โ3.125 = โ3.125 ร 10๐๐ c. 7,540,000,000 = 7.54 ร 10๐๐ d. 0.033 = 3.3 ร 10๐๐ e. 15 = 1.5 ร 10๐๐ f. 26,000 ร 200 = 5.2 ร 10๐๐ g. 3000 ร 0.0003 = 9 ร 10๐๐ h. 0.0004 ร 0.002 = 8 ร 10๐๐
i. 1600080
= 2 ร 10๐๐
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Lesson 2: Base 10 and Scientific Notation
M3 Lesson 2 ALGEBRA II
j. 5000.002
= 2.5 ร 10๐๐
4. Perform the following calculations without rewriting the numbers in decimal form. a. (2.5 ร 104) + (3.7 ร 103) b. (6.9 ร 10โ3) โ (8.1 ร 10โ3) c. (6 ร 1011)(2.5 ร 10โ5)
d. 4.5 ร 108
2 ร 1010
5. The wavelength of visible light ranges from 650 nanometers to 850 nanometers, where 1 nm = 1 ร 10โ7 cm. Express the range of wavelengths of visible light in centimeters.
6. In 1694, the Dutch scientist Antonie van Leeuwenhoek was one of the first scientists to see a red blood cell in a microscope. He approximated that a red blood cell was โ25,000 times as small as a grain of sand.โ Assume a grain
of sand is 12
mm wide, and a red blood cell is approximately 7 micrometers wide. One micrometer is 1 ร 10โ6 m.
Support or refute Leeuwenhoekโs claim. Use scientific notation in your calculations.
7. When the Mars Curiosity Rover entered the atmosphere of Mars on its descent in 2012, it was traveling roughly 13,200 mph. On the surface of Mars, its speed averaged 0.00073 mph. How many times faster was the speed when it entered the atmosphere than its typical speed on the planetโs surface? Use scientific notation in your calculations.
8. Earthโs surface is approximately 70% water. There is no water on the surface of Mars, and its diameter is roughly half of Earthโs diameter. Assume both planets are spherical. The radius of Earth is approximately 4,000 miles. The surface area of a sphere is given by the formula ๐๐๐๐ = 4๐๐๐๐2, where ๐๐ is the radius of the sphere. Which has more land mass, Earth or Mars? Use scientific notation in your calculations.
9. There are approximately 25 trillion (2.5 ร 1013) red blood cells in the human body at any one time. A red blood cell is approximately 7 ร 10โ6 m wide. Imagine if you could line up all your red blood cells end to end. How long would the line of cells be? Use scientific notation in your calculations.
10. Assume each person needs approximately 100 square feet of living space. Now imagine that we are going to build a
giant apartment building that will be 1 mile wide and 1 mile long to house all the people in the United States, estimated to be 313.9 million people in 2012. If each floor of the apartment building is 10 feet high, how tall will the apartment building be?
A STORY OF FUNCTIONS
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Lesson 3: Rational ExponentsโWhat Are 212 and 2
13?
M3 Lesson 3 ALGEBRA II
Lesson 3: Rational ExponentsโWhat are ๐๐๐๐๐๐ and ๐๐
๐๐๐๐?
Classwork
Opening Exercise
a. What is the value of 212? Justify your answer.
b. Graph ๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ for each integer ๐ฅ๐ฅ from ๐ฅ๐ฅ = โ2 to ๐ฅ๐ฅ = 5. Connect the points on your graph with a smooth curve.
A STORY OF FUNCTIONS
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Lesson 3: Rational ExponentsโWhat Are 212 and 2
13?
M3 Lesson 3 ALGEBRA II
The graph on the right shows a close-up view of ๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ for โ0.5 < ๐ฅ๐ฅ < 1.5.
c. Find two consecutive integers that are under and over estimates of the value of 212.
d. Does it appear that 212 is halfway between the integers
you specified in Exercise 1?
e. Use the graph of ๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ to estimate the value of 212.
f. Use the graph of ๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ to estimate the value of 213.
Example
a. What is the 4th root of 16?
b. What is the cube root of 125?
c. What is the 5th root of 100,000?
A STORY OF FUNCTIONS
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Lesson 3: Rational ExponentsโWhat Are 212 and 2
13?
M3 Lesson 3 ALGEBRA II
Exercise 1
Evaluate each expression.
a. โ814
b. โ325
c. โ93 โ โ33
d. โ254 โ โ1004 โ โ44
Discussion
If 212 = โ2 and 2
13 = โ23 , what does 2
34 equal? Explain your reasoning.
Exercises 2โ12
Rewrite each exponential expression as a radical expression.
2. 312
3. 1115
A STORY OF FUNCTIONS
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Lesson 3: Rational ExponentsโWhat Are 212 and 2
13?
M3 Lesson 3 ALGEBRA II
4. ๏ฟฝ14๏ฟฝ15
5. 61
10
Rewrite the following exponential expressions as equivalent radical expressions. If the number is rational, write it without radicals or exponents.
6. 232
7. 452
8. ๏ฟฝ18๏ฟฝ53
9. Show why the following statement is true:
2โ12 =
1
212
A STORY OF FUNCTIONS
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Lesson 3: Rational ExponentsโWhat Are 212 and 2
13?
M3 Lesson 3 ALGEBRA II
Rewrite the following exponential expressions as equivalent radical expressions. If the number is rational, write it without radicals or exponents.
10. 4โ32
11. 27โ23
12. ๏ฟฝ14๏ฟฝโ 12
A STORY OF FUNCTIONS
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Lesson 3: Rational ExponentsโWhat Are 212 and 2
13?
M3 Lesson 3 ALGEBRA II
Lesson Summary
๐๐TH ROOT OF A NUMBER: Let ๐๐ and ๐๐ be numbers, and let ๐๐ be a positive integer. If ๐๐ = ๐๐๐๐, then ๐๐ is a ๐๐th root of ๐๐. If ๐๐ = 2, then the root is a called a square root. If ๐๐ = 3, then the root is called a cube root.
PRINCIPAL ๐๐TH ROOT OF A NUMBER: Let ๐๐ be a real number that has at least one real ๐๐th root. The principal ๐๐th root of ๐๐ is the real ๐๐th root that has the same sign as ๐๐ and is denoted by a radical symbol: โ๐๐๐๐ .
Every positive number has a unique principal ๐๐th root. We often refer to the principal ๐๐th root of ๐๐ as just the ๐๐th root of ๐๐. The ๐๐th root of 0 is 0.
For any positive integers ๐๐ and ๐๐, and any real number ๐๐ for which the principal ๐๐th root of ๐๐ exists, we have
๐๐1๐๐ = โ๐๐๐๐
๐๐๐๐๐๐ = โ๐๐๐๐๐๐ = ๏ฟฝโ๐๐๐๐ ๏ฟฝ
๐๐
๐๐โ๐๐๐๐ =
1โ๐๐๐๐๐๐
for ๐๐ โ 0.
Problem Set 1. Select the expression from (A), (B), and (C) that correctly completes the statement.
(A) (B) (C)
a. ๐ฅ๐ฅ13 is equivalent to .
13๐ฅ๐ฅ โ๐ฅ๐ฅ3
3๐ฅ๐ฅ
b. ๐ฅ๐ฅ23 is equivalent to .
23๐ฅ๐ฅ โ๐ฅ๐ฅ23 ๏ฟฝโ๐ฅ๐ฅ๏ฟฝ
3
c. ๐ฅ๐ฅโ14 is equivalent to . โ 14 ๐ฅ๐ฅ
4๐ฅ๐ฅ
1โ๐ฅ๐ฅ4
d. ๏ฟฝ4๐ฅ๐ฅ๏ฟฝ12
is equivalent to . 2๐ฅ๐ฅ
4๐ฅ๐ฅ2
2โ๐ฅ๐ฅ
2. Identify which of the expressions (A), (B), and (C) are equivalent to the given expression.
(A) (B) (C)
a. 1612 ๏ฟฝ 116๏ฟฝ
โ12 823 64
32
b. ๏ฟฝ23๏ฟฝโ1
โ 32 ๏ฟฝ94๏ฟฝ12
27
13
6
A STORY OF FUNCTIONS
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Lesson 3: Rational ExponentsโWhat Are 212 and 2
13?
M3 Lesson 3 ALGEBRA II
3. Rewrite in radical form. If the number is rational, write it without using radicals.
a. 632
b. ๏ฟฝ12๏ฟฝ14
c. 3(8)13
d. ๏ฟฝ 64125๏ฟฝโ23
e. 81โ14
4. Rewrite the following expressions in exponent form.
a. โ5
b. โ523
c. โ53
d. ๏ฟฝโ53 ๏ฟฝ2
5. Use the graph of ๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ shown to the right to estimate the following powers of 2.
a. 214
b. 223
c. 234
d. 20.2 e. 21.2
f. 2โ15
A STORY OF FUNCTIONS
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Lesson 3: Rational ExponentsโWhat Are 212 and 2
13?
M3 Lesson 3 ALGEBRA II
6. Rewrite each expression in the form ๐๐๐ฅ๐ฅ๐๐, where ๐๐ is a real number, ๐ฅ๐ฅ is a positive real number, and ๐๐ is rational.
a. โ16๐ฅ๐ฅ34
b. 5โ๐ฅ๐ฅ
c. ๏ฟฝ1/๐ฅ๐ฅ43
d. 4
โ8๐ฅ๐ฅ33
e. 27โ9๐ฅ๐ฅ4
f. ๏ฟฝ125๐ฅ๐ฅ2 ๏ฟฝโ13
7. Find the value of ๐ฅ๐ฅ for which 2๐ฅ๐ฅ12 = 32.
8. Find the value of ๐ฅ๐ฅ for which ๐ฅ๐ฅ43 = 81.
9. If ๐ฅ๐ฅ32 = 64, find the value of 4๐ฅ๐ฅโ
34.
10. Evaluate the following expressions when ๐๐ = 19.
a. ๐๐โ12
b. ๐๐52
c. โ3๐๐โ13
11. Show that each expression is equivalent to 2๐ฅ๐ฅ. Assume ๐ฅ๐ฅ is a positive real number.
a. โ16๐ฅ๐ฅ44
b. ๏ฟฝ โ8๐ฅ๐ฅ33 ๏ฟฝ
2
โ4๐ฅ๐ฅ2
c. 6๐ฅ๐ฅ3
โ27๐ฅ๐ฅ63
12. Yoshiko said that 1614 = 4 because 4 is one-fourth of 16. Use properties of exponents to explain why she is or is not
correct.
13. Jefferson said that 843 = 16 because 8
13 = 2 and 24 = 16. Use properties of exponents to explain why he is or is
not correct.
A STORY OF FUNCTIONS
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Lesson 3: Rational ExponentsโWhat Are 212 and 2
13?
M3 Lesson 3 ALGEBRA II
14. Rita said that 823 = 128 because 8
23 = 82 โ 8
13, so 8
23 = 64 โ 2, and then 8
23 = 128. Use properties of exponents to
explain why she is or is not correct.
15. Suppose for some positive real number ๐๐ that ๏ฟฝ๐๐14 โ ๐๐
12 โ ๐๐
14๏ฟฝ
2
= 3.
a. What is the value of ๐๐? b. Which exponent properties did you use to find your answer to part (a)?
16. In the lesson, you made the following argument:
๏ฟฝ213๏ฟฝ
3= 2
13 โ 2
13 โ 2
13
= 213+
13+
13
= 21 = 2.
Since โ23 is a number so that ๏ฟฝโ23 ๏ฟฝ3
= 2 and 213 is a number so that ๏ฟฝ2
13๏ฟฝ
3
= 2, you concluded that 213 = โ23 .
Which exponent property was used to make this argument?
A STORY OF FUNCTIONS
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M3 Lesson 4 ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Lesson 4: Properties of Exponents and Radicals
Classwork
Opening Exercise
Write each exponential expression as a radical expression, and then use the definition and properties of radicals to write the resulting expression as an integer.
a. 712 โ 7
12
b. 313 โ 3
13 โ 3
13
c. 1212 โ 3
12
d. ๏ฟฝ6413๏ฟฝ
12
A STORY OF FUNCTIONS
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M3 Lesson 4 ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Examples 1โ3
Write each expression in the form ๐๐๐๐๐๐ for positive real numbers ๐๐ and integers ๐๐ and ๐๐ with ๐๐ > 0 by applying the
properties of radicals and the definition of ๐๐th root.
1. ๐๐14 โ ๐๐
14
2. ๐๐13 โ ๐๐
43
A STORY OF FUNCTIONS
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M3 Lesson 4 ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
3. ๐๐15 โ ๐๐
34
Exercises 1โ4
Write each expression in the form ๐๐๐๐๐๐ . If a numeric expression is a rational number, then write your answer without
exponents.
1. ๐๐23 โ ๐๐
12
2. ๏ฟฝ๐๐โ15๏ฟฝ
23
3. 6413 โ 64
32
A STORY OF FUNCTIONS
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M3 Lesson 4 ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
4. ๏ฟฝ93
42๏ฟฝ32
Example 4
Rewrite the radical expression ๏ฟฝ48๐ฅ๐ฅ5๐ฆ๐ฆ4๐ง๐ง2 so that no perfect square factors remain inside the radical.
Exercise 5
5. Use the definition of rational exponents and properties of exponents to rewrite each expression with rational exponents containing as few fractions as possible. Then, evaluate each resulting expression for ๐ฅ๐ฅ = 50, ๐ฆ๐ฆ = 12, and ๐ง๐ง = 3.
a. ๏ฟฝ8๐ฅ๐ฅ3๐ฆ๐ฆ2
A STORY OF FUNCTIONS
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M3 Lesson 4 ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
b. ๏ฟฝ54๐ฆ๐ฆ7๐ง๐ง23
Exercise 6
6. Order these numbers from smallest to largest. Explain your reasoning.
162.5 93.6 321.2
A STORY OF FUNCTIONS
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M3 Lesson 4 ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
Problem Set 1. Evaluate each expression for ๐๐ = 27 and ๐๐ = 64.
a. โ๐๐3 ๏ฟฝ๐๐
b. ๏ฟฝ3โ๐๐3 ๏ฟฝ๐๐๏ฟฝ2
c. ๏ฟฝโ๐๐3 + 2๏ฟฝ๐๐๏ฟฝ2
d. ๐๐โ23 + ๐๐
32
e. ๏ฟฝ๐๐โ23 โ ๐๐
32๏ฟฝ
โ1
f. ๏ฟฝ๐๐โ23 โ 18 ๐๐
32๏ฟฝ
โ1
2. Rewrite each expression so that each term is in the form ๐๐๐ฅ๐ฅ๐๐ , where ๐๐ is a real number, ๐ฅ๐ฅ is a positive real number, and ๐๐ is a rational number.
a. ๐ฅ๐ฅโ23 โ ๐ฅ๐ฅ
13
b. 2๐ฅ๐ฅ12 โ 4๐ฅ๐ฅโ
52
c. 10๐ฅ๐ฅ
13
2๐ฅ๐ฅ2
d. ๏ฟฝ3๐ฅ๐ฅ14๏ฟฝ
โ2
e. ๐ฅ๐ฅ12 ๏ฟฝ2๐ฅ๐ฅ2 โ 4๐ฅ๐ฅ๏ฟฝ
f. ๏ฟฝ27๐ฅ๐ฅ6
3
g. โ๐ฅ๐ฅ3 โ โโ8๐ฅ๐ฅ23 โ โ27๐ฅ๐ฅ43
h. 2๐ฅ๐ฅ4 โ ๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ
โ๐ฅ๐ฅ
i. โ๐ฅ๐ฅ โ 2๐ฅ๐ฅโ3
4๐ฅ๐ฅ2
Lesson Summary
The properties of exponents developed in Grade 8 for integer exponents extend to rational exponents.
That is, for any integers ๐๐, ๐๐, ๐๐, and ๐๐, with ๐๐ > 0 and ๐๐ > 0, and any real numbers ๐๐ and ๐๐ so that ๐๐1๐๐, ๐๐
1๐๐, and ๐๐
1๐๐
are defined, we have the following properties of exponents:
1. ๐๐๐๐๐๐ โ ๐๐
๐๐๐๐ = ๐๐
๐๐๐๐+
๐๐๐๐
2. ๐๐๐๐๐๐ = ๏ฟฝ๐๐๐๐๐๐
3. ๏ฟฝ๐๐1๐๐๏ฟฝ๐๐
= ๐๐
4. ๏ฟฝ๐๐๐๐๏ฟฝ1๐๐ = ๐๐
5. ๏ฟฝ๐๐๐๐๏ฟฝ๐๐๐๐ = ๐๐
๐๐๐๐ โ ๐๐
๐๐๐๐
6. ๏ฟฝ๐๐๐๐๐๐ ๏ฟฝ
๐๐๐๐
= ๐๐๐๐๐๐๐๐๐๐
7. ๐๐โ๐๐๐๐ = 1
๐๐๐๐๐๐
.
A STORY OF FUNCTIONS
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M3 Lesson 4 ALGEBRA II
Lesson 4: Properties of Exponents and Radicals
3. Show that ๏ฟฝโ๐ฅ๐ฅ+ ๏ฟฝ๐ฆ๐ฆ๏ฟฝ2 is not equal to ๐ฅ๐ฅ1 + ๐ฆ๐ฆ1 when ๐ฅ๐ฅ = 9 and ๐ฆ๐ฆ = 16.
4. Show that ๏ฟฝ๐ฅ๐ฅ12 + ๐ฆ๐ฆ
12๏ฟฝโ1
is not equal to 1
๐ฅ๐ฅ12
+1
๐ฆ๐ฆ12
when ๐ฅ๐ฅ = 9 and ๐ฆ๐ฆ = 16.
5. From these numbers, select (a) one that is negative, (b) one that is irrational, (c) one that is not a real number, and (d) one that is a perfect square:
312 โ 9
12 , 27
13 โ 144
12, 64
13 โ 64
23, and ๏ฟฝ4โ
12 โ 4
12๏ฟฝ
12
6. Show that for any rational number ๐๐, the expression 2๐๐ โ 4๐๐+1 โ ๏ฟฝ18๏ฟฝ๐๐is equal to 4.
7. Let ๐๐ be any rational number. Express each answer as a power of 10.
a. Multiply 10๐๐ by 10. b. Multiply ๏ฟฝ10 by 10๐๐. c. Square 10๐๐.
d. Divide 100 โ 10๐๐ by 102๐๐.
e. Show that 10๐๐ = 11 โ 10๐๐ โ 10๐๐+1.
8. Rewrite each of the following radical expressions as an equivalent exponential expression in which each variable occurs no more than once.
a. ๏ฟฝ8๐ฅ๐ฅ2๐ฆ๐ฆ
b. ๏ฟฝ96๐ฅ๐ฅ3๐ฆ๐ฆ15๐ง๐ง65
9. Use properties of exponents to find two integers that are upper and lower estimates of the value of 41.6.
10. Use properties of exponents to find two integers that are upper and lower estimates of the value of 82.3.
11. Keplerโs third law of planetary motion relates the average distance, ๐๐, of a planet from the sun to the time, ๐ก๐ก, it takes the planet to complete one full orbit around the sun according to the equation ๐ก๐ก2 = ๐๐3. When the time, ๐ก๐ก, is measured in Earth years, the distance, ๐๐, is measured in astronomical units (AUs). (One AU is equal to the average distance from Earth to the sun.)
a. Find an equation for ๐ก๐ก in terms of ๐๐ and an equation for ๐๐ in terms of ๐ก๐ก. b. Venus takes about 0.616 Earth year to orbit the sun. What is its average distance from the sun? c. Mercury is an average distance of 0.387 AU from the sun. About how long is its orbit in Earth years?
A STORY OF FUNCTIONS
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Lesson 5: Irrational ExponentsโWhat are 2โ2 and 2๐๐?
M3 Lesson 5 ALGEBRA II
Lesson 5: Irrational ExponentsโWhat are ๐๐โ๐๐ and ๐๐๐๐?
Classwork
Exercise 1
a. Write the following finite decimals as fractions (you do not need to reduce to lowest terms). 1, 1.4, 1.41, 1.414, 1.4142, 1.41421
b. Write 21.4, 21.41, 21.414, and 21.4142 in radical form (โ2๐๐๐๐ ).
c. Use a calculator to compute decimal approximations of the radical expressions you found in part (b) to 5 decimal places. For each approximation, underline the digits that are also in the previous approximation, starting with 2.00000 done for you below. What do you notice?
21 = 2 = 2.00000
A STORY OF FUNCTIONS
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Lesson 5: Irrational ExponentsโWhat are 2โ2 and 2๐๐?
M3 Lesson 5 ALGEBRA II
Exercise 2
a. Write six terms of a sequence that a calculator can use to approximate 2๐๐. (Hint: ๐๐ = 3.141โ59 โฆ )
b. Compute 23.14 and 2๐๐ on your calculator. In which digit do they start to differ?
c. How could you improve the accuracy of your estimate of 2๐๐?
A STORY OF FUNCTIONS
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Lesson 5: Irrational ExponentsโWhat are 2โ2 and 2๐๐?
M3 Lesson 5 ALGEBRA II
Problem Set 1. Is it possible for a number to be both rational and irrational?
2. Use properties of exponents to rewrite the following expressions as a number or an exponential expression with only one exponent.
a. ๏ฟฝ2โ3๏ฟฝโ3
b. ๏ฟฝโ2โ2๏ฟฝโ2
c. ๏ฟฝ31+โ5๏ฟฝ1โโ5
d. 31+๏ฟฝ5
2 โ 31โ๏ฟฝ5
2
e. 31+๏ฟฝ5
2 รท 31โ๏ฟฝ5
2
f. 32cos2(๐ฅ๐ฅ) โ 32sin2(๐ฅ๐ฅ)
3.
a. Between what two integer powers of 2 does 2โ5 lie?
b. Between what two integer powers of 3 does 3โ10 lie?
c. Between what two integer powers of 5 does 5โ3 lie?
4. Use the process outlined in the lesson to approximate the number 2โ5. Use the approximation โ5 โ 2.236โ067โ98.
a. Find a sequence of five intervals that contain โ5 whose endpoints get successively closer to โ5.
b. Find a sequence of five intervals that contain 2โ5 whose endpoints get successively closer to 2โ5. Write your intervals in the form 2๐๐ < 2โ5 < 2๐ ๐ for rational numbers ๐๐ and ๐ ๐ .
c. Use your calculator to find approximations to four decimal places of the endpoints of the intervals in part (b).
d. Based on your work in part (c), what is your best estimate of the value of 2โ5?
e. Can we tell if 2โ5 is rational or irrational? Why or why not?
5. Use the process outlined in the lesson to approximate the number 3โ10. Use the approximation โ10 โ3.162โ277โ7.
a. Find a sequence of five intervals that contain 3โ10 whose endpoints get successively closer to 3โ10. Write your intervals in the form 3๐๐ < 3โ10 < 3๐ ๐ for rational numbers ๐๐ and ๐ ๐ .
b. Use your calculator to find approximations to four decimal places of the endpoints of the intervals in part (a).
c. Based on your work in part (b), what is your best estimate of the value of 3โ10?
A STORY OF FUNCTIONS
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Lesson 5: Irrational ExponentsโWhat are 2โ2 and 2๐๐?
M3 Lesson 5 ALGEBRA II
6. Use the process outlined in the lesson to approximate the number 5โ7. Use the approximation โ7 โ 2.645โ751โ31.
a. Find a sequence of seven intervals that contain 5โ7 whose endpoints get successively closer to 5โ7. Write your intervals in the form 5๐๐ < 5โ7 < 5๐ ๐ for rational numbers ๐๐ and ๐ ๐ .
b. Use your calculator to find approximations to four decimal places of the endpoints of the intervals in part (a).
c. Based on your work in part (b), what is your best estimate of the value of 5โ7?
7. A rational number raised to a rational power can either be rational or irrational. For example, 412 is rational because
412 = 2, and 2
14 is irrational because 2
14 = โ24 . In this problem, you will investigate the possibilities for an irrational
number raised to an irrational power.
a. Evaluate ๏ฟฝโ2๏ฟฝ๏ฟฝโ2๏ฟฝโ
2.
b. Can the value of an irrational number raised to an irrational power ever be rational?
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M3 Lesson 6 ALGEBRA II
Lesson 6: Eulerโs Number, ๐๐
Lesson 6: Eulerโs Number, ๐๐
Classwork
Exercises 1โ3
1. Assume that there is initially 1 cm of water in the tank, and the height of the water doubles every 10 seconds. Write an equation that could be used to calculate the height ๐ป๐ป(๐ก๐ก) of the water in the tank at any time ๐ก๐ก.
2. How would the equation in Exercise 1 change ifโฆ a. the initial depth of water in the tank was 2 cm?
b. the initial depth of water in the tank was 12 cm?
c. the initial depth of water in the tank was 10 cm?
d. the initial depth of water in the tank was ๐ด๐ด cm, for some positive real number ๐ด๐ด?
3. How would the equation in Exercise 2, part (d), change ifโฆ a. the height tripled every ten seconds?
b. the height doubled every five seconds?
A STORY OF FUNCTIONS
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M3 Lesson 6 ALGEBRA II
Lesson 6: Eulerโs Number, ๐๐
c. the height quadrupled every second?
d. the height halved every ten seconds?
Example
Consider two identical water tanks, each of which begins with a height of water 1 cm and fills with water at a different rate. Which equations can be used to calculate the height of water in each tank at time ๐ก๐ก? Use ๐ป๐ป1 for tank 1 and ๐ป๐ป2 for tank 2.
a. If both tanks start filling at the same time, which one fills first?
b. We want to know the average rate of change of the height of the water in these tanks over an interval that starts at a fixed time ๐๐ as they are filling up. What is the formula for the average rate of change of a function ๐๐ on an interval [๐๐, ๐๐]?
c. What is the formula for the average rate of change of the function ๐ป๐ป1 on an interval [๐๐, ๐๐]?
d. Letโs calculate the average rate of change of the function ๐ป๐ป1 on the interval [๐๐,๐๐ + 0.1], which is an interval one-tenth of a second long starting at an unknown time ๐๐.
The height of the water in tank 2 triples every
second.
The height of the water in tank 1 doubles every
second.
A STORY OF FUNCTIONS
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M3 Lesson 6 ALGEBRA II
Lesson 6: Eulerโs Number, ๐๐
Exercises 4โ8
4. For the second tank, calculate the average change in the height, ๐ป๐ป2, from time ๐๐ seconds to ๐๐ + 0.1 second. Express the answer as a number times the value of the original function at time ๐๐. Explain the meaning of these findings.
5. For each tank, calculate the average change in height from time ๐๐ seconds to ๐๐ + 0.001 second. Express the answer as a number times the value of the original function at time ๐๐. Explain the meaning of these findings.
6. In Exercise 5, the average rate of change of the height of the water in tank 1 on the interval [๐๐,๐๐ + 0.001] can be described by the expression ๐๐1 โ 2๐๐, and the average rate of change of the height of the water in tank 2 on the interval [๐๐,๐๐ + 0.001] can be described by the expression ๐๐2 โ 3๐๐. What are approximate values of ๐๐1 and ๐๐2?
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M3 Lesson 6 ALGEBRA II
Lesson 6: Eulerโs Number, ๐๐
7. As an experiment, letโs look for a value of ๐๐ so that if the height of the water can be described by ๐ป๐ป(๐ก๐ก) = ๐๐๐ก๐ก, then the expression for the average rate of change on the interval [๐๐,๐๐ + 0.001] is 1 โ ๐ป๐ป(๐๐). a. Write out the expression for the average rate of change of ๐ป๐ป(๐ก๐ก) = ๐๐๐ก๐ก on the interval [๐๐,๐๐ + 0.001].
b. Set your expression in part (a) equal to 1 โ ๐ป๐ป(๐๐), and reduce to an expression involving a single ๐๐.
c. Now we want to find the value of ๐๐ that satisfies the equation you found in part (b), but we do not have a way to explicitly solve this equation. Look back at Exercise 6; which two consecutive integers have ๐๐ between them?
d. Use your calculator and a guess-and-check method to find an approximate value of ๐๐ to 2 decimal places.
8. Verify that for the value of ๐๐ found in Exercise 7, ๐ป๐ป๐๐(๐๐ + 0.001) โ ๐ป๐ป๐๐(๐๐)
0.001โ ๐ป๐ป๐๐(๐๐), where ๐ป๐ป๐๐(๐๐) = ๐๐๐๐.
A STORY OF FUNCTIONS
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M3 Lesson 6 ALGEBRA II
Lesson 6: Eulerโs Number, ๐๐
Lesson Summary
Eulerโs number, ๐๐, is an irrational number that is approximately equal to 2.718โ281โ828โ459โ0. AVERAGE RATE OF CHANGE: Given a function ๐๐ whose domain contains the interval of real numbers [๐๐, ๐๐] and
whose range is a subset of the real numbers, the average rate of change on the interval [๐๐, ๐๐] is defined by the number
๐๐(๐๐) โ ๐๐(๐๐)๐๐ โ ๐๐
.
Problem Set 1. The product 4 โ 3 โ 2 โ 1 is called 4 factorial and is denoted by 4!. Then 10! = 10 โ 9 โ 8 โ 7 โ 6 โ 5 โ 4 โ 3 โ 2 โ 1, and for
any positive integer ๐๐, ๐๐! = ๐๐(๐๐ โ 1)(๐๐ โ 2) โ โฏ โ 3 โ 2 โ 1.
a. Complete the following table of factorial values:
๐๐ 1 2 3 4 5 6 7 8
๐๐!
b. Evaluate the sum 1 + 11!.
c. Evaluate the sum 1 + 11! +12!.
d. Use a calculator to approximate the sum 1 + 11! +12! +
13! to 7 decimal places. Do not round the fractions
before evaluating the sum.
e. Use a calculator to approximate the sum 1 + 11! +12! +
13! +
14! to 7 decimal places. Do not round the fractions
before evaluating the sum.
f. Use a calculator to approximate sums of the form 1 + 11! +12! + โฏ+
1๐๐! to 7 decimal places for
๐๐ = 5, 6, 7, 8, 9, 10. Do not round the fractions before evaluating the sums with a calculator.
g. Make a conjecture about the sums 1 + 11! +12! + โฏ+
1๐๐! for positive integers ๐๐ as ๐๐ increases in size.
h. Would calculating terms of this sequence ever yield an exact value of ๐๐? Why or why not?
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M3 Lesson 6 ALGEBRA II
Lesson 6: Eulerโs Number, ๐๐
2. Consider the sequence given by ๐๐๐๐ = ๏ฟฝ1 +1๐๐๏ฟฝ
๐๐, where ๐๐ โฅ 1 is an integer.
a. Use your calculator to approximate the first 5 terms of this sequence to 7 decimal places. b. Does it appear that this sequence settles near a particular value? c. Use a calculator to approximate the following terms of this sequence to 7 decimal places.
i. ๐๐100 ii. ๐๐1000 iii. ๐๐10,000 iv. ๐๐100,000 v. ๐๐1,000,000 vi. ๐๐10,000,000 vii. ๐๐100,000,000
d. Does it appear that this sequence settles near a particular value? e. Compare the results of this exercise with the results of Problem 1. What do you observe?
3. If ๐ฅ๐ฅ = 5๐๐4 and ๐๐ = 2๐๐3, express ๐ฅ๐ฅ in terms of ๐๐, and approximate to the nearest whole number.
4. If ๐๐ = 2๐๐3 and ๐๐ = โ 12 ๐๐โ2, express ๐๐ in terms of ๐๐, and approximate to four decimal places.
5. If ๐ฅ๐ฅ = 3๐๐4 and = ๐ ๐ 2๐ฅ๐ฅ3 , show that ๐ ๐ = 54๐๐13, and approximate ๐ ๐ to the nearest whole number.
6. The following graph shows the number of barrels of oil produced by the Glenn Pool well in Oklahoma from 1910 to 1916.
Source: Cutler, Willard W., Jr. Estimation of Underground Oil Reserves by Oil-Well Production Curves, U.S. Department of the Interior, 1924.
0500
1000150020002500300035004000
1908 1910 1912 1914 1916 1918
Oil
prod
uctio
n pe
r yea
r, in
bar
rels
Year
Oil Well Production, Glenn Pool, Oklahoma
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M3 Lesson 6 ALGEBRA II
Lesson 6: Eulerโs Number, ๐๐
a. Estimate the average rate of change of the amount of oil produced by the well on the interval [1910, 1916], and explain what that number represents.
b. Estimate the average rate of change of the amount of oil produced by the well on the interval [1910, 1913], and explain what that number represents.
c. Estimate the average rate of change of the amount of oil produced by the well on the interval [1913, 1916], and explain what that number represents.
d. Compare your results for the rates of change in oil production in the first half and the second half of the time period in question in parts (b) and (c). What do those numbers say about the production of oil from the well?
e. Notice that the average rate of change of the amount of oil produced by the well on any interval starting and ending in two consecutive years is always negative. Explain what that means in the context of oil production.
7. The following table lists the number of hybrid electric vehicles (HEVs) sold in the United States between 1999 and 2013.
Year Number of HEVs Sold in U.S.
Year Number of HEVs Sold in U.S.
1999 17 2007 352,274 2000 9350 2008 312,386 2001 20,282 2009 290,271 2002 36,035 2010 274,210 2003 47,600 2011 268,752 2004 84,199 2012 434,498 2005 209,711 2013 495,685 2006 252,636
Source: U.S. Department of Energy, Alternative Fuels and Advanced Vehicle Data Center, 2013.
a. During which one-year interval is the average rate of change of the number of HEVs sold the largest? Explain how you know.
b. Calculate the average rate of change of the number of HEVs sold on the interval [2003, 2004], and explain what that number represents.
c. Calculate the average rate of change of the number of HEVs sold on the interval [2003, 2008], and explain what that number represents.
d. What does it mean if the average rate of change of the number of HEVs sold is negative?
Extension:
8. The formula for the area of a circle of radius ๐๐ can be expressed as a function ๐ด๐ด(๐๐) = ๐๐๐๐2. a. Find the average rate of change of the area of a circle on the interval [4, 5]. b. Find the average rate of change of the area of a circle on the interval [4, 4.1]. c. Find the average rate of change of the area of a circle on the interval [4, 4.01]. d. Find the average rate of change of the area of a circle on the interval [4, 4.001].
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M3 Lesson 6 ALGEBRA II
Lesson 6: Eulerโs Number, ๐๐
e. What is happening to the average rate of change of the area of the circle as the interval gets smaller and smaller?
f. Find the average rate of change of the area of a circle on the interval [4, 4 + โ] for some small positive number โ.
g. What happens to the average rate of change of the area of the circle on the interval [4, 4 + โ] as โ โ 0? Does this agree with your answer to part (d)? Should it agree with your answer to part (e)?
h. Find the average rate of change of the area of a circle on the interval [๐๐0, ๐๐0 + โ] for some positive number ๐๐0 and some small positive number โ.
i. What happens to the average rate of change of the area of the circle on the interval [๐๐0, ๐๐0 + โ] as โ โ 0? Do you recognize the resulting formula?
9. The formula for the volume of a sphere of radius ๐๐ can be expressed as a function ๐๐(๐๐) = 43๐๐๐๐3. As you work
through these questions, you will see the pattern develop more clearly if you leave your answers in the form of a coefficient times ๐๐. Approximate the coefficient to five decimal places.
a. Find the average rate of change of the volume of a sphere on the interval [2, 3]. b. Find the average rate of change of the volume of a sphere on the interval [2, 2.1]. c. Find the average rate of change of the volume of a sphere on the interval [2, 2.01]. d. Find the average rate of change of the volume of a sphere on the interval [2, 2.001]. e. What is happening to the average rate of change of the volume of a sphere as the interval gets smaller and
smaller?
f. Find the average rate of change of the volume of a sphere on the interval [2, 2 + โ] for some small positive number โ.
g. What happens to the average rate of change of the volume of a sphere on the interval [2, 2 + โ] as โ โ 0? Does this agree with your answer to part (e)? Should it agree with your answer to part (e)?
h. Find the average rate of change of the volume of a sphere on the interval [๐๐0, ๐๐0 + โ] for some positive number ๐๐0 and some small positive number โ.
i. What happens to the average rate of change of the volume of a sphere on the interval [๐๐0, ๐๐0 + โ] as โ โ 0? Do you recognize the resulting formula?
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M3 Lesson 7 ALGEBRA II
Lesson 7: Bacteria and Exponential Growth
Lesson 7: Bacteria and Exponential Growth
Classwork
Opening Exercise
Work with your partner or group to solve each of the following equations for ๐ฅ๐ฅ.
a. 2๐ฅ๐ฅ = 23
b. 2๐ฅ๐ฅ = 2
c. 2๐ฅ๐ฅ = 16
d. 2๐ฅ๐ฅ โ 64 = 0
e. 2๐ฅ๐ฅ โ 1 = 0
f. 23๐ฅ๐ฅ = 64
g. 2๐ฅ๐ฅ+1 = 32
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M3 Lesson 7 ALGEBRA II
Lesson 7: Bacteria and Exponential Growth
Example
The Escherichia coli bacteria (commonly known as E. coli) reproduces once every 30 minutes, meaning that a colony of E. coli can double every half hour. Mycobacterium tuberculosis has a generation time in the range of 12 to 16 hours. Researchers have found evidence that suggests certain bacteria populations living deep below the surface of the earth may grow at extremely slow rates, reproducing once every several thousand years. With this variation in bacterial growth rates, it is reasonable that we assume a 24-hour reproduction time for a hypothetical bacteria colony in this example.
Suppose we have a bacteria colony that starts with 1 bacterium, and the population of bacteria doubles every day.
What function ๐๐ can we use to model the bacteria population on day ๐ก๐ก?
๐๐ ๐ท๐ท(๐๐)
How many days will it take for the bacteria population to reach 8?
How many days will it take for the bacteria population to reach 16?
Roughly how long will it take for the population to reach 10?
We already know from our previous discussion that if 2๐๐ = 10, then 3 < ๐๐ < 4, and the table confirms that. At this point, we have an underestimate of 3 and an overestimate of 4 for ๐๐. How can we find better under and over estimates for ๐๐?
๐๐ ๐ท๐ท(๐๐)
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M3 Lesson 7 ALGEBRA II
Lesson 7: Bacteria and Exponential Growth
From our table, we now know another set of under and over estimates for the number ๐๐ that we seek. What are they?
Continue this process of โsqueezingโ the number ๐๐ between two numbers until you are confident you know the value of ๐๐ to two decimal places.
๐๐ ๐ท๐ท(๐๐) ๐๐ ๐ท๐ท(๐๐)
What if we had wanted to find ๐๐ to 5 decimal places?
To the nearest minute, when does the population of bacteria become 10?
๐๐ ๐ท๐ท(๐๐) ๐๐ ๐ท๐ท(๐๐) ๐๐ ๐ท๐ท(๐๐) ๐๐ ๐ท๐ท(๐๐)
A STORY OF FUNCTIONS
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M3 Lesson 7 ALGEBRA II
Lesson 7: Bacteria and Exponential Growth
Exercise
Use the method from the Example to approximate the solution to the equations below to two decimal places.
a. 2๐ฅ๐ฅ = 1,000
b. 3๐ฅ๐ฅ = 1,000
c. 4๐ฅ๐ฅ = 1,000
d. 5๐ฅ๐ฅ = 1,000
e. 6๐ฅ๐ฅ = 1,000
f. 7๐ฅ๐ฅ = 1,000
g. 8๐ฅ๐ฅ = 1,000
h. 9๐ฅ๐ฅ = 1,000
i. 11๐ฅ๐ฅ = 1,000
j. 12๐ฅ๐ฅ = 1,000
k. 13๐ฅ๐ฅ = 1,000
l. 14๐ฅ๐ฅ = 1,000
m. 15๐ฅ๐ฅ = 1,000
n. 16๐ฅ๐ฅ = 1,000
A STORY OF FUNCTIONS
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M3 Lesson 7 ALGEBRA II
Lesson 7: Bacteria and Exponential Growth
Problem Set 1. Solve each of the following equations for ๐ฅ๐ฅ using the same technique as was used in the Opening Exercise.
a. 2๐ฅ๐ฅ = 32 b. 2๐ฅ๐ฅโ3 = 22๐ฅ๐ฅ+5 c. 2๐ฅ๐ฅ2โ3๐ฅ๐ฅ = 2โ2
d. 2๐ฅ๐ฅ โ 24๐ฅ๐ฅโ3 = 0 e. 23๐ฅ๐ฅ โ 25 = 27 f. 2๐ฅ๐ฅ2โ16 = 1
g. 32๐ฅ๐ฅ = 27 h. 32๐ฅ๐ฅ = 81 i.
3๐ฅ๐ฅ2
35๐ฅ๐ฅ= 36
2. Solve the equation 22๐ฅ๐ฅ
2๐ฅ๐ฅ+5= 1 algebraically using two different initial steps as directed below.
a. Write each side as a power of 2. b. Multiply both sides by 2๐ฅ๐ฅ+5.
3. Find consecutive integers that are under and over estimates of the solutions to the following exponential equations.
a. 2๐ฅ๐ฅ = 20 b. 2๐ฅ๐ฅ = 100 c. 3๐ฅ๐ฅ = 50 d. 10๐ฅ๐ฅ = 432,901 e. 2๐ฅ๐ฅโ2 = 750 f. 2๐ฅ๐ฅ = 1.35
4. Complete the following table to approximate the solution to 10๐ฅ๐ฅ = 34,198 to three decimal places.
๐๐ ๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐ ๐๐ ๐๐๐๐๐๐
1 10 4.1 4.51 4.531
2 100 4.2 4.52 4.532
3 1,000 4.3 4.53 4.533
4 10,000 4.4 4.54 4.534
5 100,000 4.5 4.535
4.6
A STORY OF FUNCTIONS
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M3 Lesson 7 ALGEBRA II
Lesson 7: Bacteria and Exponential Growth
5. Complete the following table to approximate the solution to 2๐ฅ๐ฅ = 19 to three decimal places.
๐๐ ๐๐๐๐ ๐๐ ๐๐๐๐ ๐๐ ๐๐๐๐ ๐๐ ๐๐๐๐
6. Approximate the solution to 5๐ฅ๐ฅ = 5,555 to four decimal places.
7. A dangerous bacterial compound forms in a closed environment but is immediately detected. An initial detection reading suggests the concentration of bacteria in the closed environment is one percent of the fatal exposure level. This bacteria is known to double in concentration in a closed environment every hour and can be modeled by the function ๐๐(๐ก๐ก) = 100 โ 2๐ก๐ก, where ๐ก๐ก is measured in hours.
a. In the function ๐๐(๐ก๐ก) = 100 โ 2๐ก๐ก, what does the 100 mean? What does the 2 mean? b. Doctors and toxicology professionals estimate that exposure to two-thirds of the bacteriaโs fatal concentration
level will begin to cause sickness. Without consulting a calculator or other technology, offer a rough time limit for the inhabitants of the infected environment to evacuate in order to avoid sickness.
c. A more conservative approach is to evacuate the infected environment before bacteria concentration levels reach one-third of fatal levels. Without consulting a calculator or other technology, offer a rough time limit for evacuation in this circumstance.
A STORY OF FUNCTIONS
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M3 Lesson 7 ALGEBRA II
Lesson 7: Bacteria and Exponential Growth
d. Use the method of the example to approximate when the bacteria concentration will reach 100% of the fatal exposure level, to the nearest minute.
๐๐ ๐๐๐๐ ๐๐ ๐๐๐๐ ๐๐ ๐๐๐๐ ๐๐ ๐๐๐๐ ๐๐ ๐๐๐๐
A STORY OF FUNCTIONS
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M3 Lesson 8 ALGEBRA II
Lesson 8: The โWhatPowerโ Function
Lesson 8: The โWhatPowerโ Function
Classwork
Opening Exercise
a. Evaluate each expression. The first two have been completed for you. i. WhatPower2(8) = 3
ii. WhatPower3(9) = 2
iii. WhatPower6(36) = ______
iv. WhatPower2(32) =______
v. WhatPower10(1000) = ______ vi. WhatPower10(1000 000) =______
vii. WhatPower100(1000 000) =______
viii. WhatPower4(64) =______
ix. WhatPower2(64) = ______
x. WhatPower9(3) = ______
xi. WhatPower5๏ฟฝโ5๏ฟฝ = ______
xii. WhatPower12๏ฟฝ18๏ฟฝ = ______
xiii. WhatPower42(1) =______
xiv. WhatPower100(0.01) =______
xv. WhatPower2 ๏ฟฝ14๏ฟฝ = ______
A STORY OF FUNCTIONS
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M3 Lesson 8 ALGEBRA II
Lesson 8: The โWhatPowerโ Function
xvi. WhatPower14(2) = ______
b. With your group members, write a definition for the function WhatPower๐๐ , where ๐๐ is a number.
Exercises 1โ9
Evaluate the following expressions, and justify your answers.
1. WhatPower7(49)
2. WhatPower0(7)
3. WhatPower5(1)
4. WhatPower1(5)
5. WhatPowerโ2(16)
6. WhatPowerโ2(32)
7. WhatPower13(9)
8. WhatPowerโ13(27)
A STORY OF FUNCTIONS
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M3 Lesson 8 ALGEBRA II
Lesson 8: The โWhatPowerโ Function
9. Describe the allowable values of ๐๐ in the expression WhatPowerb(๐ฅ๐ฅ). When can we define a function ๐๐(๐ฅ๐ฅ) = WhatPower๐๐(๐ฅ๐ฅ)? Explain how you know.
Examples
1. log2(8) = 3 2. log3(9) = 2
3. log6(36) = ______
4. log2(32) =______
5. log10(1000) = ______
6. log42(1) =______
7. log100(0.01) =______
8. log2 ๏ฟฝ14๏ฟฝ = ______
Exercise 10
10. Compute the value of each logarithm. Verify your answers using an exponential statement. a. log2(32)
A STORY OF FUNCTIONS
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M3 Lesson 8 ALGEBRA II
Lesson 8: The โWhatPowerโ Function
b. log3(81)
c. log9(81)
d. log5(625)
e. log10(1000000000)
f. log1000(1000000000)
g. log13(13)
h. log13(1)
i. log7๏ฟฝโ7๏ฟฝ
j. log9(27)
k. logโ7(7)
l. logโ7 ๏ฟฝ1
49๏ฟฝ
m. log๐ฅ๐ฅ(๐ฅ๐ฅ2)
A STORY OF FUNCTIONS
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M3 Lesson 8 ALGEBRA II
Lesson 8: The โWhatPowerโ Function
Problem Set 1. Rewrite each of the following in the form WhatPower๐๐(๐ฅ๐ฅ) = ๐ฟ๐ฟ.
a. 35 = 243 b. 6โ3 = 1216 c. 90 = 1
2. Rewrite each of the following in the form log๐๐(๐ฅ๐ฅ) = ๐ฟ๐ฟ.
a. 1614 = 2 b. 103 = 1000 c. ๐๐๐๐ = ๐๐
3. Rewrite each of the following in the form ๐๐๐ฟ๐ฟ = ๐ฅ๐ฅ.
a. log5(625) = 4 b. log10(0.1) = โ1 c. log279 =23
4. Consider the logarithms base 2. For each logarithmic expression below, either calculate the value of the expression or explain why the expression does not make sense. a. log2(1024) b. log2(128) c. log2๏ฟฝโ8๏ฟฝ
d. log2 ๏ฟฝ1
16๏ฟฝ
e. log2(0)
f. log2 ๏ฟฝโ1
32๏ฟฝ
5. Consider the logarithms base 3. For each logarithmic expression below, either calculate the value of the expression or explain why the expression does not make sense. a. log3(243) b. log3(27) c. log3(1)
d. log3 ๏ฟฝ13๏ฟฝ
e. log3(0)
f. log3 ๏ฟฝโ13๏ฟฝ
Lesson Summary
If three numbers ๐ฟ๐ฟ, ๐๐, and ๐ฅ๐ฅ are related by ๐ฅ๐ฅ = ๐๐๐ฟ๐ฟ, then ๐ฟ๐ฟ is the logarithm base ๐๐ of ๐ฅ๐ฅ, and we write log๐๐(๐ฅ๐ฅ) = ๐ฟ๐ฟ. That is, the value of the expression log๐๐(๐ฅ๐ฅ) is the power of ๐๐ needed to obtain ๐ฅ๐ฅ.
Valid values of ๐๐ as a base for a logarithm are 0 < ๐๐ < 1 and ๐๐ > 1.
A STORY OF FUNCTIONS
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M3 Lesson 8 ALGEBRA II
Lesson 8: The โWhatPowerโ Function
6. Consider the logarithms base 5. For each logarithmic expression below, either calculate the value of the expression or explain why the expression does not make sense. a. log5(3125) b. log5(25) c. log5(1)
d. log5 ๏ฟฝ1
25๏ฟฝ
e. log5(0)
f. log5 ๏ฟฝโ1
25๏ฟฝ
7. Is there any positive number ๐๐ so that the expression log๐๐(0) makes sense? Explain how you know.
8. Is there any positive number ๐๐ so that the expression log๐๐(โ1) makes sense? Explain how you know. 9. Verify each of the following by evaluating the logarithms.
a. log2(8) + log2(4) = log2(32) b. log3(9) + log3(9) = log3(81) c. log4(4) + log4(16) = log4(64) d. log10(103) + log10(104) = log10(107)
10. Looking at the results from Problem 9, do you notice a trend or pattern? Can you make a general statement about
the value of log๐๐(๐ฅ๐ฅ) + log๐๐(๐ฆ๐ฆ)?
11. To evaluate log2(3), Autumn reasoned that since log2(2) = 1 and log2(4) = 2, log2(3) must be the average of 1 and 2 and therefore log2(3) = 1.5. Use the definition of logarithm to show that log2(3) cannot be 1.5. Why is her thinking not valid?
12. Find the value of each of the following. a. If ๐ฅ๐ฅ = log2(8) and ๐ฆ๐ฆ = 2๐ฅ๐ฅ, find the value of ๐ฆ๐ฆ. b. If log2(๐ฅ๐ฅ) = 6, find the value of ๐ฅ๐ฅ. c. If ๐๐ = 26 and ๐ ๐ = log2(๐๐), find the value of ๐ ๐ .
A STORY OF FUNCTIONS
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M3 Lesson 9 ALGEBRA II
Lesson 9: LogarithmsโHow Many Digits Do You Need?
Lesson 9: LogarithmsโHow Many Digits Do You Need?
Classwork
Opening Exercise
a. Evaluate WhatPower2(8). State your answer as a logarithm, and evaluate it.
b. Evaluate WhatPower5(625). State your answer as a logarithm, and evaluate it.
Exploratory Challenge
Autumn is starting a new club with eight members including herself. She wants everyone to have a secret identification code made up of only Aโs and Bโs, known as an ID code. For example, using two characters, her ID code could be AA.
a. Using Aโs and Bโs, can Autumn assign each club member a unique two-character ID code using only Aโs and Bโs? Justify your answer. Hereโs what Autumn has so far.
Club Member Name Secret ID Code Club Member Name Secret ID Code Autumn AA Robert
Kris Jillian
Tia Benjamin
Jimmy Scott
b. Using Aโs and Bโs, how many characters would be needed to assign each club member a unique ID code? Justify your answer by showing the ID codes you would assign to each club member by completing the table above (adjust Autumnโs ID if needed).
A STORY OF FUNCTIONS
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M3 Lesson 9 ALGEBRA II
Lesson 9: LogarithmsโHow Many Digits Do You Need?
When the club grew to 16 members, Autumn started noticing a pattern.
Using Aโs and Bโs:
i. Two people could be given a secret ID code with 1 letter: A and B. ii. Four people could be given a secret ID code with 2 letters: AA, BA, AB, BB. iii. Eight people could be given a secret ID code with 3 letters: AAA, BAA, ABA, BBA, AAB, BAB, ABB, BBB.
c. Complete the following statement, and list the secret ID codes for the 16 people:
16 people could be given a secret ID code with _________ letters using Aโs and Bโs.
Club Member Name Secret ID Code Club Member Name Secret ID Code
Autumn Gwen
Kris Jerrod
Tia Mykel
Jimmy Janette
Robert Nellie
Jillian Serena
Benjamin Ricky
Scott Mia
d. Describe the pattern in words. What type of function could be used to model this pattern?
A STORY OF FUNCTIONS
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M3 Lesson 9 ALGEBRA II
Lesson 9: LogarithmsโHow Many Digits Do You Need?
Exercises 1โ2
In the previous problems, the letters A and B were like the digits in a number. A four-digit ID code for Autumnโs club could be any four-letter arrangement of Aโs and Bโs because in her ID system, the only digits are the letters A and B.
1. When Autumnโs club grows to include more than 16 people, she will need five digits to assign a unique ID code to each club member. What is the maximum number of people that could be in the club before she needs to switch to a six-digit ID