M10 Chapter 3 (1) Exponents and Radicals PDF

CHAPTER 3 Exponents and Radicals Exponents have been used to help model and solve problems since the time of the Babylonians, about 4000 years ago. For example, Plimpton 322 is a stone tablet written in Babylonian script (about 1900 to 1600 b.c.e.). The table...

CHAPTER 3 Exponents and Radicals Exponents have been used to help model and solve problems since the time of the Babylonians, about 4000 years ago. For example, Plimpton 322 is a stone tablet written in Babylonian script (about 1900 to 1600 b.c.e.). The tablet contains part of a list of numbers now known as Pythagorean triples. These are sets of three positive numbers, such as 3, 4, and 5, that can be the measurements of the sides of a right triangle. Today, we use exponents to solve problems that range from calculating interest earned on savings to estimating how fast bacteria can grow. A growing bacterial population doubles at regular intervals. Scientists model this growth using a sequence of powers with integral exponents, 20, 21, 22, 23, …. This model can be used to estimate bacterial populations over time and make predictions. Big Ideas Algebra allows us to generalize relationships through abstract thinking. The meanings of, and connections between arithmetic operations extend to powers and polynomials. Inquire and Explore What are the similarities and differences between multiplication of numbers and multiplication of powers? How is prime factorization helpful? How can patterns in numbers lead to algebraic generalizations? Key Terms perfect square radical square root radicand perfect cube index cube root mixed radical prime factorization entire radical irrational number 88 Chapter 3 NEL  Career Link Artists create art to communicate ideas. In addition to having artistic and technical skills, artists are problem solvers who often use math concepts to represent reality. Some of these math concepts may involve the use of exponents or radicals. Artists use a variety of methods and materials to create their works. Some artists use concrete materials to create their designs. Multimedia artists and animators use computer design software to model objects in 3-D. At the end of the chapter, you will create your own work of art that incorporates the mathematical concepts from this chapter. NEL Chapter 3   89  Get Ready Area and Volume Write Powers 1. What is the area of each square? 5. Write each expression as a power. a) a) (6)(6)(6)(6) b) (−4)(−4)(−4)(−4)(−4) c) (3.2)(3.2)(3.2) d) (0.5)(0.5)(0.5)(0.5)(0.5)(0.5) 5 cm b) 6. Write each power using repeated multiplication. Then, evaluate. a) 34 9m b) 53 c) (−2)2 d) −34 2 e) ​​​(_ ​  1 ​) ​​​  ​​ c) a square with sides of length 3 km 4 2. What is the side length of each square? f) 0.43 a) 7. Evaluate. a) 92 A = 81 mm2 b) −52 c) (−2)3 2 ​​  ​3​​   ​​​ d) _ b) a square with an area of 36 cm2 4 c) a square rocket landing pad with an area e) _ ​​  53 ​​ of 900 m2 ​2​​  ​ 2 f) ​​​(_ ​  2 ​) ​​​  ​​ 3. What is the side length of each cube? 3 a) 8. The area of a square on grid paper is 49 square units. Draw the square, and label its area and side length. 9. A cube has an edge length of 5 cm. What V = 27 cm3 is its volume a) in repeated multiplication form? b) in exponential form? b) a cube with a volume of 125 mm3 Prime Factors 5 cm Show all your work. 4. What are the prime factors of each number? a) 54 b) 864 c) 7203 d) 900 90 Chapter 3 NEL Exponent Laws Modelling with Equations 10. Write each expression using repeated 15. Write an equation for each phrase. multiplication. Then, write as a a) double a number is 14 single power. b) a number decreased by 6 is 5 a) 86 ÷ 84 b) 55 ÷ 53 c) one third of a number is 2 c) 77 ÷ 72 d) 48 ÷ 45 d) 1 more than triple a number is 8 e) (−9)7 ÷ (−9)6 f) 0.16 ÷ 0.14 e) a number squared plus 3 is 7 5 3 g) (−0.3)4 ÷ (−0.3) h) ( ​  2 ​) ​​​  ​  ÷ ​​(_ ​​​ _ ​  2 ​) ​​​  ​​ f) a number cubed plus 1 is 9 3 3 g) 2 less than the square root of a number 11. Describe how you would use the exponent is 1 laws to simplify each expression. h) three times the square of a number is 27 Then, evaluate. a) 43 × 44 ÷ 45 Substituting into Formulas b) 87 ÷ 87 × 8 6 3 16. Evaluate each expression. ​​  ​9​​  ​  × ​9​​  c) _  ​​ ​ ​9​​  7​ a) 2x − 3 for x = 4 5 2 d) _ ​​  ​6​​  ​  × ​6​​ 3 ​​​ b) 3y + 2 for y = 7 6 × ​6​​  ​ c) r2 − r + 1 for r = 6 e) (24)2 × 23 d) a2 − 2b2 for a = 3 and b = 1 ​( ​3​​  2​  )​​  4​  × ​3​​  3​ f) ___________ ​​   ​​ e) p2 + 2p − 3 for p = 4 ​3​​  8​ f) 4x2 − y − 2 for x = 2 and y = 1 12. Write each expression as a single power. a) (−3)2(−3)5 17. The stopping distance, d, in metres, for a car is given by the formula d = 0.008s2, b) (−22)4 9 where s is the speed of the car when it starts ​​  72 ​​ c) __ to brake, in kilometres per hour. 7 d) (43)(82) a) What is the stopping distance if the car 4 3 is travelling at 105 km/h when it starts (−6 )(−6 ​ ) e)​​  ________ 2 3  ​ to brake? (−6 ) b) What is the speed of the car if it takes 13. Evaluate each expression. 51.2 m to come to a stop? a) 83 18. The formula for the volume of a sphere is b) (34)(25) V = ​​ _43 ​​πr3, where V is the volume and r is the ( −3 ) 4 0 ​  9 4 ​  ​​​  ​​ c) ​​​ ___ radius of the sphere. What is the volume 2 of a sphere with a radius of 4 cm? Give the d) ​50 − ( ​  1 ​) ​​​  ​​ ​​ __ answer to the nearest cubic centimetre. 2 14. Write the calculator key sequence you r = 4 cm would use to evaluate each expression. Then, evaluate. a) ​​(4 − 1)​​  2​  × ​ _ 1 ​​ 2 b) _ ​​  2 ​(3 + ​3​​  2​)​ 3 NEL Get Ready 91  3.1 Square Roots and Cube Roots Focus on … determining the square root of a perfect square and explaining the process determining the cube root of a perfect cube and explaining the process solving problems involving square roots or cube roots Workers apply what they know about surface area and volume when working with square shapes and cubes. A house painter must calculate the surface area of the walls of a house when preparing a cost estimate. If you know the area of a square wall, how can you calculate the side lengths? A designer must calculate the size of the case required to enclose a speaker for a sound system. If you know the volume of a cube-shaped box, how can you calculate the edge lengths? Explore and Analyze Materials 1. a) Determine the area of each square dot paper square shown. Record the isometric dot paper information in a table. t e sen cat nd pre ni lve ta fle t Re mu Moason Re nec So ers ct l de m d n Un Re Co Co Side Length Area in Exponential Form Area b) Extend the pattern for squares with side lengths of 4, 5, and 6 units. c) What is the relationship between the side length of a square and the area of the square? 92 Chapter 3 NEL  2. a) Determine the volume of each cube shown. Record the information in a table. Edge Length Volume in Exponential Form Volume b) Extend the pattern for cubes with edge lengths of 4, 5, and 6 units. c) What is the relationship between the edge length of a cube and the volume of the cube? Reflect and Respond 3. Discuss with a partner. a) What strategy could you use to find the side length of a square if you were given the area? b) What strategy could you use to find the edge length of a cube if you were given the volume? c) Explain, using a diagram, how you could predict the side length of a square with an area of 64 square units the edge length of a cube with a volume of 343 cubic units Develop Understanding Perfect squares and square roots are related to each other. The number 25 perfect square is a perfect square. It is formed by multiplying two factors of 5 together. a number that is the __ product of the same (5)(5) or 52 = 25 The symbol for square root is ​​ X ​​. √ two factors ___ ______ it has an even number The square root of 25 is 5, or √ ​​​ 25 ​   ​ ​ ​  = ​√ ​​ (5)(5) ​ of each prime factor ___ = ​√52 ​ 25 = (5)(5) or 52 =5 36 = (2)(2)(3)(3) or 62 square root a factor of a number that when squared gives the number for___ example, _____ ​​​ 49 ​​  ​  = √ √ ​  ​​ (7)(7) ​​ =7 NEL 3.1 Square Roots and Cube Roots 93 perfect cube Perfect cubes and cube roots are related to each other. The number 27 is a a number that is the perfect cube. It is formed by multiplying three factors of 3 together. product of the same 3 __ three factors (3)(3)(3) or 33 = 27 The symbol for cube root is ​​ √X ​​. for example, ___ ________ The cube root of 27 is 3, or √ ​​​ 3 27 ​   ​  = √ ​   ​​ (3)(3)(3) ​ 3 ​ ​ 64 = (4)(4)(4) or 43 ___ 3 ​  33 ​ =√ cube root =3 the number that, when multiplied three Some numbers are both perfect squares and perfect cubes. times, results in the perfect cube ​​64​ ​  = (8)(8)​ ​  = (4)(4)(4)​  ​​  and ​​64​     ​​ for example, = 82 = 43 3 ____ ________ 3 √ ​  = √ ​​​  125 ​​    ​   ​​ (5)(5)(5) ​​ Therefore, 64 is a perfect square and a perfect cube. =5 You can distinguish a square root from a cube root by the symbol. While the symbol for square root does not contain a number, the symbol for cube root contains ___ the number___ 3. 3 ​​√ 64 ​ = 8​  and ​​ √ 64 ​ = 4​ Example 1 Identify Perfect Squares and Perfect Cubes State whether each number is a perfect square, a perfect cube, both, or neither. a) 121 b) 729 c) 356 Solution a) To decide whether 121 is a perfect square, you might use a diagram. 102 = 100 Too low A = 121 units2 122 = 144 Too high 112 = 121 Correct! s = √121 A square with side lengths of 11 units has an area of 121 units2. (11)(11) = 121 Therefore, 121 is a perfect square. To decide whether 121 is a perfect cube, you could use guess and check. No whole number cubed results in a product of 121. 43 = 64 Too low 53 = 125 Too high Therefore, 121 is not a perfect cube. 94 Chapter 3 NEL  b) For 729, you might use prime factorization. Prime factorization prime factorization involves writing a number as the product of its prime factors. the process of writing a A factor tree helps organize the prime factors. number as a product of its prime factors Record the prime factorization for 729 the prime factorization 729. Then, identify the factors that of 24 is 2 × 2 × 2 × 3 can be squared or cubed to form the 3 243 product 729. 3 3 81 3 3 3 27 These two groups indicate Why is prime factorization the square root of 729. an efficient method for 3 3 3 3 9 determining whether 729 These three groups indicate is a perfect square or the cube root of 729. 3 × 3 × 3 × 3 × 3 × 3 cube? 27 27 or 3 × 3 × 3 × 3 × 3 × 3 9 9 9 You can write 729 as the product (27)(27) = 272. Therefore, 729 is a perfect square. You can write 729 as the product (9)(9)(9) = 93. Therefore, 729 is a perfect cube. D i d You K n ow? c) For 356, you might use a calculator. Key sequences vary among __ calculators. Check the key sequence Between 1850 C 356 ​​√x ​​ 18.867962 for determining square roots and and 1750 b.c.e., the __ cube roots of numbers on the Babylonians used C 356 2nd ​​  x y ​​ √ 3 = 7.0873411 calculator you are using. Record the what is now known correct sequence for the calculator. as the Pythagorean Since the square root is not a whole relationship. They number, 356 is not a perfect square. recorded tables of Since the cube root is not an integer, square roots and cube roots on clay tablets. 356 is not a perfect cube. This was even before The number 356 is neither a perfect Pythagoras was born. square nor a perfect cube. Why is it more efficient to use a calculator to determine if 356 is a perfect square or cube? Your Turn State whether each number is a perfect square, a perfect cube, both, or neither. Use a variety of methods. Which method do you prefer? Why? a) 125 b) 196 c) 4096 NEL 3.1 Square Roots and Cube Roots 95  Example 2 Solve Problems Involving Square Roots and Cube Roots Did Yo u Know ? The daily volume of gold produced in a British Columbia mine is approximately 512 cm3. If this volume were made into a single cube, The Brucejack mine, north of Stewart, BC, is what would the dimensions of the cube be? estimated to produce 7.3 million ounces Solution of high-grade gold The volume of a cube of length x is given by V = x3. worth approximately $18.3 billion Determine the dimensions__ of the cube, x, by calculating the cube root of 3 (2018 dollars). the volume, or x = √ ​​  V ​​. Method 1: Use Prime Factorization Determine the cube root of 512. 512 Record the prime factorization for 2 256 512. Then, identify the factor that can be cubed to form 512. 2 2 128 2 2 2 64 2 2 2 2 32 2 2 2 2 2 16 2 2 2 2 2 2 8 Since there are three equal groups, you know that 512 2 2 2 2 2 2 2 4 is a perfect cube. How do you know that 512 2 ×2 × 2 × 2 ×2 × 2 × 2 × 2× 2 is not a perfect square? 8 8 8 The cube root of 512 is 8. The cube would be 8 cm in length, height, and width. Method 2: Use a Calculator __ C 512 2nd ​​  x y ​​ √ 3 = 8 The cube would be 8 cm in length, height, and width. Your Turn a) A floor mat for gymnastics is a square with an area of 196 m2. What is its side length? b) The volume of a cubic box is 27 000 cm3. Use two methods to determine its dimensions. 96 Chapter 3 NEL  Connect and Reflect Key Ideas A perfect square is the product of two equal factors. One of these factors is called the square root. ___ ​​ 36 ​​ = 6 because 62 = 36. 36 is a perfect square: √ A perfect cube is the product of three equal factors. One of these factors is called the cube root. _____ 3 ​​  −125 ​​ = −5 because (−5)3 = −125. −125 is a perfect cube: √ Numbers can be both perfect squares and perfect cubes. 15 625 is a perfect square: 1252 = 15 625 15 625 is a perfect cube: 253 = 15 625 You can use diagrams or manipulatives, factor trees, or a calculator to solve problems involving square roots and cube roots. Determine the cube root of 64. ◾ Use a diagram. ◾ Use prime factorization. 64 2 32 s = 4 units 2 2 16 V = 64 units3 2 2 2 8 2 2 2 2 4 The edge lengths represent the 2× 2 × 2× 2× 2× 2 cube root: There are three equal groups of (4)(4)(4) = 64 4. Therefore, the cube root of 64 is 4. ◾ Use a calculator. __ C 64 2nd ​​  x y ​​ √ 3 = 4 NEL 3.1 Square Roots and Cube Roots 97  Practise 1. What is the value of each expression? Express your answers as integers or fractions in lowest terms. a) 72 b) −502 c) (−3)2 d) ​​  _ ​​ e) ​​  _ ​​ 2 f) ​​​(__ ​  3 ​) ​​​  ​​ 42 3 5 22 4 2. Evaluate. Give your answers as integers or fractions in lowest terms. a) 23 b) −43 c) (−5)3 _ 23 d) ​​   ​​ e) ​​  _ 3  ​​ f) ​​​(__ 3 ​  2 ​) ​​​  ​​ 4 63 3 3. What is the value of each expression? ___ ____ _______ a) ​​√49 ​​ b) ​​√169 ​​ c) ​​√(25)(4) ​​ ___ ____ 16 d) ​​ ____ ___ ​​ ​√36 ​ e) ​​ _____  ​​ ​​ 9x2 ​​ f) √ ​ 64 ​ √ 3 4. Evaluate. __ _______ _____ 3 3 3 a) ​​ √1 ​​ b) ​​ √(8)(27) ​​ ​​  8000 ​​ c) √ 3 ___ ____ _____ ​  64 ​ √ 3 27 d) ​​ ____ √ 125 e) ​​  ____ 3  ​​ ​   ​ ​​ ​​  64a3 ​​ f) √ 2 5. Identify each number as a perfect square, a perfect cube, or both. Support your answers using a diagram or a factor tree. a) 1 b) 1000 c) 81 d) 169 e) 216 f) 1024 6. State whether each of the following numbers is a perfect square, a perfect cube, both, or neither. a) 144 b) 2197 c) 16 d) 225 e) 15 625 f) 117 649 7. Evaluate using prime factorization. Explain the process. ____ 3 __ ___ a) ​​√100 ​​ b) ​​ √8 ​​ c) ​​√81 ​​ 3 ___ ____ ____ d) ​​ √27 ​​ e) ​​√144 ​​ f) ​​√576 ​​ 8. Calculate. ____ 3 _____ 3 _____ a) ​​√196 ​​ b) ​​ √4096 ​​ ​​  9261 ​​ c) √ 3 _____ ____ 3 _____ d) ​​ √3375 ​​ e) ​​√961 ​​ ​​  4913 ​​ f) √ 9. Connor needs to replace the edging on a square rug. If the rug has an area of 25 m2, what length of edging does he need? 10. Serena is watching the sea life in an aquarium. The volume of water in the cubic tank is 343 m3. What is the edge length of the cube? 98 Chapter 3 NEL Apply 11. A square-shaped region of a gymnasium is used for wrestling. This t e sen cat floor space has an area of 1444 m2. nd pre ni lve ta fle t Re mu Moason Re ec So ders ct l de nn m Un Re Co Co a) Before calculating the side length of the square-shaped region, estimate two whole numbers between which the answer falls. Which number do you think the answer is closer to? b) Calculate the side length. c) How does your estimate compare to the calculated answer? D i d You K n ow? 12. Star quilts are in the shape of a The star quilt is a square with a minimum area of pattern used by many cultures including 1 m2 and a maximum area of the Lakota, Dakota, 9 m2. What are the possible other Sioux nations, whole-number dimensions of and Europeans. It such a quilt? was inspired from the design for buffalo robes. When buffalo were no longer available, the star quilt replaced the buffalo robe in Aboriginal traditions. 13. Competency Check In 2009, the Greater Victoria Harbour Authority t e sen cat nd sponsored the creation of Na’Tsa’Maht—The Unity Wall Mural. This pre ni lve ta fle t Re mu Moason Re ec So ders ct l de nn m Un Re Co Co project transformed the Ogden Point breakwater into an enormous canvas. The mural is dedicated to the Esquimalt and Songhee Nations and celebrates their culture, history, and art. Your art class decides to create a mural mosaic. Your mosaic will highlight the regions of the province or territory where you live. a) The class mosaic will be composed of 15 cm by 15 cm squares. How many squares will be needed to create a mural that covers an area of 2.7 m2? b) How is the class mosaic a geometric representation of square roots? NEL 3.1 Square Roots and Cube Roots 99  14. A recycling depot compresses cardboard into cubic bales. If each bale has a volume of 46 656 cm3, what are its edge lengths? 15. The cubic sculpture shown here is t e sen cat nd pre ni made of steel with copper leaf. It was lve ta fle t Re mu Moason Re nec So ers ct l de m d n Un Re Co Co created by Tony Bloom, an artist from Canmore, AB. a) The sculpture has a volume of 4913 in.3. What is the length of one edge of the cube? b) Explain how the sculpture is a geometric representation of a cube root. 16. The surface area of a cubic box is 600 mm2. What is the volume of t e sen cat nd the box? pre ni lve ta fle t Re mu Moason Re nec So ers ct l de m d n Un Re Co Co Extend 17. Meteorologists use the formula D 3 = 684t 2 to describe the area covered by violent storms, such as tornadoes and hurricanes. D is the diameter of the storm, in kilometres, and t is the time, in hours, it will last. a) If a storm lasts for 4 h, what is its diameter? b) If the diameter of a hurricane is 30 km, how long will it last? 18. A cube has a volume of 3375 cm3. What is the diagonal distance through the cube from one corner to the opposite corner? 19. A manufacturer is designing a cube-shaped box t e sen cat nd with no lid to hold a basketball. The basketball pre ni lve ta fle t Re mu Moason Re nec So ers ct l de m d Basketball n has a volume of 2304π cm3. Un Re Co Co a) How much cardboard is needed to create the smallest box possible using the least amount of material? Do not include seam overlap in your calculations. b) What is the volume of the box? What are its dimensions? 100 Chapter 3 NEL Create Connections 20. The following graph can be used to determine squares and square roots. t e sen cat nd pre ni lve ta fle t Re mu 60 Moason Re ec So ders ct l de nn m Un Re Co Co 50 40 30 20 10 0 1 2 3 4 5 6 7 8 a) Use the graph to complete the following table of values. Number 0 1 2 3 6 7 Number Squared 0 1 16 25 64 b) Based on the table, how would you label the axes on the graph? c) What does each small unit represent on the horizontal axis? vertical axis? d) Explain how you could use the graph to find the value of 52. ___ e) How could you use the graph to evaluate √ ​​ 49 ​​? f) Show how you ___ could use the graph to determine the approximate value for √ ​​ 18 ​​. Multiply your answer by itself. How close is your product to 18? g) What is an approximation for 6.22? 21. a) Write an arithmetic expression, involving a square root, that has a value of _​​  23 ​​. b) Write an arithmetic expression, involving a cube root, that has a value of _​​  23 ​​. NEL 3.1 Square Roots and Cube Roots 101  3.2 Integral Exponents Focus on … applying the exponent laws to expressions using rational numbers or variables as bases and integers as exponents converting a power with a negative exponent to an equivalent power with a positive exponent solving problems that involve powers with integral exponents The Rhind mathematical papyrus (RMP) is a valuable source of information about ancient Egyptian mathematics. This practical handbook includes problems that illustrate how Egyptians solved problems related to surveying, building, and accounting. The RMP was written in approximately 1650 b.c.e. How do archaeologists know this? One way to determine the age of organic matter is by using carbon-14 dating. All living things absorb radioactive carbon-14. Papyrus is made from papyrus plants. As soon as the papyrus dies, it stops taking in new carbon. The carbon-14 decays at a constant, known rate and is not replaced. Scientists can measure the amount of carbon-14 remaining. They use a formula involving exponents to accurately assess the age of the papyrus. Is the quantity of carbon-14 increasing or decreasing? Do you think the exponent in the formula would be positive or negative? Why? Did Yo u Know ? Carbon-14 dating is accurate for dating artifacts up to about 60 000 years old. 102 Chapter 3 NEL  Explore and Analyze 1. On a sheet of paper, draw a line 16 cm long and mark it as shown. Materials ruler 0 16 24 2. Mark a point halfway between 0 and 16. Label the point with its value and its equivalent value in exponential form (2x). Repeat this procedure until you reach a value of 1 cm. a) How many times did you halve the line segment to reach 1 cm? b) What did you notice about the exponents as you kept reducing the line segment by half? 3. a) Mark the halfway point between 0 and 1. What fraction does this represent? b) Using the pattern established in step 2, what is the exponential form of the fraction? c) Halve the remaining line segment two more times. 4. Use a table to summarize the line segment lengths and the equivalent exponential form with a base of 2. Reflect and Respond 5. a) Describe the pattern you observe in the exponents as the distance is halved. b) Is there a way to rewrite each fraction so that it is expressed as a power with a positive exponent? Try it. Compare this form to the equivalent power with a negative exponent. What is the pattern? c) Create a general form for writing any power with a negative exponent as an equivalent power with a positive exponent. 6. a) Carbon-14 has a half-life of 5700 years. This means the rate of D i d You K n ow? decay is _​​  12 ​​ or 2−1 every 5700 years. What fraction of carbon-14 The half-life of a would be present in organic material that is 11 400 years old? radioactive element is 17 100 years old? Express each answer as a power with a negative the amount of time it exponent. Explain how you arrived at your answers. takes for half of the b) Suggest other types of situations when a negative exponent might atoms in a sample to be used. decay. The half-life of a radioactive element is constant. It does not depend on the quantity of material. NEL 3.2 Integral Exponents 103  Develop Understanding You can use the exponent laws to help you simplify expressions with integral exponents. Exponent Law Note that a and b are rational or variable bases and m and n are integral exponents. Product of Powers (am)(an) = am+n Quotient of Powers ​​  _ am What are the differences  ​​ = am−n, a ≠ 0 an between rational and Power of a Power (am)n = amn integral numbers? m Power of a Product (ab) = (am)(bm) ​  a ​) ​​​  ​​ = ​​  _n ​​  , b ≠ 0 n an Power of a Quotient ​​​(__ b b Zero Exponent a 0 = 1, a ≠ 0 To simplify expressions with integral exponents, you can use the following principle as well as the exponent laws. A power with a negative exponent can be written as a power with a positive exponent. ​​  1n ​​, a ≠ 0 2­−3 = ​​  _ 1  ​​  How is this related to (​​​ __ ​  12 ​) ​​​  ​​? 3 a−n = ___ a 23 ​​  _1  ​​ = an, a ≠ 0 ​​  _ 1  ​​ = 23 a−n 2−3 Example 1 Powers of 10 Express each number as a power of 10 with a positive or a negative exponent. a) 1000 ​​  1  ​​ b) _ c) 100 000 d) 0.001 10 000 Solution Rewrite each number as a power of 10. ​  = 10 × a) ​​1000​     ​​ 10 × 10​ = 103 Therefore, 1000 can be expressed as a power of 10 with a positive exponent of 3. ​​​  1    b) _ ​ ​​ 1  ​​  = ​ _____________     ​​ 10 000 10 × 10 × 10 × 10 1  ​ = ​ _ ​ 0​​  4​ 1 = ​10​​  −4​ Therefore, _____ ​​  1  ​​can be expressed as a power of 10 with a negative 10 000 exponent of −4. 104 Chapter 3 NEL  ​  = 10 × c) ​​100 000​      ​​ 10 × 10 × 10 × 10​ = 105 Therefore, 100 000 can be expressed as a power of 10 with a positive exponent of 5. d) ​​0.001​    ​  1 ​  ​​ ​​ ​  = _ ​ 1000 = ____________ ​  1  ​ 10 × 10 × 10 =_ ​  1 3 ​ ​10​​  ​ = ​10​​  −3​ Therefore, 0.001 can be expressed as a power of 10 with a negative exponent of −3. Your Turn Express each number as a power of 2 with a positive or a negative exponent. a) 16 ​​  1  ​​ b) _ c) 256 d) 0.125 32 Example 2 Multiply or Divide Powers with the Same Base Write each product or quotient as a power with a single exponent. b) (0.8−2)(0.8−4) c) ​​  _ ​​ d) ​​  __ ​​ x5 (2x)3 a) (58)(5−3) x −3 (2x)−2 Solution Use the exponent laws for multiplying or dividing powers with the same base and integral exponents. a) Method 1: Add the Exponents ​​(58)(5−3)​ ​  = 58+(−3)  ​​ ​ How do you know that you can add the exponents? = 55 Method 2: Use Positive Exponents Convert the power with a negative exponent to one with a positive exponent. Rewrite as a division statement. ​​(58)(5−3)​    ​  = (58 ​​​)​(​ __ ​  13 ​) ​​ 5 _ 5 8 = ​  3 ​ 5 = 58−3 How do you know that you can subtract the exponents? = 55 NEL 3.2 Integral Exponents 105  b) Method 1: Add the Exponents ​​(0.8−2)(0.8−4)​    ​  = 0.8−2+(−4)  ​​ ​ = 0.8−6 Which method do Method 2: Use Positive Exponents you prefer? Why? ​  = ( ​​(0.8−2)(0.8−4)​       ​  1 2 ​​  ​) ​​( ​ ____ ​  1 4​ ​) ​​ ​ ____ 0.8 0.8 = ​  __ 1  ​ (0.82)(0.84) = ​  __ 1  ​ 0.82+4 = ​  _ 1  ​ 0.86 _5 x c) ​ ​​ −3 ​​  = x5−(−3)  ​​  ​ ​ What strategy was used? x Could you use a different strategy? = x5+3 = x8 d) Method 1: Subtract Method 2: Use the Exponents Positive Exponents ​​​  __ ​​​  __ (2x)3 (2x)3  ​​ ​  = (2x)3−(−2)     ​​ ​  ​​ ​  = (2x)3 ​​(2x)    ​ 2​ (2x)−2 (2x)−2 = (2x)5 = (2x)3+2 = (2x)5 Your Turn Simplify each product or quotient. b) ​​  _ ​​ c) ​​  __ ​​ __ 7−5 (−3.5)4 (3y)2 a) (2−3)(25) d) ​​   ​​ 73 (−3.5)−3 (3y)−6 Example 3 Powers of Powers Write each expression as a power with a single, positive exponent. Then, evaluate where possible. 4 −2 c) ​​​(___ ​  26 ​) ​​​  ​​ d) ​​​[​​(__ ​  3 ​) ​​​  ​]​​​  ​​ 4 −3 −2 a) (43)−2 b) [(a−2)(a0)]−1 ​  3 ​) ​​​  ​​​(__ 2 4 4 Solution a) Multiply the exponents. Then, rewrite with a positive exponent. ​​(43)−2​  = 4(3)(−2) ​​  ​​ ​ = 4−6 = ​  _ 1  ​ If you rewrite this as (​​​ __ ​  41 ​) ​​​  ​​, does the answer change? 6 46 = ​  _ 1  ​ 4096 106 Chapter 3 NEL  b) Since the bases are the same, you can multiply the powers by adding D i d You K n ow? the exponents. Raise the result to the exponent −1. Then, multiply John Wallis was the exponents. a professor of ​​[(a−2)(a0)]−1   ​  = (a−2+0     ​​  )−1 ​​ ​ How could you use your knowledge of the geometry at Oxford −2 −1 exponent laws for zero exponents to help = (a ) University in England simplify the original expression? = a(−2)(−1) in 1655. He was the first to explain = a2 the significance of c) Method 1: Simplify within the Brackets zero and negative Since the bases are the same, you can subtract the exponents. exponents. He also Raise the result to the exponent −3. Then, multiply. introduced the current ​​​​(___ ​  26 ​) ​​​     4 −3 symbol for infinity, ∞. ​ ​​ )−3​ ​ ​ ​​  = (24−6 2 = (2−2)−3 = 2(−2)(−3) = 26 = 64 Method 2: Raise Each Power to an Exponent Raise each power to the exponent −3. Then, divide the resulting powers by subtracting the exponents, since they have the same base. ​​​​(___ ​ ​​  = ​  _ ​  26 ​) ​​​     4 −3 (24)−3 ​ ​​ ​  ​​ ​ ​ 2 (26)−3 = ​  __ 2 (4)(−3)  ​ 2(6)(−3) Which method do you = ​  _ prefer? Why? 2−12 ​ 2−18 = 2−12−(−18) = 26 = 64 d) Add the exponents. Raise the resulting power to the exponent −2. ​ ​​  = [​​ ​​(__ ​ ​​​  ​​ ​]​ ​​​  ​​ −2 −2 ​​​​[​​(__ ​  3 ​) ​​​  ​]​​​     −2 4 −2+4 ​  3 ​) ​​​  ​​​(__       ​  3 ​)  ​​  How do you know that you can add 4 4 4 the exponents? = [​​ ​​(__ ​  3 ​) ​​​  ​]​​​  ​ 2 −2 4 (2)(−2) = ​​(__ ​  3 ​) ​​​  ​ 4 −4 = ​​(​   ​) ​​​  ​ __ 3 4 = ​  1 4 ​ _____ ​​(__ ​  3 ​) ​​​  ​ 4 4 Why is the base now __ = ​​(__​  4 ​) ​​​  ​ ​​ 43 ​​ instead of 3 the original base of __ ​​ 34 ​​? 256 ​ = ​ ____ 81 Your Turn Simplify and evaluate where possible. 2 0 −3 [ (y) ] c) ​​​(___ ​  x4 ​) ​​​  ​​ 6 −2 (y ) a) [(0.63)(0.6−3)]−5 b) [(t−4)(t3)]−3 d) ​​​ _____ ​  3 ​ ​​​  ​​ x NEL 3.2 Integral Exponents 107  Example 4 Apply Powers with Integral Exponents Did Yo u Know ? It is estimated that during a moderate infestation there are 639 billion grasshoppers in the Thompson Okanagan region, which has an area of The clear-winged 71 000 km2. Approximately how many grasshoppers are there per grasshopper is a pest of grasses and square kilometre? cereal grain crops. These insects can Solution completely destroy Method 1: Use Arithmetic barley and wheat fields early in the Divide the number of grasshoppers by the total area. season. Agricultural ​  = ​  ___ ​​grasshoppers per square kilometre​    639 000       000 000  ​​  ​​ field workers 71 000 conduct grasshopper = 9 000 000 surveys and produce There are approximately 9 000 000 grasshoppers per square kilometre. forecasts to help assess the need for Method 2: Use Exponent Rules control measures to Since you cannot enter numbers as large as 639 billion directly into some protect crops. calculators, rewrite them using exponential form. Then, use the exponent rules to calculate the power of 10. ​  = ​  __ (639)(109) ​​grasshoppers per square kilometre​        ​​ ​ ​​ Is it possible to enter (71)(103) numbers expressed = (9)(109−3) using exponential 6 form directly into = (9)(10 ) your calculator? How There are approximately 9 000 000 grasshoppers per would doing this help square kilometre. calculate the answer? Your Turn British Columbia Ministry of Agriculture staff conducted a grasshopper count. In one 25 km2 area, there were 401 000 000 grasshoppers. Use the following table to assess the degree of grasshopper infestation in this area. Remember that one square kilometre is equal to 1 000 000 m2. Grasshopper Density 0—4 per square metre = very light 4—8 per square metre = light 8—12 per square metre = moderate 12—24 per square metre = severe 24 per square metre = very severe 108 Chapter 3 NEL  Connect and Reflect Key Ideas A power with a negative exponent can be written as a power with a positive exponent. 3−4 = ​​  _ _ −2 (​  2 ​) ​​​  ​ ​​  = _____ ​  1 2 ​​    ​​  1  ​​    ​​​​ __  ​ ​ 1  ​​ = 25 3 −5 ​​(__​   ​) ​​​  ​ 3 4 2 2 3 2 = ​​(__​   ​) ​​​  ​ 3 2 You can apply the above principle to the exponent laws. Exponent Law Example Note that a and b are rational or variable bases and m and n are integral exponents. Product of Powers ​​(3−2)(34)​  ​  = 3−2+4  ​​​ (am)(an) = am+n = 32 ​​​  _ Quotient of Powers 3 x  ​​  ​  = x3−(−5)  ​​​ ​​  _ am x−5  ​​ = am−n, a ≠ 0 an = x8 Power of a Power ​​(0.754)−2    ​  = 0.75(4)(−2)​ ​​ (am)n = amn = 0.75−8 or ​ __1  ​ 0.758 ​​(4z)−3​  = ​  _ Power of a Product 1 ​ ​​ ​​ (ab)m = (am)(bm) (4z)3 ​  31 3 ​ = _____ 4 z ​  1 3 ​ = _____ 64z Power of a Quotient (​  3 ​) ​​​  ​ = ​  ​3​​  −2 ​​ t −2​  ____ ​​ __ ​t​​  −2 ​​​​​ a ​  ​​​  n​​ = ​​  _ an ( b) ​​​ ​ __  ​​  , b ≠ 0 = ​  _ ​ bn 32 t2 = ​  _ ​ 9 t2 0 Zero Exponent (4y 2) = 1 0 a0 = 1, a ≠ 0 −(4y 2) = −1 Practise 1. a) Write two numbers that can be expressed as a power of 5 with a positive exponent. b) Write two numbers that can be expressed as a power of 3 with a negative exponent. NEL 3.2 Integral Exponents 109  2. For each situation, identify when a positive and when a negative exponent would be used. a) calculating the population growth of a city since 2005 using the expression 150 000(1.005)n b) calculating the amount of a radioactive substance remaining from a known sample amount using the expression 25​​​(_​  12 ​) ​​​ n​​ c) determining how many bacteria are present in a culture after h hours using the expression 500(2)h 3. Write each expression with positive exponents. a) b −3 b) xy −4 c) 2x −2 d) 2x2y−1 e) −4x−5 f) −2x−3y−4 4. Simplify the quotient __ ​​ xx5 ​​using two different methods to show that __ 3 ​​ x12 ​​ is −2 the same as x. 5. Daniel was rewriting the expression ____ −3 ​​ 2xy5 ​​with positive exponents. He ____ 2 quickly recorded ​​  x3y5 ​​. Is his answer correct? Justify your answer. 6. Simplify each expression. State the answer using positive exponents. a) (43)(4−5) b) ​​  _ 3−4 ​​ c) ​​  _ 123 ​​ 3−2 127 (8 ) 3 8−1 d) ​​​ ____ ​  0 ​  ​​​  ​​ e) (54)−2 f) [(32)(2−5)]3 g) ​​​(___ ​  52 ​) ​​​  ​​ 2 −1 h) (3.2−2)−3 i) 4[(2−1)(2−2)]−1 4 7. Simplify each expression by restating it using positive exponents only. a) ​​  _ 1  ​​ b) [(h7)(h−2)]−2 c) ​​  _ 8t  ​​ s2t−6 t−3 e) ​​​(___ ​  n−4 ​) ​​​  ​​ 4 −3 d) (2x−4)3 f) [(xy4)−3]−2 n 8. Simplify and then evaluate. Express your answers to four decimal places, where necessary. 3 −3 a) (0.52)−3 b) ​​​[​​(__ ​  2 ​) ​​​  ​]​​​  ​​ c) [(5)(53)]−1 3 2 −1 d) ​​​(___ ​  64 ​) ​​​  ​​ f) ​​​[​​(__ ​  3 ​) ​​​  ​]​​​  4 −3 −4 −4 6 (8 ) ​  83 ​  ​​​  ​​ e) ​​​ __ ​  3 ​) ​​​  ​ ÷ ​​(__ 4 4 ​​ 9. Is it true in all cases that you can express a rational number with a negative exponent as its reciprocal with a positive exponent? If so, create a mental math shortcut for this situation. 110 Chapter 3 NEL 10. A mountain pine beetle population can double every year if conditions are ideal. Assume the forest in Tweedsmuir Provincial Park, BC, has a population of 20 000 beetles. The formula P = 20 000(2)n can model the population, P, after n years. a) How many beetles were there in the forest 4 years ago? 8 years ago? b) If the conditions remain ideal, how many beetles will there be 2 years from now? 11. French-language publishing sales in Canada increased at a rate of 1.05 per year from 2010 to 2014. There were sales of $300 000 in 2010. The formula S = 300 000(1.05)n models the sales, S, after n years. Assume that the growth rate stays constant. What would be the projected sales for 2024? Apply 12. The bacterium Escherichia coli is commonly found in the human intestine. A single bacterium has a width of 10−3 mm. The head of a pin has a diameter of 1 mm. How many Escherichia coli bacteria can fit across the diameter of a pin? NEL 3.2 Integral Exponents 111  13. Competency Check A culture of bacteria in a lab contains t e sen cat nd 2000 bacteria cells. The number of cells doubles every day. This pre ni lve ta fle t Re mu Moason Re nec So ers ct l de m d n relationship can be modelled by the equation N = 2000(2)t, where N Un Re Co Co is the estimated number of bacteria cells and t is the time, in days. a) How many cells were present for each amount of time? i) after 2 days ii) after 1 week iii) 2 days ago b) What does t = 0 indicate? 14. The Great Galaxy in Andromeda is about 2 200 000 light years from Earth. Light travels 9 500 000 000 000 km in a year. How many kilometres is the Great Galaxy in Andromeda from Earth? 15. A red blood cell is about 0.0025 mm in diameter. How large would it appear if it were magnified 108 times? 16. There are approximately (3.2)(1024) atoms in 1 kg of lead. How many atoms are there in a milligram of lead? Hint: 1 kg = 106 mg. 17. Over time, all rechargeable batteries lose their charge, even when not in use. A 12 V nickel-metal hydride (NiMH) battery, commonly used in power tools, will lose approximately 30% of its charge every month if not recharged. This situation can be modelled by the formula V = 12(0.70)m, where V is the estimated voltage of the battery in volts (V), and m is the number of months the battery is not used. What is the estimated voltage of an unused battery after 3 months? Assume the battery was initially fully charged. 112 Chapter 3 NEL 18. Wildlife biologists are tracking the sandhill crane population growth in D i d You K n ow? the Lulu Islands Wetlands. The crane population increased by a growth Even though sandhill rate of 7.3% per year from 2012 to 2018. There were 174 sandhill crane populations cranes in 2012. The rate of growth can be modelled using the formula suffered significant P = 174(1.073)n, where P is the estimated population and n is the declines in the last number of years since 2012. If conditions remain constant, what is the century, the species is projected crane population? making a comeback in some areas of coastal British Columbia. a) in 2021? b) in 2024? 19. From 2011 to 2016, the population of Prince George increased at an average annual rate of 0.92%. This can be modelled using the formula P = 62 623(1.0092)n, where P is the estimated population and n is the number of years since 2011. a) What was the population of Prince George in 2014? b) If this rate of increase stays the same, what will the population be in 2022? 20. Following the 1989 Exxon Valdez oil spill, 100 km of Arctic shoreline t e sen cat nd was contaminated. Crude oil is made up of thousands of compounds. pre ni lve ta fle t Re mu Moason Re ec So ders ct l de nn m Un Re Co Co It takes many different kinds of naturally occurring bacteria to break the oil down. Lab technicians identified and counted the bacteria. They monitored how well the oil was degrading. More bacteria and less oil were signs that the shoreline was recovering. The number of bacteria needed to effectively break down an oil spill is 1 000 000 per millilitre of oil. The bacteria double in number every 2 days. The starting concentration of bacteria is 1000 bacteria per millilitre. This situation can be modelled by the equation C = 1000(2)d , where C is the estimated concentration of bacteria and d is the number of 2 day periods the bacteria grow. Approximately how long would it take for the bacteria to reach the required concentration to break down 1 mL of oil? NEL 3.2 Integral Exponents 113  21. The fraction of the surface area of a pond covered by algae cells doubles every week. Today, the pond surface is fully covered with algae. This situation can be modelled by the formula C = ​​​(_​  21 ​) ​​​ t​​, where C is the fraction of the surface area covered by algae t weeks ago. When was 25% of the pond covered? Extend 22. Calculate the value of x that makes each statement true. a) x −4 = ​​  _ 81 ​​ 16 x b) ​​​(__ ​  1 ​) ​​​  ​​ = 81 3 x c) ​​​(__ ​  3 ​) ​​​  ​= ___