UNIT 14 Exponents. Scientific Notation

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1 Unit 14 CCM6+/7+ Page 1 UNIT 14 Exponents and Scientific Notation CCM6+/7+ Name Math Teacher Projected Test Date Main Ideas Page(s) Unit 14 Vocabulary 2 Exponent Basics, Zero & Negative Exponents 3 6 Multiplying, Dividing, and Raising a Power to a Power 7 13 Laws of Exponents Review Intro to Scientific Notation Operations with Scientific Notation Real World Scientific Notation Problems Study Guide Page 1

2 Unit 14 CCM6+/7+ Page 2 CCM6+/7+ Plus Unit 14: Exponents and Scientific Notation Base Dividing Powers with the Same Base Property Exponent Exponential Form Irrational Numbers Laws of Exponents Multiplication Property of Exponents Perfect Cube Perfect Square Power Raising a Power to a Power Property Raising a Product to a Power Property Raising a Quotient to a Power Property When a number is raised to a power, the number that is used as a factor. For every nonzero number a and integers m and n, The number that indicates how many times the base is used as a factor. A number is written in exponential form when it has a base and an exponent. A number that cannot be expressed as a ratio of two integers (or as a repeating or terminating decimal) The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas: The exponent says how many times to use the number in a multiplication. For any nonzero number a and integers m and n, a m a n =a m+n The cube of a rational number The square of a rational number The power of a number says how many times to use the number in a multiplication. For every nonzero number a and integers m and n, (a m ) n =a mn For every nonzero number a and b integer n, (ab) n =a n b n For every nonzero numbers a and b and integer n, Rational Numbers Scientific Notation A number expressible in the form of a/b or -a/b for some fraction a/b. Rational numbers include integers A method of writing very large or very small numbers by using a number written between 1 and 10 multiplied by a power of 10. A number written as the product of two factors in the form, where n is an integer and Standard Form of a Standard form is a way of writing down very large or very small numbers easily Number Zero Exponent For every non-zero number a, a 0 = 1. Page 2

3 Unit 14 CCM6+/7+ Page 3 Page 3 Exponential Form and Properties of Exponents Vocabulary Labeled Example Base Exponent Write each of these expressions in exponential form. a. ( 6) ( 6) ( 6) ( 6) ( 6) b. x x x x x x x x x c d

4 Unit 14 CCM6+/7+ Page 4 Zero Property of Exponents and Negative Exponents The exponent pattern: 2 4 = = 2 3 = = 2 2 = 2 2 = 2 1 = 2 0 = = 2-1 = = 2-2 = = What happens to the product when you increase the exponent by one? What happens to the product when you decrease the exponent by one? Predict the answer for 2 0, 2-1, and 2-2. Be sure to follow your rule! When finished, discuss your ideas with a partner. Let s try a different base: 5 4 = = 5 3 = = 5 2 = 5 5 = 5 1 = 5 0 = 5-1 = 5-2 = 5-3 = Page 4

5 Unit 14 CCM6+/7+ Page 5 Zero Property of Exponents and Negative Exponents For every nonzero number x, x 0 = Rules For every nonzero number x, x -a = Examples. Simplify each expression completely (-7.8) 0 3. b b a m a m 3 1 m a Simplify each expression Zero and Negative Exponents 1) 4x ) 3-2 3) ) ( 1 5 )-3 5) -3a -2 b -4 6) 5x 0 p -3 Page 5

6 1) Simplify: Unit 14 CCM6+/7+ Page 6 a) 6 b) 4 x y c) 5x d) x e) -3 2) Evaluate when a 2, b 1, c 3 a) 4 ab 2 0 b) 5a c c) d) 2 b -1 b 3-2 a 3) Simplify a n a -n. What is the mathematical relationship of a n and a -n? Justify your answer. 4) Are 3x -2 and 3x 2 reciprocals? Explain. 5) Choose a fraction to use as a value for the variable a. Find the values of a -1, a 2, and a -2. Page 6

7 Unit 14 CCM6+/7+ Page 7 Ex. Ex. Ex. Multiplication Property of Exponents = = 6 5 e e = = e f f f = = f Rule To multiply numbers or variables that are raised to a power, the exponents of the numbers or variables with the. Examples. Simplify each expression completely m 5 m 7 m a 5 a b 2 a x 2 y 4 x 3 y 5. (3 2 )(3)(2 3 ) 6. c 4 d 7 c 17 What do you do when there are coefficients? Example: 6a 3 3a 2a y 2 3y 3 2y y 3 7x 2 2y m 2p 4 3m 8 Page 7

8 Unit 14 CCM6+/7+ Page 8 Multiplication with Exponents 5 5 (5 5) (5 5 5) 5 2 3? Rule: x a x b x? Example 1: Rewrite using one base Example 2: Simplify a) a a a 4 2 b) 4x 3x 3 5 c) a b a b d) 4x 3y 6x y Page 8

9 Unit 14 CCM6+/7+ Page 9 Let s Try! Simplify: 1) a a 2 4 2) 2x 3x ) (6 x y)(2 xy ) ) (5 x y)(6 x y ) Find the area of each figure below. Write your answer is simplest exponential form. 4 3x 3 9x 4 3x 2 6x Page 9

10 Unit 14 CCM6+/7+ Page 10 Division Property of Exponents Ex Ex. 4 x x x x x 2 x x x x Rule To divide numbers or variables with the same non-zero base, the exponents. Or, look for where the base is heavier and leave the remainder. Examples. Simplify each expression completely x x m 3m b xy 5 6 xy Page 10

11 Unit 14 CCM6+/7+ Page 11 Exponents: Powers of a Power You can use what you learned about multiplying numbers with powers to find a shortcut for simplifying expressions with powers. Complete each statement by showing equivalent expressions. Let your final answer be written as a base raised to a single power (exponential form). 1.) (3 6 ) 2 = = 2. (5 4 ) 3 = = 3.) (2 7 ) 4 = = 4. (4 5 ) 5 = = 5.) (1 4 ) 6 = 6.) (6 2 ) 4 = Look at your answers. What do you notice about the two exponents in the original expression as compared to the value of the exponent in the final expression? What operation would allow you to go straight from the original two exponents to the final one? Rule To simplify a power to a power, the exponents. Examples. Simplify each expression completely. 1. (2 2 ) 3 2. (c 5 ) 4 3. (3 a) (-5) 2 6. (-4) Page 11

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16 Unit 14 CCM6+/7+ Page 16 Scientific Notation Notes (with powerpoint) How wide is our universe? 210,000,000,000,000,000,000,000 miles ( zeros) This number is written in. When numbers get this large, it is easier to write them in. A number is expressed in scientific notation when it is in the form a x 10 n where is between 1 and 10 and is an integer Write the width of the universe in scientific notation: 210,000,000,000,000,000,000,000 miles Where is the decimal point now? Where would you put the decimal to make this number be between 1 and 10? How many decimal places did you move the decimal? When the original number is more than 1, the exponent is. The answer in scientific notation is: Express in scientific notation. Where would the decimal go to make the number be between 1 and 10? The decimal was moved how many places? When the original number is less than 1, the exponent is. 1) Write in scientific notation. (choose one) x x x x 10 5 Page 16

17 Unit 14 CCM6+/7+ Page 17 2) Express 1.8 x 10-4 in standard notation. 3) Express 4.58 x 10 6 in standard notation. Determine whether each of the following numbers is written in scientific notation? Explain. Write each number in scientific notation. 4) 62,400 5) ) 1,602,000 Write each number in standard notation. What does Scientific Notation look like on a calculator? Enter any 8-digit number into your calculator. Next, multiply by a 4-digit number. What do you see? Page 17

18 Unit 14 CCM6+/7+ Page 18 Scientific Notation Homework 1. Is each number in scientific notation? If not, put the number in scientific notation. 1.6 x x x x Write 75,000,000,000 in scientific notation. 3. Write in scientific notation. 4. Express 2.45 x 10 5 in standard form. 5. How much larger is 6 x 10 5 compared to 2 x Which is the larger value: 2 x 10 6 or 9 x 10 5? Page 18

19 Unit 14 CCM6+/7+ Page A sample of bacteria triples every month. The expression 300 x 3 m models a population of 300 bacteria after m months of growth. Evaluate the expression for m = 0, 3, -2 and describe what each value of the expression represents in the situation. 8. Recently, scientists have discovered 2 new moons. One moon s distance from the sun is 234,000,000 miles, while the other moon is 345,000,000 miles from the sun. a. Write each number in scientific notation. b. How many times closer to the sun is the first moon than the second moon? Write your answer in scientific notation. 9. A person s heart beats about 35 million beats in a year. If there are about 530 thousand minutes in a year, what is the average heart rate in beats per minute? 10. The populations for four states are given below. List the states in order of their populations from least to greatest. Alaska: 6.19 x 10 5 Connecticut: 3.28 x 10 6 Hawaii: 1.18 x 10 6 North Carolina: 7.65 x 10 6 Page 19

20 Unit 14 CCM6+/7+ Page 20 Operations with Scientific Notation Page 20

21 Unit 14 CCM6+/7+ Page 21 Multiplying with Scientific Notation Example 1: Example 2: *The answer then must be changed to scientific notation. Example 3: Example 4: Dividing with Scientific Notation Example 1: Example 2: *The answer then must be changed to scientific notation. Example 3: Example 4: How do the numbers compare to one another? A B C How does B compare to A? How does C compare to A? How does A compare to C? Page 21

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23 Unit 14 CCM6+/7+ Page 23 REAL WORLD APPLICATION 1) a. You are supposed to go to Idaho. It is 50 miles from here to Ogden. Then it is 90 miles to Pocatello Idaho from Ogden. How far must you go? b. You are supposed to go Venus. The earth is 9.3 x 10 7 miles from the sun. Venus is 8.5 x miles from the sun. How far is it to Venus? 2) a. You can travel 70 miles in one hour. How many hours will it take to get to Pocatello from Salt Lake City? b. You can travel 5.88 x miles in one light year. How many years will it take you to get to Venus? 3) a. The teeth of a comb are 3 millimeters wide. There are 45 teeth. How long is the comb? b. A centipede s leg is 7.23 x 10-2 cm. There are 50 legs on a side. How long is the centipede? 4) a. A bracelet weighs 8 oz. How many bracelets are in box which weighs a pound? b. A grasshopper weighs 5.88 X 10-2 ounces. How many grasshoppers are in a pound? (a pound has 16 ounces) 5) Some stars in the Milky Way are 8 x 10 4 light years away. Write this number in standard (expanded) form. Why might scientists prefer to use this number in scientific notation? 6) A light year is 5.88 x miles. Write this number in standard form. Page 23

24 Unit 14 CCM6+/7+ Page 24 7) How many miles is it to the stars in the Milky Way: You ll need the information in questions 1 and 2 to answer this question. Show your work, including what you make the calculator do. Write your answer in scientific notation. 8) If one eyelash measures 1.19 x 10-2 cm in diameter, and if your eyelashes lined up side by side in your eyelid which measures 3 cm, how many eyelashes could fit on one eyelid? Write your answer in standard form. Write your answer in scientific notation. If you lose 5 eyelashes per day, per eye, what percent of your total eyelashes are you losing per day? 9) A house spider weighs 4.22 x 10-3 ounces. How many house spiders are there in a pound? Note: there are 16 oz. in one pound. Show your work, including what you make the calculator do. Page 24

25 Unit 14 CCM6+/7+ Page 25 Unit 13 Study Guide EXPONENTS REVIEW 1. Any number to the power of zero always equals because. 2. If a number has a negative exponent, just. 3. If two numbers with the same base are multiplying, just the exponents. 4. If two numbers with the same base are dividing, just the exponents. 5. If an exponent is beside a set of parentheses, just it with the exponents inside the parentheses. 6. If a negative sign is in front of parentheses that have an exponent outside, where does it fall in the order of operations? 7. If a negative sign is inside parentheses that have an exponent outside, where does it fall in the order of operations? Multiple Choice Identify the choice that best completes the statement or answers the question. 8. a. 1 b. 0 c. 8.6 d a. b. c. d. 10. a. b. 16 c. d a. b. c. d. Page 25

26 Unit 14 CCM6+/7+ Page a. b. c. d. 13. a. b. c. d. 14. a. b. c. d. 15. a. b. c. d. 16. a. b. c. d. 17. a. b. c. d. SIMPLIFY. Pay attention to what you wrote above in the exponents review! a 2 b a 3 b m n 2 m n 5 2 x y 21. (r 5 s 4 ) x y ( 2x 3 y 1 exponent 4x 2 y 3 )3 Page 26

27 Unit 14 CCM6+/7+ Page 27 SCIENTIFIC and STANDARD NOTATION write the number in its equivalent other form ,030,000,000 = x 10-7 = x 10 5 = = Put in order from least to greatest x 10 5, 6.9 x 10 6, 23 x x 10-3, 2.5 x 10-4, 1.89 x 10-4 Solve. Express your result in scientific notation x x x x (1.5 x 10 5 )(4 x 10 9 ) 32. (5.1 x 10 3 )(1.63 x 10-5 ) 33. Which number is written in scientific notation? a. b. c. d. 34. Which number is NOT written in scientific notation? a. b. c. d a. b. c. d. Page 27

28 Unit 14 CCM6+/7+ Page a. 9,000 b. c. 90,000 d Order from least to greatest. a. c. b. d. 38. Which list shows the numbers in order from least to greatest? a. c. b. d. 39. a b c d a. b. c. d. 41. The diameter of Mercury is about miles. The diameter of Jupiter, the largest planet, is about miles. What is the difference between the diameters of these planets expressed in scientific notation? a. miles c. miles b. miles d. miles 42. The masses of four objects were measured during a physics experiment. The first and the last objects each had a mass of g. The second and the third objects each had a mass of g. Find the total mass of the four objects. Write your answer in scientific notation. a. g c. g b. g d. g Page 28