Warm-up Adding Like Terms Simplify each expression and write a general rule for adding like terms. Start with teams Pong bit.

Transcription

1 Chapter 8: Eponents and Eponential Functions Section 8.1: Appl Eponents Properties Involving Products Name: Warm-up Adding Like Terms Simplif each epression and write a general rule for adding like terms = = = = Rule: March Madness Video Games Consoles Start with teams Pong bit teams Atari bit teams NES bit teams PS1/Sega/SNES bit teams PS/Nintendo 6 bit teams team 1

2 Multipling Terms Product of Powers Work with our partner to determine how each epression is simplified. After completing all problems in this section, tr to determine a general rule Rule: Power of a Power Work with our partner to determine how each epression is simplified. After completing all problems in this section, tr to determine a general rule Rule:

3 What is the difference between the Product of Powers rule and the Power of a Power rule? Power of a Product Work with our partner to determine how each epression is simplified. After completing all problems in this section, tr to determine a general rule Rule: Simplif the epression write our answer using eponents. 1. =. w 9 w w =. (-6) (-6) =. 8 8 =. ( ) = 6. (v 7 ) =

4 7. [(-) ] 8. [(z ) ] 9. ( 9) 7 = 10. ( ) 11. (q ) 1. (8cd) 1. (-7) 8 (-7) 1. k k k Simplif the epression. 1. (-rs) 16. -(rs) 17. ( ) 18. (-s)(-r st) (-r st 7 )

5 Common Errors Warm up Find and fi the errors m 1m

6 6 Section 8.: Appl Eponent Properties Involving Quotients Power of a Quotient z w Rule: Quotient of Powers Rule:

7 Eample 1: Simplif the epression write our answer using eponents () 7 () r s 6. w 7. (8) 8 (8) 8. t Simplif the epression. 9. a 10. b s t t s 11. 6m n 1. mn a b a b a 7

8 Eample : Solve each problem a. The order of magnitude of the luminosit of the Milk Wa gala is 10 6 watts. The order of magnitude of the luminosit of a gamma ra burster is 10 watts. How man times as luminous is the gamma ra burster as the Milk Wa gala? b. Cell Phone Subscribers The table shows the approimate number of cell phone subscribers in selected countries in 001. Countr Number of subscribers Algeria Dominican Republic Poland Solomon Islands I. How man times greater is the number of cell phone subscribers in Poland than in the Solomon Islands? II. How man times greater is the number of cell phone subscribers in the Dominican Republic than in the Solomon Islands? 8

9 Section 8.: Define and Use Zero and Negative Eponents Fill in the following tables starting from the right and moving to the left. Write answers as whole numbers or fractions Use the tables about to determine the following two definitions: 0 a n a Notice: when we crossed the line, we changed the sign. What would ou predict the following definition would be? 1 n a Tr it out! 1) ) 0 1 ) ) 1 ) 6) 9 9

10 1 1 7) 8) 8 7 9) 10) 11) a b b = Turn it up a notch!! 1) a a ) 7 ) ) ab 8 ab Take it up a notch! Rewrite each of the following epressions with positive eponents. 1) ) z 1 ) 8 1 )

11 Section 8.: Scientific Notation Evaluate the following epressions with a necessar. if Eponent Form Standard Form Words Q.) What is scientific notation used for? A) IT s used to represent a REALLY number and a REALLY number! A number is written in scientific notation if it is in the form: Scientific Notation: This is NOT scientific notation: This is IS scientific notation: Write the following numbers in scientific notation. a. 9,000,000 (distance to the sun in miles) b. 8,000 (distance to the moon in miles) c (diameter of an atom in cm) d (wavelength of an -ra in meters) Write the following numbers in standard form: e (speed of light mi/sec) f. g (Total debt in the U.S. in 009) (Population of the U.S.) h.. 10 (diameter of a grain of sand) 11

12 Practice with our partner: Words Standard Form Scientific Notation two million two thousandths 00,000,000, Betelgeuse (a star in the constellation Orion) is 6 quadrillion miles awa. Write this number: Write this number in scientific notation: The following numbers are NOT in scientific notation. Rewrite them in scientific notation Order the numbers from least to greatest. 1. 9,000,000, , and 9,08, , 0.000, and

13 Evaluate the epression. Write our answer in scientific notation. 1. ( )( ). (.7 10 )(.6 10 ). (. 10 ). ( )

14 Comparing Linear and Eponential Functions 1. = (Linear Model). = (Eponential Model) Domain: Range: Domain: Range: = Domain: Range: Domain: Range: 1

15 Compare the graphs and tables on the previous page: How are the similar? How are the different? (Hint: How does each graph grow ) Similarities Differences ( b) The BASIC form of an eponential function is a b BIG IDEA: If b >, then it is eponential, If 0 < b < 1, then it is eponential. Think about it: What would an eponential function look like if b is one? Make a table to help ou answer. X 0 1 Challenge: What would an eponential function look like if b is negative? Make a table to help ou answer. X 0 1 1

16 Determine if each of the following represents GROWTH or DECAY 1 1) ) ) 6 ) 1 1 ) 6) 1 7) 8) (.99) Linear or eponential? Decide if the equation is linear or eponential and then write an equation that represents the rule for the table: Linear Functions Equation Form: Pattern: Eponential Functions Equation Form: Pattern:

17 8. and 8.6 Writing and Graphing Eponential Functions (Growth and Deca) Eample 1 Rabbit Population A population of 10 rabbits is released into a wild-life region. The population triples each ear for ears. Fill in the table below to figure out how man rabbits there would be after ears: Make a Table: time, t 0 1 population, P Graph: A) What is the rabbit population after ears? B) Did the rabbit population grow CONSTANTLY? C) Can ou come up with an equation that relates to the population of rabbits in terms of the time that has passed? D) Do ou think this equation will continue to hold true for an value of? Eample Bacteria Scientists start a bacterial colon with 1000 bacteria. The population is being cut in half ever hour. Make a Table: time, t 0 1 population, P Graph: A) How man bacteria are left after hours? B) Did the bacteria population decrease CONSTANTLY? C) Can ou come up with an equation that relates to the number of bacteria in terms of the time that has passed? D) Use our equation to determine the number of bacteria after 9 hours

18 Eample - Gossip The resident high school gossip girl starts a rumor and tells one person. That person tells two people, each of those people tell two people, and so on. Make a Table: Times people 0 1 talk, t People who know, P Graph: A) After people have spread the rumor how man people know? B) How man times would people have to talk in order for 18 people to know? C) Come up with an equation that represents how man people know the rumor? D) Use our equation to determine the number of people who know after 11 hours. Wh might we have to restrict, t, in this situation? Eample : Graph the eponential function Graph the function =. Identif its domain and range Domain: Range:

19 Eample : Graph the functions and compare = = = - = = - = Eample 6: Graph the eponential function Graph the function = 1. Identif its domain and range Domain: Range:

20 Eample 7: Graph the functions and compare 1 = = 1 = = - = 1 = Eample 8: Tell whether the graph represents eponential growth or eponential deca. Then write a rule for the function. a. b.

21 Application of Eponential Growth/Deca Models Eponential Growth/Deca Model: = C(1 ± r) t OR in other words Make sure our rate is a decimal Final amount = initial amount (1 ± rate) time + if Growing if Decreasing Eample 1 Compound Interest You deposit $00 in an account that pas 8% annual interest compounded earl. What is the account balance after 6 ears? A) In this situation would we add the rate (growth) or subtract the rate (deca)? B) Using the formula, write the equation that represents this equation: Eample Cell phone value: You purchase a cell phone for $1. The value of the cell phone decreases b about 0% annuall. A) In this situation would we add the rate (growth) or subtract the rate (deca)? B) Using the formula, write the equation that represents this equation: C) How much would be in the account after 6 ears? C) How much would the cell phone be worth after 6 ears? D) How much would the $00 be worth after ears? D) Would the value of the phone ever be $0? Eplain wh or wh not

22 Eample Computer growth: One computer industr epert reported that there were about 600 million computers in use worldwide in 001 and that the number was increasing at an annual rate of 10%. a. Write a function that models the number of computers in use over time b. Use the function to predict the number of computers that will be in use worldwide in 01 if the growth rate continues. c. What could cause this rate of increase to change? Do ou think it is greater or less than 10% toda? Eample Tennis Tournament Each ear the local countr club sponsors a tennis tournament. Pla starts with 18 participants. During each round, half of the plaers are eliminated. a. Write a function that models the number of participants as each round is plaed. b. How man plaers would there be after rounds? Eample Home purchase You have inherited land that was purchased for $0,000 in The value of the land increased approimatel % per ear. a. Write a function that models the value of the house over time b. How much would ou epect the land to be worth in 011? c. The actual value of the land in 011 was actuall $0,000. What could have caused the difference?