Simplify each expression:

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1 Warm Up Simplify each epression: 1. rs 3 (r 3 4rs r s 3 ) 4. n 1 n. 7a 3 b c 5 45a 4 c 3 5. n + n 3. 3a 3 3a y y 9

2 Chapter 5 Eponents and Logarithms

3 5.1 Growth & Decay: Integral Eponents 5. Growth & Decay: Rational Eponents 1. Eponent Rules. Growth and Decay Eponential Functions 3. Solving Equations With Eponents

4 Laws of Eponents Same Bases Same Eponents b b y b y ( ab) a b b b y y b b 0 a a ( ) ( b 0) b b b b y If and only if =y b 0, 1,1 E: 5 = 5 16 means = 16

5 y b b y ) ( b 0 =1 b b 1 Laws of Eponents q p q p b b p q q p b b or 3 = 3 or = 4 3 = 64 = 8

6 f() = ab a = starting value b = multiplier = time A t = A 0 (1 + r) t Eponential Equations Eponential growth and decay- given a rate A 0 = the initial amount, r = the rate as a decimal, t = time r is positive for growth, negative for decay t is positive for the future, negative for the past

7 5.1 Growth & Decay: Integral Eponents The cost of hamburgers has been increasing by 9% per year Currently, a hamburger costs $4.00. Ct ( ) 4(1.09) t t 0 gives future costs t 0 gives past costs C(t) is an eponential function Find the cost of a hamburger 5 years from now. Find the cost of a hamburger 5 years ago. $6.15 $.60

8 5.1 Growth & Decay: Integral Eponents Set up a function for the cost of a $70 calculator that goes down in price by 9% each year. Ct ( ) 70(0.91) t

9 Eponential Growth Initial Amount A( t) A (1 r) t 0 Growth Rate Eponential Decay r is positive for growth r is negative for decay

10 5.1 Growth & Decay: Integral Eponents Suppose radioactivity decreases by 15% per day. If 40 kg are present now, find the amount present ( a) 6 days from now and ( b) 6 days ago. A( t) A (1 r) t 0 A0 40, r 0.15 A(6) 40(0.85) 15.1 kg 6 6 A( 6) 40(0.85) kg

11 5. Growth & Decay: Rational Eponents Suppose a car presently worth $800 depreciates 0% per year. About how much will it be worth in years and 3 months? Ct ( ) 800(0.80).5 C(.5) 800(0.80) $4963 t

12 A house bought 5 years ago for $100,000 was just sold for $135,000. To the nearest tenth of a percent what was the annual growth rate? (1 r) / r r 5 r 1/ r % 5

3) A population of 10000 frogs decreases at an annual rate of %.

13 3) A population of frogs decreases at an annual rate of %. How many frogs were there in 5 years ago? a) 1000(0.78) 5 b) 1000(1.) 5 c) 1000(1.) 5 d) 1000(0.78) 5

14 ) Given the equation y = (0.63), what is true? a) the starting point is smaller than the growth factor b) the equation is growing at 63% c) this is a linear equation d) the rate is -0.37

15 5) A type of bacteria has a very high eponential growth rate at 80% every hour. If there are 10 bacteria, determine how many there will be in 5 hours. a) 189 b) 180 c) 18.9 d) 18

16 6) A species of etremely rare, deep water fish rarely have children. If there are a 81 of this type of fish and their growth rate is % each month, how many will there be in half of a year? a) 81 b) 5544 c) d) 94

17 7) Given this table, what s the equation? y a) y = 60(1.) b) y = 50() c) y = 50(1.) d) y = 60()

18 8) A culture of bacteria contained 3,84,700 cells on one day and is growing at a daily rate of 6.8%. How many cells would be present days and 9 hours later? a) 4,650,430 b) 13,174,860 c) 4,49,55 d) 15,370,800

19 9) If there are 0 foes in the forest this year, and 1 after one year, what is the growth rate of the foes? a) 1% b).5% c).95% d) 5%

20 1) If the starting population of 5 rabbits grows at 00% each year, how many will there be in 0 years? a) 5() 0 b) (5) 0 c) 5(3) 0 d) 00(5) 0

21 ) If the starting population of 5 rabbits grows at 00% each year, how many will there be in 50 years? a) b) c) 1600 d) too many to fit on my calculator

22 Solve Solve / /

23 3/ Solve 4 3 3/ /3 3/ / / 4 1/ 4 Solve / ( 1)

24 Class Eercises Pg 177 #1,,5-9,13,17

25 Simplify each epression 1/ 1/ 3/ 3/ 1. a. 4 b. 4.a. 4 b

26 Simplify each epression 1/ 3/ /6 3/ 3/ / / / 1/ 64

27 Simplify each epression 1/

28 Simplify each epression 13. The cost in $$$$ of a new pair of running shoes t years from now is C t t a. What is the cost now? $6.5 b. To find the cost in.5 years, use t. c. To find the cost 9 months ago, use t. 3/ 4

29 Solve. 1/ /

30 Homework Page 173 #9,13,17,1,5,9,33,34,35 Page 178 #1,5,7,9,13,15,17,9,31,35,37

31 Simplify Simplify Simplify 3 b a a b Simplify (3y) 5

32 5.1 Growth & Decay: Integral Eponents An epression is simplified when it contains neither negative eponents nor powers of powers. Simplify Simplify

33 5.1 Growth & Decay: Integral Eponents An epression is simplified when it contains neither negative eponents nor powers of powers. Simplify 3 b a a b b a a b a b b a ab

34 5.1 Growth & Decay: Integral Eponents An epression is simplified when it contains neither negative eponents nor powers of powers. Simplify 1 a b a b Common Mistake b a ab a b a b b a Positive Eponents Common Denominator

35 5.1 Growth & Decay: Integral Eponents An epression is simplified when it contains neither negative eponents nor powers of powers. 1 Simplify a b ab

f ( x) a, where a 0 and a 1. (Variable is in the exponent. Base is a positive number other than 1.)

f ( x) a, where a 0 and a 1. (Variable is in the exponent. Base is a positive number other than 1.) MA 590 Notes, Lesson 9 Tetbook (calculus part) Section.4 Eponential Functions In an eponential function, the variable is in the eponent and the base is a positive constant (other than the number ). Eponential