Exponents and Exponential Functions

Chapter 8

8-1 Zero and Negative ExponentsAll nonzero numbers raised to the zero power =

1e.g. 80 = 1, 324939874830 = 1, (-3422)0 = 1

A negative exponent does NOT make anything negative, it takes the entire power and moves it to the other half of a fractione.g. and

Simplify each expression (write without negative exponents)

Ex1. Ex2. Ex3.

33

1 14

4 64 2

2

15 25

5

4 5 23a b c 3 4

6 5

5

4

d e

f g

4 5 33 x y z

8-2 Scientific NotationScientific notation is a common way to write very

large and/or very small numbersTo write a number in scientific notation: write as

the product of two factors in the form a x 10n where n is an integer and 1 < a < 10

The n represents the number of spaces the decimal point needs to move to return to its original place

If the original number is < 1, then n will be negative

If the original number is > 1, then n will be positive

Ex1. Are the numbers in scientific notation. If not, why?A) 45.342 x 105 B) .83 x 10-4

Write each number in scientific notationEx2. 83,800,000,000Ex3. .000000456Numbers, as we are used to looking at them, are

said to be in standard formEx4. Write 2.35 x 10-7 in standard formEx5. Write the numbers in order from least to

greatest 3 5 2 6.24 10 ,2.4 10 ,.0024 10 ,24 10x x x x

8-3 Multiplication Properties of ExponentsIf you multiply nonzero powers with the SAME

BASE, you add the exponentse.g. and

Simplify each expression, write without negative exponents

Ex1. Ex2.To multiply two numbers in scientific notation

Multiply the coefficientsMultiply the powers of tenConvert to scientific notation

4 3 75 5 5 7 8 15v v v

5 4 7 93 2a b a b 4 2 6 5 7x y z x y z

Simplify each expression. Write each answer in scientific notation.

Ex3. Ex4. Ex5. Complete the equation

4 25 10 7 10x x 7 33 10 8 10x x

5 17a a a

8-4 More Multiplication Properties of ExponentsWhen you raise a power to a power, multiply the

exponents togethere.g. and

Follow the order of operations if there are multiple steps

Simplify each expression.Ex1. Ex2.

Ex3. Ex4.

42 83 3 53 15m m

64a 54n

73 4x x 3 84 2w w

If you raise a product to a power, raise each base to the power outside of the parenthesese.g. and

Simplify each expressionEx5. Ex6.

Ex7. Ex8.

5 5 5ab a b 3 3 3 32 2 8x x x

43y 434v

2 44 3 53a b a b 356 10x

8-5 Division Properties of ExponentsWhen you divide powers with the SAME base,

subtract the exponentse.g. and

Simplify each expression (no negative exponents)

Ex1. Ex2.

If you raise a quotient to a power, raise each base to the power outside of the parenthesese.g. and

118

3

55

5

54

9 4

1xx

x x

3 8

9 5

12

16

a b

a b

5 8

9 6

c d

c d

4 4

4

2 2 16

3 3 81

5 5

5

m m

n n

Simplify each expression (no negative exponents)Ex3. Ex4.

Ex5.

32

3

45

2

3x

y

25

8

4a

a

8-6 Geometric SequencesA sequence is geometric if you can multiply by

the SAME number each time to get the next number This number may be an integer, but it doesn’t have

to beThe number you multiply by each time is called

the common ratioTo find the common ratio, divide the 2nd number

by the 1st numberCheck this by dividing the 3rd number by the 2nd,

etc.A sequence is arithmetic if you can add the

SAME number each time to get the next number (see section 5-6)

Ex1. 81, 27, 9, 3, …A) find the common ratioB) find the next two terms

Formula for a geometric sequencen is the term positiona is the first term (some books use a1)r is the common ratio

Ex2. A(n) = 3(-2)n-1

A) find the sixth termB) find the twelfth term

Ex3. 200, 40, 8, …A) find the next three termsB) write a rule for the sequence

1( ) nA n a r

8-7 Exponential FunctionsAny function that is in the form y = a • bx where

a is a nonzero constant, b > 0, b ≠ 1, and x is a real number is an exponential function

Ex1. Evaluate f(x) = 2 ∙ 3x for the domain {-4, 0, 3}

If |b|>1, then the graph is an exponential growth curve

If |b|<1, then the graph is an exponential decay curve

Exponential growth Exponential decay

When graphing exponential curves, make a table of values and connect (at least 4 points)

Ex2. Suppose an investment of $2000 doubles in value every 15 years. How much is the investment worth after 45 years? Show your set up and answer.

Ex3. Suppose two mice live in a barn. If the number of mice quadruples every 3 months, how many mice will live in the barn after two years? Show your set up and answer.

8-8 Exponential Growth and DecayBoth exponential growth and decay are in the

form It is growth if |b|>1It is decay if |b|<1The base b is the growth factorThe starting amount is aWhen writing your equation, remember to define

your variables firstWhen dealing with interest:

Add 100% to the interest rate and then change to a decimal

That is your growth factor (b)

xy a b

Ex1. Suppose you deposited $800 in an account paying 3.4% interest compounded annually when you were born. Find the account balance after 18 years.

If the account is compounded more than once a year, it will change b and xDivide the interest rate by the number of

compoundings per yearMake sure the exponent reflects the number of

times it is compounded totalEx2. Suppose you deposit $800 in an account

paying 3.4% interest compounded monthly when you were born. Find the account balance after 18 years.

If the initial amount is decaying, subtract the percent of decay from 100%, change it to a decimal, and then use it as the growth factor

Ex3. Suppose the population of a certain endangered species has decreased 2.4% each year. Suppose there were 60 of these animals in a given area in 1999.A) Write an equation to model the number of

animals in this species that remain alive in that area

B) Use your equation to find the approximate number of animals remaining in 2005.