1. Chapter 8 Exponents and Exponential Functions
Section 8 – 1 Zero and Negative Exponents
Any nonzero number raised to the zero power equals a0 = 1
For every nonzero number a and integer n,
Notice that a negative exponent does NOT make anything negative, it moves the base and the exponent to the other part of the fraction

2. Ex1. Write as a simple fraction 5-3
Ex2. Write as a simple fraction with no negative exponents
Ex3. Write with no negative exponents
Ex4. Evaluate 5x2y-3z4 when x = 2, y = 6, and z = -3 (write answer as a simple fraction)

3. Section 8 – 2 Scientific Notation
Scientific notation is used to write large and small numbers in a way that is easier to read
A number in scientific notation is written as the product of two factors in the form a x 10n, where n is an integer and 1< a < 10
It is ok to use x for multiplication when writing numbers in scientific notation
The exponent indicates how many places the decimal point was moved and in which direction

4. Ex1. Are the following numbers in scientific notation? If no, why not?
A) 54.2 x 10-6
B) 4.32 x 108
C) x 1023
D) .045 x 1011
Ex2. Write each of the following numbers in scientific notation
45,600,000,000
459.2 x 1012

5. The way we typically write numbers is standard notation
Ex3. Write 5.82 x 108 in standard notation

6. Section 8-3 Multiplication Properties of Exponents
You can multiply numbers that have the same base by adding their exponents: am · an = am+n
Simplify. Write without negative exponents
Ex1. x3 · x5 · x · x-2
Ex2. 4a4 · 5a-3 · a6
Ex3. 2g-3 · 4h6 · g-2 · h-4

7. To multiply numbers in scientific notation: 1) multiply the coefficients (the a) ) multiply the powers of 10 by adding their exponents ) convert to scientific notation
Ex4. Simplify. Write the answer in scientific notation. (5 x 106)(7 x 108)

8. Section 8-4 More Multiplication Properties of Exponents
Raising a power to a power: For every nonzero number a and integers m and n, (am)n = amn
Simplify. Write without negative exponents.
Ex1. (x5)3 Ex2. a3(a5)-4
Raising a product to a power: For every nonzero number a and b and integer n (ab)n = anbn

9. Notice with a product to a power, everything in the parentheses is raised to the exponent, whether it is a number or variable
Simplify. Write without negative exponents.
Ex3. (5x3y2)4
Ex4. (3a2)-4(2a3b4)2
Ex5. m-6(3m5)2
Ex6. (4 x 105)3

10. Section 8 – 5 Division Properties of Exponents
When you divide powers with the same bases, subtract their exponents
Simplify. Write without negative exponents
Ex1. Ex Ex3.

11. Ex4. (6.8 x 109) ÷ (4 x 106)
Raising a quotient to a power: For every nonzero a and b and integer n,
Just like with products, everything in the parentheses is raised to the power
Simplify. Write without negative exponents
Ex Ex6.
Ex Ex8.

12. Section 8 – 6 Geometric Sequences
A geometric sequence is one in which you can find consecutive numbers by multiplying by a common ratio
Remember that with arithmetic sequences you were adding a common difference, now you will be multiplying by a common number (called the common ratio)
If you cannot easily identify the common ratio, divide the 2nd number by the 1st and that is your common ratio

13. To test that it is a geometric sequence and that the ratio is constant, also divide the 3rd number by the 2nd (you should get the same number)
Find the common ratio
Ex1. 6, -18, 54, -162, …
Ex2. 64, 48, 36, 27, …
Ex3. Find the next two terms 8, 56, 392, 2744, …
The formula for a geometric sequence is A(n) = a · rn–1

14. A(n) = nth term a = first term r = common ratio n = term #
Ex4. Find the 5th and 12th terms A(n) = 4 · (-2)n–1
Ex5. Determine whether the sequence is arithmetic or geometric
A) 40, 20, 10, 5, …
B) 40, 20, 0, -20, …
Read example 5 on page 426

15. Section 8 – 7 Exponential Functions
Exponential functions are in the form y = a·bx, where a is a nonzero constant, b is greater than 0 and b ≠ 1, and x is a real number
The graphs will be curves
Ex1. Evaluate y = 5 · 2x for x = -2, 2, 4
Open your book to page 431 and look at Objective 2 Graphing Exponential Functions
When graphing, it is easiest to use a graphing calculator to create these tables for you

16. Ex2. Suppose 2 mice live in a barn
Ex2. Suppose 2 mice live in a barn. If the number of mice quadruples every 3 months, how many mice will be in the barn after 2 years?
Ex3. Graph y = 2 · 3x

17. Section 8 – 8 Exponential Growth and Decay
Exponential growth is y = a·bx with a > 0 and b >1 (a is still the starting amount and b is called the growth factor)
The growth factor must be greater than 1 for it to be exponential growth (this means that the amount is increasing)
Compound interest is a type of exponential growth (a = initial deposit, b = 100% + interest rate, x = # of interest periods)

18. Exponential decay uses the same model as exponential growth, except the growth factor b is between 0 and 1 (so it is called the decay factor)
The amount is decreasing in exponential decay (100% – rate of decrease)
Ex1. a) Suppose you deposit $1000 in a college fund that pays 7.2% interest compounded annually. Find the account balance after 5 years b) find the account balance if interest is paid quarterly instead of yearly

19. Ex2. 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 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.