Exponents and Exponential Functions

1. Exponents and Exponential Functions Chapter 8

2. 8-1 Zero and Negative Exponents • All nonzero numbers raised to the zero power = 1 • e.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 fraction • e.g. and • Simplify each expression (write without negative exponents) • Ex1. Ex2. Ex3.

3. 8-2 Scientific Notation • Scientific notation is a common way to write very large and/or very small numbers • To 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

4. Write each number in scientific notation • Ex2. 83,800,000,000 • Ex3. .000000456 • Numbers, as we are used to looking at them, are said to be in standard form • Ex4. Write 2.35 x 10-7 in standard form • Ex5. Write the numbers in order from least to greatest

5. 8-3 Multiplication Properties of Exponents • If you multiply nonzero powers with the SAME BASE, you add the exponents • e.g. and • Simplify each expression, write without negative exponents • Ex1. Ex2. • To multiply two numbers in scientific notation • Multiply the coefficients • Multiply the powers of ten • Convert to scientific notation

6. Simplify each expression. Write each answer in scientific notation. • Ex3. Ex4. • Ex5. Complete the equation

7. 8-4 More Multiplication Properties of Exponents • When you raise a power to a power, multiply the exponents together • e.g. and • Follow the order of operations if there are multiple steps • Simplify each expression. • Ex1. Ex2. • Ex3. Ex4.

8. If you raise a product to a power, raise each base to the power outside of the parentheses • e.g. and • Simplify each expression • Ex5. Ex6. • Ex7. Ex8.

9. 8-5 Division Properties of Exponents • When you divide powers with the SAME base, subtract the exponents • e.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 parentheses • e.g. and

10. Simplify each expression (no negative exponents) • Ex3. Ex4. • Ex5.

11. 8-6 Geometric Sequences • A 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 be • The number you multiply by each time is called the common ratio • To find the common ratio, divide the 2nd number by the 1st number • Check 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)

12. Ex1. 81, 27, 9, 3, … • A) find the common ratio • B) find the next two terms • Formula for a geometric sequence • n is the term position • a is the first term (some books use a1) • r is the common ratio • Ex2. A(n) = 3(-2)n-1 • A) find the sixth term • B) find the twelfth term • Ex3. 200, 40, 8, … • A) find the next three terms • B) write a rule for the sequence

13. 8-7 Exponential Functions • Any 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 decay Exponential growth

14. 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.

15. 8-8 Exponential Growth and Decay • Both exponential growth and decay are in the form • It is growth if |b|>1 • It is decay if |b|<1 • The base b is the growth factor • The starting amount is a • When writing your equation, remember to define your variables first • When dealing with interest: • Add 100% to the interest rate and then change to a decimal • That is your growth factor (b)

16. 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 x • Divide the interest rate by the number of compoundings per year • Make sure the exponent reflects the number of times it is compounded total • Ex2. Suppose you deposit $800 in an account paying 3.4% interest compounded monthly when you were born. Find the account balance after 18 years.

17. 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.