1. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 1 Chapter 4 Exponential Functions

2. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 2 4.1 Properties of Exponents

3. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 3 Exponent Definition For any counting number n, We refer to b n as the power, the nth power of b, or b raised to the nth power. We call b the base and n the exponent.

4. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 4 Properties of Exponents If m and n are counting numbers, then

5. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 5 Example: Meaning of Exponential Properties 1. Show that b 2 b 3 = b 5. 2. Show that b m b n = b m + n, where m and n are counting numbers 3. Show that where n is a counting number and c ≠ 0.

6. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 6 Solution 1. By writing b 2 b 3 without exponents, we see We can verify that this result is correct for various constant bases by examining graphing calculator tables for both y = x 2 x 3 and y = x 5.

7. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 7 Solution 2. Write b m b n without exponents:

8. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 8 Solution 3. Write where c ≠ 0, without exponents:

9. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 9 Simplifying Expressions Involving Exponents An expression involving exponents is simplified if 1. It includes no parentheses. 2. Each variable or constant appears as a base as few times as possible. For example, we write x 2 x 4 = x 6. 3. Each numerical expression (such as 7 2 ) has been calculated and each numerical fraction has been simplified. 4. Each exponent is positive.

10. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 10 Example: Simplifying Expressions Involving Exponents Simplify. 1. (2b 2 c 3 ) 5 2. (3b 3 c 4 )(2b 6 c 2 ) 3. 4.

11. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 11 Solution 1. 2.

12. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 12 Solution 3.4.

13. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 13 Simplifying Expressions Involving Exponents Warning The expressions 3b 2 and (3b) 2 are not equivalent expressions:

14. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 14 Zero Exponent Definition For b ≠ 0, b 0 = 1

15. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 15 Negative integer exponent Definition If b ≠ 0 and n is a counting number, then b -n In words, to find b -n, take its reciprocal and switch the sign of the exponent.

16. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 16 Negative Exponent in a Denominator If b ≠ 0 and n is a counting number, then In words, to find take its reciprocal and switch the sign of the exponent.

17. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 17 Example: Simplifying Expressions Involving Exponents Simplify. 1. 9b -7 2.3. 3 -1 + 4 -1

18. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 18 Solution 1. 2. 3.

19. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 19 Properties of Integer Exponents If m and n are integers, b ≠ 0, and c ≠ 0, then

20. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 20 Example: Simplifying Expressions Involving Exponents Simplify. 1.2.

21. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 21 Solution 1.

22. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 22 Solution 2.

23. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 23 Exponential function Definition An exponential function is a function whose equation can be put into the form f(x) = ab x Where a ≠ 0, b > 0, and b ≠ 1. The constant b is called the base.

24. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 24 Example: Evaluating Exponential Functions For f(x) = 3(2) x and g(x) = 5 x, find the following. 1. f(3) 2. f(–4) 3. g(a + 3)4. g(2a)

25. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 25 Solution 1. 2. 3. 4. f(x) = 3(2) x and g(x) = 5 x

26. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 26 Exponential Functions Warning It is a common error to confuse exponential functions such as E(x) = 2 x with linear functions such as L(x) = 2x. For the exponential function, the variable x is the exponent. For the linear function, the variable x is a base.

27. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 27 Scientific notation Definition A number is written in scientific notation if it has the form where k is an integers and either –10 < N ≤ –1 or 1 ≤ N < 10.

28. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 28 Converting from Scientific Notation to Standard Decimal Notation To write the scientific notation in standard decimal notation, we move the decimal point of the number N as follows: If k is positive, we multiply N by 10 k times; hence, we move the decimal point k places to the right. If k is negative, we divide N by 10 k times; hence, we move the decimal k places to the left.

29. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 29 Example: Converting to Standard Decimal Notation Write the number in standard decimal notation. 1.2.

30. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 30 Solution 1. We multiply 3.462 by 10 five times; hence, we move the decimal point of 3.462 five places to the right: 2. We divide 7.38 by 10 four times; hence, we move the decimal point of 7.38 four places to the left:

31. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 31 Converting from Standard Decimal Notation to Scientific Notation To write a number in scientific notation, count the number of places k that the decimal point must be moved so the new number N meets the condition –10 < N ≤ –1 or 1 ≤ N < 10: If the decimal point is moved to the left, then the scientific notation is written as If the decimal point is moved to the right, then the scientific notation is written as

32. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 32 Example: Converting to Scientific Notation Write the number in scientific notation. 1. 6,257,000,0002. 0.00000721

33. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 33 Solution 1. In scientific notation, we would have We must move the decimal point of 6.257 nine places to the right to get 6,257,000,000. So, k = 9 and the scientific notation is

34. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 34 Solution 2. In scientific notation, we would have We must move the decimal point of 7.21 six places to the left to get 0.00000721. So, k = –6 and the scientific notation is