1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 1.3a: Properties of Exponents

2. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Objectives o Natural number exponents. o Integer exponents. o Properties of exponents and their use.

3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Natural Number Exponents If a is any real number and if n is any natural number then In the expression a is called the base and n is the exponent. Note: Read “a to the power” baseexponent

4. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 1: Natural Number Exponents ExpressionBase(s)Exponent(s)Solution

5. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Integer Exponents If n and m are natural numbers, then Thus,. n factors m factors

6. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. 0 as an Exponent If n=0, then. This suggests the following: For any real number, we define

7. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Integer Exponents Consider the following: The exponent is decreased by one at each step. The result is 1/2 of the result from the previous line. What, then, do you know about the following expressions with negative exponents?

8. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Integer Exponents Negative Integer Exponents For any real number and for any natural number n, ( simply to avoid the possibility of division by 0.)

9. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 2: Integer Exponents Simply the following. a. b. c. Continued on the next slide…

10. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 2: Integer Exponents (cont.) d. e.

11. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Properties of Exponents and Their Use Properties of Exponents Throughout this table, a and b may be taken to represent constants, variables or more complicated algebraic expressions. The letters n and m represent integers. Property 1. 2. 3. Example

12. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Properties of Exponents and Their Use Property 4. 5. 6. Example

13. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 3: Properties of Exponents and Their Use Simplify the following by using the properties of exponents. Write the final answer with only positive exponents. a. Any non-zero expression with an exponent of 0 is 1. Continued on the next slide…

14. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 3: Properties of Exponents and Their Use (cont.) b. We could have used the same properties in a different order to achieve the same result. Ex:

15. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 4: Properties of Exponents and Their Use Simplify the following by using the properties of exponents. Write the final answer with only positive exponents.

16. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Properties of Exponents and Their Use The first column contains example of common errors and the second column contains the corrected statements. Incorrect StatementsCorrect Statements