1. Section 10.2
Rational Exponents

2. Definition of Rational Exponents
Definition of Rational Expressions
Introduction
How should we define , n is a counting number?
Exponent property
(–3)2 = 9 and 32 = 9 suggest (32) ½ = 9 ½ = 3
The nonnegative number 3 is the principal second root, or principle square root, of 9, written
If m = , n = 3:

3. Definition of Rational Exponents
Definition of Rational Expressions
Introduction
23 = 8
Suggests that a good meaning of is 2
The number 2 is called the third root, or cube root, of 8 written

4. Definition of Rational Exponents
Definition of Rational Expressions
Definition
For the counting number n, where n ≠ 1,
If n is odd, then is the number whose nth power is b, and we call the nth root of b.
If n is even and , then is the nonnegative number whose nth power is b, and we call the principle n root of b.
If n is even and b < 0, then is not a real number
may be represented by

5. Example Solution Simplifying Expressions Involving Rational Exponents
Definition of Rational Expressions
Example
Simplify.
Solution

6. Simplifying Expressions Involving Rational Exponents
Definition of Rational Expressions
Solution Continued
is not a real number, since the fourth power of any real number is nonnegative.
Graphing calculator checks
problems 1, 2 and 3

7. Definition: Rational Exponent
Definition of Rational Expressions
Definition
For the counting numbers m and n, where n ≠ 1 and b is any real number for which is a real number,
A power of the form or is said to have a rational exponent.

8. Simplifying Expressions Involving Rational Expressions
Definition of Rational Expressions
Example
Simplify.
Solution

9. Graphing calculator checks problems 1, 2 and 3
Simplifying Expressions Involving Rational Expressions
Definition of Rational Expressions
Solution Continued
Graphing calculator checks
problems 1, 2 and 3

10. For find the following:
Simplifying Expressions Involving Rational Expressions
Definition of Rational Expressions
Example
For
find the following:
Solution

11. Solution Continued Properties
Simplifying Expressions Involving Rational Expressions
Definition of Rational Expressions
Solution Continued
If m and n are real rational numbers and b and c are any real number for which bm, bn and cn are real numbers
Properties

12. Properties of Rational Expressions
Properties Continued

13. Simplify. Assume that b is positive.
Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Example
Simplify. Assume that b is positive.
Solution

14. Simplify. Assume that b is positive
Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Solution Continued
Example
Simplify. Assume that b is positive

15. Solution Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Solution

16. Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Solution Continued

17. Simplify. Assume that b and c are constants.
Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Solution Continued
Example
Simplify. Assume that b and c are constants.

18. Solution Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Solution

19. Simplifying Expressions Involving Rational Exponents
Properties of Rational Expressions
Solution Continued