1. Section 4.2
Rational Exponents

2. Definition of Rational Exponents
Introduction
How should we define , n is a counting number?
Exponent property is true if m = ½ n = 2
(–3)2 = 9 and 32 = 9
Suggests that 9½ = 3
The nonnegative number 3 is the principal second root, or principle square root, of 9, written
If m = , n = 3:
Section 4.2
Slide 2

3. Definition of Rational Exponents
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
Section 4.2
Slide 3

4. Definition of Rational Exponents
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 square root of b.
If n is even and b < 0, then is not a real number
may be represented by
Section 4.2
Slide 4

5. Example Solution Simplifying Expressions Involving Rational Exponents
Definition of Rational Exponents
Example
Simplify.
Solution
Section 4.2
Slide 5

6. Simplifying Expressions Involving Rational Exponents
Definition of Rational Exponents
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
Section 4.2
Slide 6

7. Definition: Rational Exponent
Definition of Rational Exponents
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.
Section 4.2
Slide 7

8. Example Solution Simplifying Expressions Involving Rational Exponents
Definition of Rational Exponents
Example
Simplify.
Solution
Section 4.2
Slide 8

9. Graphing calculator checks problems 1, 2 and 3
Simplifying Expressions Involving Rational Exponents
Definition of Rational Exponents
Solution Continued
Graphing calculator checks
problems 1, 2 and 3
Section 4.2
Slide 9

10. For find the following:
Simplifying Expressions Involving Rational Exponents
Definition of Rational Exponents
Example
For
find the following:
Solution
Section 4.2
Slide 10

11. Solution Continued Properties
Simplifying Expressions Involving Rational Exponents
Definition of Rational Exponents
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
Section 4.2
Slide 11

12. Properties of Rational Exponents
Properties Continued
Section 4.2
Slide 12