Rational Exponents

1. Section 4.2 Rational Exponents

2. 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 Definition of Rational Exponents Definition of Rational Exponents

3. 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 Definition of Rational Exponents Definition of Rational Exponents

4. 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 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 Definition of Rational Exponents Definition of Rational Exponents

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

6. 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 Simplifying Expressions Involving Rational Exponents Definition of Rational Exponents

7. 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 Definition: Rational Exponent Definition of Rational Exponents

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

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

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

11. Solution Continued If m and n are real rational numbers and b and c are any real number for which bm, bnand cn are real numbers Properties Section 4.2 Slide 11 Simplifying Expressions Involving Rational Exponents Definition of Rational Exponents

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