Section 10.2 Rational Exponents.

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Presentation transcript:

Section 10.2 Rational Exponents

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:

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

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

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

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

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.

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

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

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

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

Properties of Rational Expressions Properties Continued

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

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

Solution Simplifying Expressions Involving Rational Exponents Properties of Rational Expressions Solution

Simplifying Expressions Involving Rational Exponents Properties of Rational Expressions Solution Continued

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.

Solution Simplifying Expressions Involving Rational Exponents Properties of Rational Expressions Solution

Simplifying Expressions Involving Rational Exponents Properties of Rational Expressions Solution Continued