Copyright © 2011 Pearson Education, Inc. Rational Exponents and Radicals Section P.3 Prerequisites

P.3 Copyright © 2011 Pearson Education, Inc. Slide P-3 Definition: nth Roots If n is a positive even integer and a n = b, then a is called an nth root of b. If a 2 = b, then a is a square root of b. If a 3 = b, then a is a cube root of b. If n is even (or odd) and a is an nth root of b, then a is called an even (or odd) root of b. Every positive real number has two real even roots, a positive root and a negative root. Every real number has exactly one real odd root. Roots

P.3 Copyright © 2011 Pearson Education, Inc. Slide P-4 Definition: Exponent 1/n If n is a positive even integer and a is positive, then a 1/n denotes the positive real nth root of a and is called the principal nth root of a. If n is a positive odd integer and a is any real number, then a 1/n denotes the real nth root of a. If n is a positive integer, then 0 1/n = 0. Definition: Rational Exponents If m and n are positive integers, then a m/n = (a 1/n ) m, provided that a 1/ n is a real number. Note that a 1/ n is not real when a is negative and n is even. Roots and Rational Exponents

P.3 Copyright © 2011 Pearson Education, Inc. Slide P-5 Procedure: Evaluating a –m/n To evaluate a –m/n mentally, 1. find the nth root of a, 2. raise it to the m power, and 3. find the reciprocal. Rational Exponents

P.3 Copyright © 2011 Pearson Education, Inc. Slide P-6 The following rules are valid for all real numbers a and b and rational numbers r and s, provided that all indicated powers are real and no denominator is zero Rules for Rational Exponents

P.3 Copyright © 2011 Pearson Education, Inc. Slide P-7 Definition: Radical If n is a positive integer and a is a number for which a 1/n is defined, then the expression is called a radical, and If n = 2, we write rather than Rule: Converting a m/n to Radical Notation If a is a real number and m and n are integers for which is real, then Radical Notation

P.3 Copyright © 2011 Pearson Education, Inc. Slide P-8 For any positive integer n and real numbers a and b (b ¹ 0), 1. Product rule for radicals 2. Quotient rule for radicals provided that all of the roots are real. The Product and Quotient Rules for Radicals

P.3 Copyright © 2011 Pearson Education, Inc. Slide P-9 An expression that is the square of a term that contains no radicals is called a perfect square. An expression that is the cube of a term that contains no radicals is called a perfect cube. In general, an expression that is the nth power of an expression that contains no radicals is a perfect nth power. The product and quotient rules for radicals are used to simplify radicals containing perfect squares, perfect cubes, and so on. The Product and Quotient Rules for Radicals

P.3 Copyright © 2011 Pearson Education, Inc. Slide P-10 The product rule is used to remove the perfect nth powers that are factors of the radicand, and the quotient rule is used when fractions occur in the radicand. The process of removing radicals from the denominator is called rationalizing the denominator. Since radical expressions with the same index are added in the same manner as variable like terms, they are called like terms or like radicals. Rationalizing the Denominator

P.3 Copyright © 2011 Pearson Education, Inc. Slide P-11 Definition: Simplified Form for Radicals of Index n A radical of index n in simplified form has 1. no perfect nth powers as factors of the radicand, 2. no fractions inside the radical, and 3. no radicals in a denominator. Theorem: mth Root of an nth Root If m and n are positive integers for which all of the following roots are real, then Simplified Form and Rationalizing the Denominator