CHAPTER R: Basic Concepts of Algebra R.1 The Real-Number System R.2 Integer Exponents, Scientific Notation, and Order of Operations R.3 Addition, Subtraction, and Multiplication of Polynomials R.4 Factoring R.5 The Basics of Equation Solving R.6 Rational Expressions R.7 Radical Notation and Rational Exponents Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

R.7 Radical Notation and Rational Exponents Simplify radical expressions. Rationalize denominators or numerators in rational expressions. Convert between exponential and radical notation. Simplify expressions with rational exponents. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Notation A number c is said to be a square root of a if c2 = a. Similarly, c is a cube root of a if c3 = a. nth Root A number c is said to be an nth root of a if cn = a. The symbol denotes the nth root of a. The symbol is called a radical. The number n is called the index. Any positive number has two square roots, one positive and one negative. For any even index, a positive number has two real-number roots. The positive root is called the principal root. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Simplify each of the following: a) = 7, because 72 = 49. b) = 7, because 72 = 49 and c) because d) because (4)3 = 64. e) is not a real number. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Properties of Radicals Let a and b be any real numbers or expressions for which the given roots exist. For any natural numbers m and n (n  1): 1. If n is even, 2. If n is odd, 3. 4. 5. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples a) b) c) d) e) f) g) h) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Another Example Perform the operation. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

The Pythagorean Theorem The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse: a2 + b2 = c2. b c a Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Juanita paddled her canoe across a river 525 feet wide. A strong current carried her canoe 810 feet downstream as she paddled. Find the distance Juanita actually paddled, to the nearest foot. Solution: 525 ft 810 ft c Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Rationalizing Denominators or Numerators Rationalizing the denominator (or numerator) is done by multiplying by 1 in such a way as to obtain a perfect nth power. Example Rationalize the denominator. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Rationalizing Denominators or Numerators Conjugates Pairs of expressions of the form are called conjugates. The product of such a pair contains no radicals and can be used to rationalize a denominator or numerator. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Rationalizing Denominators or Numerators Example Rationalize the numerator. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Rational Exponents For any real number a and any natural numbers m and n, for which exists, Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Examples Convert to radical notation and, if possible, simplify. a) b) c) Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley More Examples Convert each to exponential notation. a) b) Simplify. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley