Slide 7- 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Exponents and Radicals 7.1Radical Expressions and Functions 7.2Rational Numbers as Exponents 7.3Multiplying Radical Expressions 7.4Dividing Radical Expressions 7.5Expressions Containing Several Radical Terms 7.6Solving Radical Equations 7.7Geometric Applications 7.8 The Complex Numbers 7

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rational Numbers as Exponents Rational Exponents Negative Rational Exponents Laws of Exponents Simplifying Radical Expressions 7.2

Slide 7- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley In Chapter 1, our discussion of exponents included all integers. We now expand the study to include all rational numbers.

Slide 7- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rational Exponents Consider a 1/2 a 1/2. If we still want to add exponents when multiplying, it must follow that a 1/2 a 1/2 = a 1/2 + 1/2, or a 1. This suggests that a 1/2 is a square root of a.

Slide 7- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When a is nonnegative, n can be any natural number greater than 1. When a is negative, n must be odd. Note that the denominator of the exponent becomes the index and the base becomes the radicand.

Slide 7- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write an equivalent expression using radical notation.

Slide 7- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution The denominator of the exponent becomes the index. The base becomes the radicand.

Slide 7- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution Write an equivalent expression using exponential notation. The index becomes the denominator of the exponent. The radicand becomes the base.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Positive Rational Exponents For any natural numbers m and n (n not 1) and any real number a for which exists,

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution Write an equivalent expression using radical notation and simplify.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write an equivalent expression using exponential notation. Example Solution

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Negative Rational Exponents For any rational number m/n and any nonzero real number a for which exists,

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Caution A negative exponent does not indicate that the expression in which it appears is negative:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write an equivalent expression with positive exponents and simplify, if possible.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution 8 -2/3 is the reciprocal of 8 2/3.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Laws of Exponents The same laws hold for rational exponents as for integer exponents.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Laws of Exponents For any real numbers a and b and any rational exponents m and n for which a m, a n, and b m are defined:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Use the laws of exponents to simplify

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplifying Radical Expressions Many radical expressions contain radicands or factors of radicands that are powers. When these powers and the index share a common factor, rational exponents can be used to simplify the expression.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To Simplify Radical Expressions 1. Convert radical expressions to exponential expressions. 2. Use arithmetic and the laws of exponents to simplify. 3. Convert back to radical notation as needed.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Use rational exponents to simplify. Do not use exponents that are fractions in the final answer.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Convert to exponential notation Simplify the exponent and return to radical notation