Chapter 8 Section 7

Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Using Rational Numbers as Exponents Define and use expressions of the form a 1/n. Define and use expressions of the form a m/n. Apply the rules for exponents using rational exponents. Use rational exponents to simplify radicals

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 1 Define and use expressions of the form a 1/n. Slide 8.7-3

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Define and use expressions of the form a 1/n. Now consider how an expression such as 5 1/2 should be defined, so that all the rules for exponents developed earlier still apply. We define 5 1/2 so that 5 1/2 · 5 1/2 = 5 1/2 + 1/2 = 5 1 = 5. This agrees with the product rule for exponents from Section 5.1. By definition, Since both 5 1/2 · 5 1/2 and equal 5, this would seem to suggest that 5 1/2 should equal Similarly, then 5 1/3 should equal Review the basic rules for exponents: Slide 8.7-4

Copyright © 2012, 2008, 2004 Pearson Education, Inc. a 1/n If a is a nonnegative number and n is a positive integer, then Slide Define and use expressions of the form a 1/n. Notice that the denominator of the rational exponent is the index of the radical.

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify. 49 1/ /3 81 1/4 Solution: Slide EXAMPLE 1 Using the Definition of a 1/n

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 2 Define and use expressions of the form a m/n. Slide 8.7-7

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Define and use expressions of the form a m/n. Now we can define a more general exponential expression, such as 16 3/4. By the power rule, (a m ) n = a mn, so However, 16 3/4 can also be written as Either way, the answer is the same. Taking the root first involves smaller numbers and is often easier. This example suggests the following definition for a m/n. a m/n If a is a nonnegative number and m and n are integers with n > 0, then Slide 8.7-8

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Evaluate. 9 5/2 8 5/3 –27 2/3 Solution: Slide EXAMPLE 2 Using the Definition of a m/n

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Earlier, a –n was defined as for nonzero numbers a and integers n. This same result applies to negative rational exponents. Using the definition of a m/n. a −m/n If a is a positive number and m and n are integers, with n > 0, then A common mistake is to write 27 –4/3 as –27 3/4. This is incorrect. The negative exponent does not indicate a negative number. Also, the negative exponent indicates to use the reciprocal of the base, not the reciprocal of the exponent. Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Evaluate. 36 –3/2 81 –3/4 Slide EXAMPLE 3 Using the Definition of a −m/n

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 3 Apply the rules for exponents using rational exponents. Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Apply the rules for exponents using rational exponents. All the rules for exponents given earlier still hold when the exponents are fractions. Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Simplify. Write each answer in exponential form with only positive exponents. Slide EXAMPLE 4 Using the Rules for Exponents with Fractional Exponents

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify. Write each answer in exponential form with only positive exponents. Assume that all variables represent positive numbers. Solution: Slide EXAMPLE 5 Using Fractional Exponents with Variables

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 4 Use rational exponents to simplify radicals. Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use rational exponents to simplify radicals. Sometimes it is easier to simplify a radical by first writing it in exponential form. Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify each radical by first writing it in exponential form. Solution: Slide EXAMPLE 6 Simplifying Radicals by Using Rational Exponents