1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 3 4 Rational Exponents Use exponential notation for nth roots. Define and use expressions.

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Objectives

2

3

4

Rational Exponents

Use exponential notation for nth roots.

Define and use expressions of the form am/n.

Convert between radicals and rational exponents.

Use the rules for exponents with rational exponents.

8.2



Objective 1

Slide 8.2- 2


a1/n

If is a real number, then 1 / .n na an a

Slide 8.2- 3


Notice that the denominator of the rational exponent is the index of the radical.


Evaluate each exponential.

1/5321/ 264

1/ 4811/ 4( 81)

1/3( 64)1/3

1

27

Slide 8.2- 4

CLASSROOM EXAMPLE 1

Evaluating Exponentials of the Form a1/n



Objective 2

Slide 8.2- 5


am/n

If m and n are positive integers with m/n in lowest terms, then

provided that a1/n is a real number. If a1/n is not a real number, then am/n is not a real number.

1/ / ,mm n na a

Slide 8.2- 6



Evaluate each exponential.

3/ 225 2/327

Slide 8.2- 7

CLASSROOM EXAMPLE 2

Evaluating Exponentials of the Form am/n

3/ 216 2/364

3/ 236


a–m/n

If am/n is a real number, then 1

0 //

( ).m nm n

a aa

Slide 8.2- 8


A negative exponent does not necessarily lead to a negative result. Negative exponents lead to reciprocals, which may be positive.


Evaluate each exponential.3/ 481

3/ 264

25

3/ 236

Slide 8.2- 9

CLASSROOM EXAMPLE 3

Evaluating Exponentials with Negative Rational Exponents


am/n

If all indicated roots are real numbers, then

11 // / .

m nm n n ma a a

Slide 8.2- 10



Radical Form of am/n

If all indicated roots are real numbers, then

That is, raise a to the mth power and then take the nth root, or take the nth root of a and then raise to the mth power.

/ .m

nm n m na a a

Slide 8.2- 11



Convert between radicals and rational exponents.

Objective 3

Slide 8.2- 12


Write each exponential as a radical. Assume that all variables, represent positive real numbers.

1/ 219 3/ 411

2/314x 3/53/55 2x x

Slide 8.2- 13

CLASSROOM EXAMPLE 4

Converting between Rational Exponents and Radicals

5/ 7x 1/32 2x y


37

84 9

8 8z

Write each radical as an exponential.

Slide 8.2- 14

CLASSROOM EXAMPLE 4

Converting between Rational Exponents and Radicals (cont’d)



Objective 4

Slide 8.2- 15


Rules for Rational Exponents

Let r and s be rational numbers. For all real numbers a and b for which the indicated expressions exist,

r s r sa a a

sr rsa a

1 rr

aa

r r rab a b

r

r ss

aa

b

r r

r

a b

b a

r r

r

a a

b b

1

r

raa

Slide 8.2- 16



Write with only positive exponents. Assume that all variables represent positive real numbers.

1/ 2 1/33 3

2/3

4/3

7

7

61/3 2/3a b

b

Slide 8.2- 17

CLASSROOM EXAMPLE 5

Applying Rules for Rational Exponents


1/ 23 4

2 1/5

a b

a b

2/5 3/5 8/5r r r

CLASSROOM EXAMPLE 5

Applying Rules for Rational Exponents (cont’d)

Write with only positive exponents. Assume that all variables represent positive real numbers.


Write all radicals as exponentials, and then apply the rules for rational exponents. Leave answers in exponential form. Assume that all variables represent positive real numbers.

34 5x x 5

3

x

x

3 6 x

Slide 8.2- 19

CLASSROOM EXAMPLE 6

Applying Rules for Rational Exponents

1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 3 4 Rational Exponents Use exponential notation for nth roots. Define and use expressions.

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