10.2 Rational Exponents

10.2 Rational Exponents

Objective 1

Use exponential notation for nth roots.

Slide 10.2- 2

a1/n

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

Slide 10.2- 3

Use exponential notation for nth roots.

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

Evaluate each exponential.

1/532 5 32 2

1/ 264 2 64 64 8

1/ 481 4 81 3

1/ 4( 81) 4 81 Is not a real number because the radicand, – 81, is negative and the index, 4, is even.

1/3( 64) 3 64 4

1/31

27

31

27

1

3

Slide 10.2- 4

CLASSROOM EXAMPLE 1

Evaluating Exponentials of the Form a1/n

Solution:

Objective 2

Define and use expressions of the form am/n.

Slide 10.2- 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 10.2- 6

Define and use expressions of the form am/n.

Evaluate each exponential.

3/ 225 31/ 225

2/327

325

23 27 9

35 125

21/327 23

Slide 10.2- 7

CLASSROOM EXAMPLE 2

Evaluating Exponentials of the Form am/n

Solution:

3/ 216 31/ 216 316 34 64

2/364 23 64 16 21/3

64 24

3/ 236 is not a real number, since or is not a real number.

1/ 236 , 36,

a–m/n

If am/n is a real number, then 1

0 //

( ).m nm n

a aa

Slide 10.2- 8

Define and use expressions of the form am/n.

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

Evaluate each exponential.

3/ 4813/ 4

1

81

31/ 4

1

81

3/ 264

25

34

1

81

3

1

3 1

27

3/ 2363/ 2

1

36

31/ 2

1

36

31

36

3

1

6

1

216

3/ 225

64

3

25

64

35

8

125

512

Slide 10.2- 9

CLASSROOM EXAMPLE 3

Evaluating Exponentials with Negative Rational Exponents

Solution:

am/n

If all indicated roots are real numbers, then

11 // / .

m nm n n ma a a

Slide 10.2- 10

Define and use expressions of the form am/n.

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 10.2- 11

Define and use expressions of the form am/n.

Objective 3

Convert between radicals and rational exponents.

Slide 10.2- 12

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

1/ 219 12 119 9 3/ 411 34 112/314x 2314 x

3/53/55 2x x 3 35 55 2x x

Slide 10.2- 13

CLASSROOM EXAMPLE 4

Converting between Rational Exponents and Radicals

Solution:

5/ 7x

5/ 7 57

11

x x

1/32 2x y 2 23 x y

371/ 237

84 9 4 28/ 89 9 1

8 8z = |z|

Write each radical as an exponential.

Slide 10.2- 14

CLASSROOM EXAMPLE 4

Converting between Rational Exponents and Radicals (cont’d)

Solution:

Objective 4

Use the rules for exponents with rational exponents.

Slide 10.2- 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

a

r r

r

a b

b a

r r

r

a a

b b

1

r

raa

Slide 10.2- 16

Use the rules for exponents with rational exponents.

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

1/ 2 1/33 3 1/ 2 1/33 2/3

4/3

7

7

3/ 6 2/ 63 5/ 63

2/3 4/3 /3

22/

37 71

7

61/3 2 /3a b

b

61/3 2/3 1a b 61/3 1/3a b 6 61/3 1/3a b

1/3 1/36 6a b 6/3 6 /3a b 2 2a b2

2

a

b

Slide 10.2- 17

CLASSROOM EXAMPLE 5

Applying Rules for Rational Exponents

Solution:

1/ 23 4

2 1/5

a b

a b

1/ 23 ( 2) 4 1/5a b

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

1/ 25 21/5a b

1/ 2 1/ 25 21/5a b 5/ 2 21/10a b

21/10

5/ 2

b

a

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

2/5 3/5 2/5 8/5r r 5/5 10/5r r 2r r Slide 10.2- 18

CLASSROOM EXAMPLE 5

Applying Rules for Rational Exponents (cont’d)

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

Solution:

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 3/ 4 1/5x x 3/ 4 1/5x 15/ 20 4/ 20x 19/ 20x

5

3

x

x

5/ 25/ 2 1/3 15/ 6 2 / 6

1/13/ 6

3

xx x

xx

3 6 x 1/3 1/11/ 6 1/33 1/ 86 1/ 6x x x x

Slide 10.2- 19

CLASSROOM EXAMPLE 6

Applying Rules for Rational Exponents

Solution: