Rational Exponents - OK

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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 1

9.2 Rational Exponents

Objectives

1. Use exponential notation for nth roots.

2. Define am/n.

3. Convert between radicals and rational exponents.4. Use the rules for exponents with rational exponents.





Exponents of the Form a1/ n

a1/ n

If is a real number, then a n

a1/ n

= . a

n




Evaluate each expression.

(a)

EXAMPLE 1 Evaluating Exponentials of the Form a1 /n


271/3 27 3

= 3

(b) 641/2 64 = 8

=

=

(c) –6251/4 625 4= –5= –

(d) ( –625)1/4 –6254

is not a real number because the radicand,

–625, is negative and the index is even.

=





Caution on Roots

CAUTION

Notice the difference between parts (c) and (d) in Example 1. The radical

in part (c) is the negative fourth root of a positive number, while the radical

in part (d) is the principal fourth root of a negative number, which is not a real number.

(c) –6251/4 625 4 = –5= –

(d) ( –625)1/4 –6254

is not a real number because the radicand,

–625, is negative and the index is even.

=

EXAMPLE 1




(e) ( –243)1/5 –2435

= –3=

Evaluate each expression.

EXAMPLE 1 Evaluating Exponentials of the Form a1 /n


(f) 1/2 4

25=

425

=25





Exponents of the Form a m/ n

a m/ n

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

a m/n

= ( a1 / n

) m

,

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

is not a real number.


http://slidepdf.com/reader/full/rational-exponents-ok 7/22Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sec 9.2 - 7

Evaluate each exponential.

(a) 253/2

EXAMPLE 2 Evaluating Exponentials of the Form a m/n


= ( 251/2 )3 = 53 = 125

(b) 322/5 = ( 321/5 )2 = 22 = 4

(c) –274/3 = –( 271/3 )4 = –(3)4 = –81

(d) ( –

64)2/3 = [( –

64)1/3 ]2 = ( –

4)2 = 16

(e) ( –16)3/2 is not a real number, since ( –16)1/2 is not a real number.

= –( 27)4/3




(a) 32 –4/5

EXAMPLE 3 Evaluating Exponentials with Negative

Rational Exponents


1

324/5

By the definition of a negative exponent,

32 –4/5 = .

Since 324/5 =4

= 24 = 16,5 32

= 32 –4/5 = .1

324/5=

116




(b)

EXAMPLE 3 Evaluating Exponentials with Negative

Rational Exponents


–4/3 827 4/3 8

27

1= =

827

34

1

23

= 4

11681

= 1 81

16=

We could also use the rule = here, as follows. –m b

a

m a

b

–4/3 827

4/3 278

=278

3= 4

= 4 3

28116

=




Caution on Roots

CAUTION

When using the rule in Example 3 (b), we take the reciprocal only of the

base, not the exponent. Also, be careful to distinguish between exponential

expressions like –

321/5, 32 –1/5, and –

32 –1/5.

–321/5 = –2, 12

32 –1/5 = , 12

and –32 –1/5 = – .




Alternative Definition of a m/ n

a m/ n

If all indicated roots are real numbers, then

a m/n

= ( a1 / n

) m

= ( a m

)1 / n

.




Radical Form of a m/ n

Radical Form of a m/ n

If all indicated roots are real numbers, then

In words, we can raise to the power and then take the root, or take the

root and then raise to the power.

a m/n = = ( ) . n a m n a m



Write each exponential as a radical. Assume that all variables represent

positive real numbers. Use the definition that takes the root first.

(a) 151/2

EXAMPLE 4 Converting between Rational Exponents

and Radicals


15= (b) 105/6 = ( )510 6

(c) 4n2/3 = 4( )2n3

(d) 7h3/4

–

(2h)2/5

= 7( )3

h4

(e) g –4/5 = 1

g4/5=

1

( )4g5

–( )

252h



In (f) – (h), write each radical as an exponential. Simplify. Assume that all

variables represent positive real numbers.

EXAMPLE 4 Converting between Rational Exponents

and Radicals


(f) 33 = 331/2

= 76/3(g) 763 = 72 = 49

= m, since m is positive. (h) m55





Rules for Rational Exponents

Rules for Rational Exponents

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

indicated expressions exist:

a r · a s = a r + s

( a r

) s

= a r s

( ab ) r

= a r

b

r

a – r =

1 a

r = a r – s a

r a

s = a b

– r b r

a r

= a

b

r a r

b r a

– r

=

1

a

r

.




Write with only positive exponents. Assume that all variables represent

positive real numbers.

(a) 63/4 · 61/2

EXAMPLE 5 Applying Rules for Rational Exponents


= 63/4 + 1/2 = 65/4 Product rule

Quotient rule = 32/3 – 5/6(b) 32/3

35/6= 3 –1/6 1

31/6=








Power rule

(c) m1/4 n –6

m –8 n2/3

–3/4=

( m –8) –3/4 (n2/3) –3/4(m1/4) –3/4 (n –6) –3/4

=

m6 n –1/2 m –3/16 n9/2

= m –3/16 – 6 n9/2 – ( –1/2) Quotient rule

= m –99/16 n5

Definition of negative

exponent =

m99/16 n5








(d) x3/5( x –1/2 – x3/4) = x3/5 · x –1/2

– x3/5 · x3/4 Distributive property

= x3/5 + ( –1/2) – x3/5 + 3/4 Product rule

= x1/10 – x27/20

Do not make the common mistake of multiplying exponents in the

first step.





Caution on Converting Expressions to Radical Form

CAUTION

Use the rules of exponents in problems like those in Example 5. Do not

convert the expressions to radical form.




Rewrite 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.



Convert to rational exponents. (a) · 4 a3 3 a2 = a3/4 · a2/3

= a3/4 + 2/3

= a

9/12 + 8/12

= a17/12

Product rule

Write exponents with a commondenominator









Convert to rational exponents.

Quotient rule

Write exponents with a common

denominator

(b)

4

c

c3= c

1/4

c3/2

= c1/4 – 3/2

= c1/4 – 6/4

= c –5/4

=

c5/4

1

Definition of negative exponent



Copyright © 2006 Pearson Education Inc Publishing as Pearson Addison-Wesley Sec 9.2 - 22






(c) 3 x25 x2/35=

( x2/3 )1/5=

x2/15=

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