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C. Addition and Subtraction of Rational Expressions

C. Addition and Subtraction of Rational Expressions

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Section R.5 Rational Expressions and Equations



EXAMPLE 5







Adding and Subtracting Rational Expressions

Compute as indicated:

7

3

a.

ϩ

10x

25x2



Solution







57



b.



5

10x

Ϫ

xϪ3

x Ϫ9

2



a. The LCM for 10x and 25x2 is 50x2.

3

3 2

7

7 # 5x

#

ϩ

ϩ

ϭ

2

10x

10x 5x

25x

25x2 2

35x

6

ϭ

ϩ

2

50x

50x2

35x ϩ 6

ϭ

50x2



find the LCD

write equivalent expressions



simplify



add the numerators and write the result over the LCD



The result is in simplest form.

b. The LCM for x2 Ϫ 9 and x Ϫ 3 is 1x Ϫ 32 1x ϩ 32.

5

5

10x

10x

#xϩ3

Ϫ

Ϫ

ϭ

2

xϪ3

xϪ3 xϩ3

1x Ϫ 321x ϩ 32

x Ϫ9

10x Ϫ 51x ϩ 32

ϭ

1x Ϫ 321x ϩ 32

10x Ϫ 5x Ϫ 15

ϭ

1x Ϫ 321x ϩ 32

5x Ϫ 15

ϭ

1x Ϫ 32 1x ϩ 32



find the LCD

write equivalent expressions



subtract numerators, write

the result over the LCD

distribute



combine like terms



1



ϭ



51x Ϫ 32



1x Ϫ 321x ϩ 32



ϭ



5

xϩ3



factor and reduce



Now try Exercises 43 through 48



EXAMPLE 6











Adding and Subtracting Rational Expressions

Perform the operations indicated:

nϪ3

c

5

b2

a.

b.

Ϫ 2

Ϫ

2

a

nϩ2

n Ϫ4

4a



Solution







a. The LCM for n ϩ 2 and n2 Ϫ 4 is 1n ϩ 22 1n Ϫ 22.

5

nϪ3

nϪ3

5

#nϪ2Ϫ

ϭ

Ϫ 2

nϩ2

1n ϩ 22 n Ϫ 2

1n ϩ 22 1n Ϫ 22

n Ϫ4

51n Ϫ 22 Ϫ 1n Ϫ 32

ϭ

1n ϩ 221n Ϫ 22

5n Ϫ 10 Ϫ n ϩ 3

ϭ

1n ϩ 221n Ϫ 22

4n Ϫ 7

ϭ

1n ϩ 221n Ϫ 22



write equivalent

expressions

subtract numerators, write

the result over the LCD

distribute



result



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b. The LCM for a and 4a2 is 4a2:



b2

c

c 4a

b2

Ϫ

Ϫ #

ϭ

a

a 4a

4a2

4a2

b2

4ac

ϭ 2Ϫ 2

4a

4a

b2 Ϫ 4ac

ϭ

4a2



write equivalent

expressions

simplify

subtract numerators, write

the result over the LCD



Now try Exercises 49 through 64







CAUTION







When the second term in a subtraction has a binomial numerator as in Example 6a, be

sure the subtraction is applied to both terms. It is a common error to write

51n Ϫ 22

nϪ3

5n Ϫ 10 Ϫ n Ϫ 3 X

in which the subtraction is applied

Ϫ

ϭ

1n ϩ 221n Ϫ 22

1n ϩ 221n Ϫ 22

1n ϩ 221n Ϫ 22

to the first term only. This is incorrect!



C. You’ve just seen how

we can add and subtract

rational expressions



D. Simplifying Compound Fractions

Rational expressions whose numerator or denominator contain a fraction are called

3

2

Ϫ

3m

2

compound fractions. The expression

is a compound fraction with a

3

1

Ϫ

4m

3m2

2

3

3

1

numerator of

Ϫ and a denominator of

Ϫ

. The two methods commonly

3m

2

4m

3m2

used to simplify compound fractions are summarized in the following boxes.

Simplifying Compound Fractions (Method I)

1. Add/subtract fractions in the numerator, writing them as a

single expression.

2. Add/subtract fractions in the denominator, also writing them

as a single expression.

3. Multiply the numerator by the reciprocal of the denominator

and simplify if possible.

Simplifying Compound Fractions (Method II)

1. Find the LCD of all fractions in the numerator and denominator.

2. Multiply the numerator and denominator by this LCD and simplify.

3. Simplify further if possible.

Method II is illustrated in Example 7.



EXAMPLE 7







Simplifying a Compound Fraction

Simplify the compound fraction:

2

3

Ϫ

3m

2

3

1

Ϫ

4m

3m2



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Section R.5 Rational Expressions and Equations



Solution







The LCD for all fractions is 12m2.

2

3

3 12m2

2

Ϫ

Ϫ ba

b

a

3m

2

3m

2

1

ϭ

1

1

3

3

12m2

Ϫ

Ϫ

b

a

ba

4m

4m

1

3m2

3m2

2

12m2

3 12m2

a ba

b Ϫ a ba

b

3m

1

2

1

ϭ

12m2

1

3

12m2

b Ϫ a 2 ba

b

a ba

4m

1

1

3m

8m Ϫ 18m2

ϭ

9m Ϫ 4



multiply numerator and

denominator by 12m 2 ϭ



59



12m 2

1



distribute



simplify



Ϫ1



ϭ



2m14 Ϫ 9m2

9m Ϫ 4



ϭ Ϫ2m



D. You’ve just seen how

we can simplify compound

fractions



factor and write in lowest terms



Now try Exercises 65 through 74







E. Solving Rational Equations

In Section R.3 we solved linear equations using basic properties of equality. If any

equation contained fractional terms, we “cleared the fractions” using the least common

denominator (LCD). We can also use this idea to solve rational equations, or equations that contain rational expressions.

Solving Rational Equations

1. Identify and exclude any values that cause

a zero denominator.

2. Multiply both sides by the LCD and simplify

(this will eliminate all denominators).

3. Solve the resulting equation.

4. Check all solutions in the original equation.



EXAMPLE 8







Solving a Rational Equation

Solve for m:



Solution







4

1

2

.

ϭ 2

Ϫ

m

mϪ1

m Ϫm



Since m2 Ϫ m ϭ m1m Ϫ 12, the LCD is m1m Ϫ 12, where m 0 and m 1.

4

1

2

d multiply by LCD

b ϭ m1m Ϫ 12 c

m1m Ϫ 12a Ϫ

m

mϪ1

m1m Ϫ 12

m1m Ϫ 12 2

m1m Ϫ 12

m1m Ϫ 12

1

4

a bϪ

a



a

b distribute and simplify

m

1

1

mϪ1

1

m1m Ϫ 12

denominators are eliminated

21m Ϫ 12 Ϫ m ϭ 4

2m Ϫ 2 Ϫ m ϭ 4

distribute

mϭ6

solve for m



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Checking by substitution we have:

2

4

1

ϭ 2

original equation

Ϫ

m

mϪ1

m Ϫm

1

4

2

Ϫ

ϭ

substitute 6 for m

2

162

162 Ϫ 1

162 Ϫ 162

1

4

1

Ϫ ϭ

simplify

3

5

30

3

2

5

Ϫ

ϭ

common denominator

15

15

15

2

2



result

ϭ

15

15

A calculator check is shown in the figure.

Now try Exercises 75 through 80 ᮣ

Multiplying both sides of an equation by a variable sometimes introduces a solution that satisfies the resulting equation, but not the original equation—the one we’re

trying to solve. Such “solutions” are called extraneous roots and illustrate the need to

check all apparent solutions in the original equation. In the case of rational equations,

we are particularly aware that any value that causes a zero denominator is outside the

domain and cannot be a solution.

EXAMPLE 9







Solving a Rational Equation

Solve: x ϩ



Solution







4x

12

ϭ1ϩ

.

xϪ3

xϪ3



The LCD is x Ϫ 3, where x 3.

multiply both

4x

12

b ϭ 1x Ϫ 32a1 ϩ

b

1x Ϫ 32ax ϩ

sides by LCD

xϪ3

xϪ3

12

xϪ3

4x

xϪ3

distribute and

ba

ba

b ϭ 1x Ϫ 32 112 ϩ a

b simplify

1x Ϫ 32x ϩ a

1

xϪ3

1

xϪ3

x2 Ϫ 3x ϩ 12 ϭ x Ϫ 3 ϩ 4x denominators are eliminated

set equation equal to zero

x2 Ϫ 8x ϩ 15 ϭ 0

factor

1x Ϫ 321x Ϫ 52 ϭ 0

zero factor property

x ϭ 3 or x ϭ 5

Checking shows x ϭ 3 is an extraneous root, and x ϭ 5 is the only valid solution.



E. You’ve just seen how we

can solve rational equations



Now try Exercises 81 through 86 ᮣ



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Section R.5 Rational Expressions and Equations



61



R.5 EXERCISES





CONCEPTS AND VOCABULARY



Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary.



1. In simplest form, 1a Ϫ b2/1a Ϫ b2 is equal to

, while 1a Ϫ b2/1b Ϫ a2 is equal to

3. As with numeric fractions, algebraic fractions

require a

for addition and

subtraction.



.



2. A rational expression is in

when the

numerator and denominator have no common

factors, other than

.

4. Since x2 ϩ 9 is prime, the expression 1x2 ϩ 92/

1x ϩ 32 is already written in

.



State T or F and discuss/explain your response.



5.





x

xϩ1

1

Ϫ

ϭ

xϩ3

xϩ3

xϩ3



6.



1x ϩ 321x Ϫ 22



1x Ϫ 221x ϩ 32



ϭ0



DEVELOPING YOUR SKILLS



Reduce to lowest terms.



7. a.

8. a.



aϪ7

Ϫ3a ϩ 21



b.



xϪ4

Ϫ7x ϩ 28



b.



x2 Ϫ 5x Ϫ 14

9. a. 2

x ϩ 6x Ϫ 7



2x ϩ 6

4x2 Ϫ 8x

3x Ϫ 18

6x2 Ϫ 12x



a2 ϩ 3a Ϫ 28

b.

a2 Ϫ 49



16. a.

c.



x3 ϩ 4x2 Ϫ 5x

x3 Ϫ x

12y2 Ϫ 13y ϩ 3

27y3 Ϫ 1



b.

d.



5p2 Ϫ 14p Ϫ 3

5p2 ϩ 11p ϩ 2

ax2 Ϫ 5x2 Ϫ 3a ϩ 15

ax Ϫ 5x ϩ 5a Ϫ 25



Compute as indicated. Write final results in lowest terms.



17.



a2 Ϫ 4a ϩ 4 a2 Ϫ 2a Ϫ 3

#

a2 Ϫ 9

a2 Ϫ 4



10. a.



r2 ϩ 3r Ϫ 10

r2 ϩ r Ϫ 6



b.



m2 ϩ 3m Ϫ 4

m2 Ϫ 4m



18.



11. a.



xϪ7

7Ϫx



b.



5Ϫx

xϪ5



b2 ϩ 5b Ϫ 24

# 2 b

2

b Ϫ 6b ϩ 9 b Ϫ 64



19.



12. a.



v2 Ϫ 3v Ϫ 28

49 Ϫ v2



b.



u2 Ϫ 10u ϩ 25

25 Ϫ u2



x2 Ϫ 7x Ϫ 18 # 2x2 ϩ 7x ϩ 3

x2 Ϫ 6x Ϫ 27 2x2 ϩ 5x ϩ 2



20.



13. a.



Ϫ12a3b5

4a2bϪ4



b.



7x ϩ 21

63



6v2 ϩ 23v ϩ 21 4v2 Ϫ 25

#

3v ϩ 7

4v2 Ϫ 4v Ϫ 15



21.



m3n Ϫ m3

d. 4

m Ϫ m4n



22.



a3 Ϫ 4a2

a2 ϩ 3a Ϫ 28

Ϭ 3

2

a ϩ 5a Ϫ 14

a Ϫ8



y2 Ϫ 9

c.

3Ϫy

14. a.

c.

15. a.

c.



p3 Ϫ 64

p3 Ϫ p2



Ϭ



p2 ϩ 4p ϩ 16

p2 Ϫ 5p ϩ 4



5mϪ3n5

Ϫ10mn2



b.



Ϫ5v ϩ 20

25



23.



n2 Ϫ 4

2Ϫn



d.



w4 Ϫ w4v

w3v Ϫ w3



3Ϫx

3x Ϫ 9

Ϭ

4x ϩ 12

5x ϩ 15



24.



2n3 ϩ n2 Ϫ 3n

n3 Ϫ n2



b.



6x2 ϩ x Ϫ 15

4x2 Ϫ 9



2Ϫb

5b Ϫ 10

Ϭ

7b Ϫ 28

5b Ϫ 20



25.



x3 ϩ 8

x2 Ϫ 2x ϩ 4



d.



mn2 ϩ n2 Ϫ 4m Ϫ 4

mn ϩ n ϩ 2m ϩ 2



a2 ϩ a 3a Ϫ 9

#

a2 Ϫ 3a 2a ϩ 2



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CHAPTER R A Review of Basic Concepts and Skills

2

p2 Ϫ 36

# 2 4p

2p

2p ϩ 12p



Compute as indicated. Write answers in lowest terms

[recall that a ؊ b ‫ ؍‬؊1(b ؊ a)].



8

# 1a2 Ϫ 2a Ϫ 352

27. 2

a Ϫ 25



43.



3

5

ϩ

2

2x

8x



44.



15

7

Ϫ 2

16y

2y



m2 Ϫ 5m

m2 Ϫ m Ϫ 20



45.



7

1

Ϫ

4x2y3

8xy4



46.



5

3

ϩ

6a3b

9ab3



28. 1m2 Ϫ 162

29.



#



xy Ϫ 3x ϩ 2y Ϫ 6

x2 Ϫ 3x Ϫ 10



xy Ϫ 3x

xy Ϫ 5y



Ϭ



47.



4p

p Ϫ 36

2



Ϫ



2

pϪ6



48.



3q

q Ϫ 49

2



Ϫ



3

2q Ϫ 14



30.



ab Ϫ 2a

2a Ϫ ab ϩ 7b Ϫ 14

Ϭ

2

ab Ϫ 7a

b Ϫ 14b ϩ 49



49.



4

m

ϩ

4Ϫm

m Ϫ 16



50.



p

2

ϩ

2

pϪ2

4Ϫp



31.



m2 Ϫ 16

m2 ϩ 2m Ϫ 8

Ϭ

m2 Ϫ 2m

m2



51.



2

Ϫ5

mϪ7



52.



4

Ϫ9

xϪ1



32.



2x2 Ϫ 18

18 Ϫ 6x

Ϭ 3

2

x Ϫ 25

x Ϫ 2x2 Ϫ 25x ϩ 50



53.



33.



yϩ3



2



#y



3y2 ϩ 9y



ϩ 7y ϩ 12

y2 Ϫ 16



Ϭ



y2 ϩ 4y



2



yϩ1

y ϩ y Ϫ 30

2



Ϫ



2

yϩ6



54.



3

4n

Ϫ

4n Ϫ 20

n Ϫ 5n



y2 Ϫ 4y



2



34.



x2 Ϫ 1 x ϩ 1

x2 ϩ 4x Ϫ 5

#

Ϭ

x2 Ϫ 5x Ϫ 14

x2 Ϫ 4 x ϩ 5



55.



1

a

ϩ 2

aϩ4

a Ϫ a Ϫ 20



35.



x2 Ϫ x ϩ 0.21

x2 Ϫ 0.49

Ϭ

x2 ϩ 0.5x Ϫ 0.14

x2 Ϫ 0.09



56.



xϪ5

2x Ϫ 1

Ϫ 2

x ϩ 3x Ϫ 4

x ϩ 3x Ϫ 4



36.



x2 Ϫ 0.8x ϩ 0.15

x2 Ϫ 0.25

Ϭ

x2 ϩ 0.1x Ϫ 0.2

x2 Ϫ 0.16



57.



4

4

n2 ϩ n ϩ

3

9

37.

Ϭ

13

1

2

n2 Ϫ n ϩ

n2 Ϫ

15

15

25

n2 Ϫ



q2 Ϫ

38.

q2 Ϫ



40.

41.



9

25



1

3



10

10



3

17



20

20

1

q2 Ϫ

16



q2 ϩ

Ϭ



2



p3 ϩ p2 Ϫ 49p Ϫ 49

p2 ϩ 6p Ϫ 7



42. a



4x Ϫ 25

2x Ϫ x Ϫ 15 # 4x ϩ 25x Ϫ 21

Ϭ 2

b

x Ϫ 11x ϩ 30

x Ϫ 9x ϩ 18

12x2 Ϫ 5x Ϫ 3

2



2



y ϩ 2y ϩ 1



mϪ5

2

ϩ 2

m Ϫ9

m ϩ 6m ϩ 9



60.



mϩ6

mϩ2

Ϫ 2

2

m Ϫ 25

m Ϫ 10m ϩ 25



2



yϩ2

5y ϩ 11y ϩ 2

2



ϩ



5

y ϩyϪ6

2



m

mϪ4

ϩ

2

3m Ϫ 11m ϩ 6

2m Ϫ m Ϫ 15

2



Write each term as a rational expression. Then compute

the sum or difference indicated.



p3 Ϫ 1



4n2 Ϫ 1

6n2 ϩ 5n ϩ 1 # 12n2 Ϫ 17n ϩ 6

#

12n2 Ϫ 5n Ϫ 3

2n2 ϩ n

6n2 Ϫ 7n ϩ 2

2



2y Ϫ 5

2



59.



62.



p2 ϩ p ϩ 1



y ϩ 2y ϩ 1



Ϫ



Ϫ2

7

Ϫ 2

3a ϩ 12

a ϩ 4a



2



Ϭ



3y Ϫ 4

2



58.



61.



6a Ϫ 24

3a Ϫ 24a Ϫ 12a ϩ 96

Ϭ 3

2

a Ϫ 11a ϩ 24

3a Ϫ 81

3



39.



4

9



2



2



63. a. pϪ2 Ϫ 5pϪ1



64. a. 3aϪ1 ϩ 12a2 Ϫ1



b. xϪ2 ϩ 2xϪ3



b. 2yϪ1 Ϫ 13y2 Ϫ1



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Simplify each compound fraction. Use either method.



5

1

Ϫ

a

4

65.

25

1

Ϫ

2

16

a



1

8

Ϫ

3

27

x

66.

2

1

Ϫ

x

3



1

pϪ2

67.

1



pϪ2



3

yϪ6

68.

9



yϪ6



3

2

ϩ

3Ϫx

xϪ3

69.

4

5

ϩ

x

xϪ3



2

1

Ϫ

yϪ5

5Ϫy

70.

3

2

Ϫ

y

yϪ5



2

y Ϫ y Ϫ 20

71.

3

4

Ϫ

yϩ4

yϪ5



2

x Ϫ 3x Ϫ 10

72.

6

4

Ϫ

xϩ2

xϪ5







2







73. a.

74. a.



1 ϩ 3mϪ1

1 Ϫ 3mϪ1



b.



4 Ϫ 9aϪ2

3aϪ2



b.



Solve each equation. Identify any extraneous roots.



75.



2

5

1

ϭ 2

ϩ

x

xϩ1

x ϩx



76.



1

5

3

ϭ

Ϫ 2

m

mϩ3

m ϩ 3m



77.



3

21

ϭ

aϩ2

aϪ1



78.



4

7

ϭ

2y Ϫ 3

3y Ϫ 5



79.



1

1

1

Ϫ

ϭ 2

3y

4y

y



81. x ϩ



80.



1

1

3

Ϫ

ϭ 2

5x

2x

x



2x

14

ϭ1ϩ

xϪ7

xϪ7



82.



2x

10

ϩxϭ1ϩ

xϪ5

xϪ5



83.



5

20

6

ϭ

ϩ 2

nϩ3

nϪ2

n ϩnϪ6



84.



1 ϩ 2xϪ2

1 Ϫ 2xϪ2



2

1

7

ϭϪ

Ϫ 2

pϩ2

p

ϩ

3

p ϩ 5p ϩ 6



85.



3 ϩ 2nϪ1

5nϪ2



3

2a2 ϩ 5

a

ϭ

Ϫ 2

2a ϩ 1

aϪ3

2a Ϫ 5a Ϫ 3



86.



Ϫ18

3n

4n

ϩ

ϭ

2n Ϫ 1

3n ϩ 1

6n Ϫ n Ϫ 1



2



Rewrite each expression as a compound fraction. Then

simplify using either method.







63



Section R.5 Rational Expressions and Equations



2



WORKING WITH FORMULAS



87. Cost to seize illegal drugs: C ‫؍‬



450P

100 ؊ P



The cost C, in millions of

450P

P

dollars, for a government to find

100 ؊ P

and seize P% 10 Յ P 6 1002 of

40

a certain illegal drug is modeled

60

by the rational equation shown.

80

Complete the table (round to the

nearest dollar) and answer the

90

following questions.

93

a. What is the cost of seizing

95

40% of the drugs? Estimate

98

the cost at 85%.

100

b. Why does cost increase

dramatically the closer you get to 100%?

c. Will 100% of the drugs ever be seized?



88. Chemicals in the bloodstream: C ‫؍‬



200H2

H3 ؉ 40



Rational equations are often used

200H2

to model chemical concentrations H

H3 ؉ 40

in the bloodstream. The percent

0

concentration C of a certain drug

1

H hours after injection into

muscle tissue can be modeled by

2

the equation shown (H Ն 0).

3

Complete the table (round to the

4

nearest tenth of a percent) and

5

answer the following questions.

6

a. What is the percent

7

concentration of the drug

3 hr after injection?

b. Why is the concentration virtually equal at

H ϭ 4 and H ϭ 5?

c. Why does the concentration begin to decrease?

d. How long will it take for the concentration to

become less than 10%?



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APPLICATIONS



89. Stock prices: When a hot new stock hits the market,

its price will often rise dramatically and then taper

5017d 2 ϩ 102

off over time. The equation P ϭ

d3 ϩ 50

models the price of stock XYZ d days after it has

“hit the market.” (a) Create a table of values showing

the price of the stock for the first 10 days (rounded to

the nearest dollar) and comment on what you notice.

(b) Find the opening price of the stock. (c) Does the

stock ever return to its original price?

90. Population growth: The Department of Wildlife

introduces 60 elk into a new game reserve. It is

projected that the size of the herd will grow

1016 ϩ 3t2

, where

according to the equation N ϭ

1 ϩ 0.05t

N is the number of elk and t is the time in years.

(a) Approximate the population of elk after 14 yr.

(b) If recent counts find 225 elk, approximately

how many years have passed?

91. Typing speed: The number of words per minute

that a beginner can type is approximated by the

60t Ϫ 120

, where N is the number

equation N ϭ

t







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CHAPTER R A Review of Basic Concepts and Skills



of words per minute after t weeks, 3 6 t 6 12.

Use a table to determine how many weeks it takes

for a student to be typing an average of forty-five

words per minute.

92. Memory retention: A group of students is asked

to memorize 50 Russian words that are unfamiliar

to them. The number N of these words that the

average student remembers D days later is modeled

5D ϩ 35

1D Ն 12. How many

by the equation N ϭ

D

words are remembered after (a) 1 day? (b) 5 days?

(c) 12 days? (d) 35 days? (e) 100 days? According

to this model, is there a certain number of words

that the average student never forgets?

How many?

93. Pollution removal: For a steel mill, the cost C (in

millions of dollars) to remove toxins from the

22P

resulting sludge is given by C ϭ

, where

100 Ϫ P

P is the percent of the toxins removed. What percent

can be removed if the mill spends $88,000,000 on

the cleanup? Round to tenths of a percent.



EXTENDING THE CONCEPT



94. One of these expressions is not equal to the others.

Identify which and explain why.

20n

a.

b. 20 # n Ϭ 10 # n

10n

20 n

1

#

c. 20n #

d.

10n

10 n

95. The average of A and B is x. The average of C, D,

and E is y. The average of A, B, C, D, and E is:

3x ϩ 2y

2x ϩ 3y

a.

b.

5

5

21x ϩ y2

31x ϩ y2

c.

d.

5

5



3

2

and , what is the

5

4

reciprocal of the sum of their reciprocals? Given

c

a

that and are any two numbers—what is the

b

d

reciprocal of the sum of their reciprocals?



96. Given the rational numbers



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R.6



Radicals, Rational Exponents, and Radical Equations



LEARNING OBJECTIVES

In Section R.6 you will review how to:



A. Simplify radical



B.



C.



D.

E.



F.



expressions of the form

n

1a n

Rewrite and simplify

radical expressions using

rational exponents

Use properties of radicals

to simplify radical

expressions

Add and subtract radical

expressions

Multiply and divide

radical expressions; write

a radical expression in

simplest form

Solve equations and use

formulas involving

radicals



Square roots and cube roots come from a much larger family called radical expressions. Expressions containing radicals can be found in virtually every field of mathematical study, and are an invaluable tool for modeling many real-world phenomena.

n



A. Simplifying Radical Expressions of the Form 1an

In previous coursework, you likely noted that 1a ϭ b only if b2 ϭ a. This definition

cannot be applied to expressions like 1Ϫ16, since there is no number b such that

b2 ϭ Ϫ16. In other words, the expression 1a represents a real number only if a Ն 0

(for a full review of the real numbers and other sets of numbers, see Appendix I at

www.mhhe.com/coburn). Of particular interest to us now is an inverse operation for a2.

In other words, what operation can be applied to a2 to return a? Consider the following.



EXAMPLE 1







Evaluating a Radical Expression



Evaluate 2a2 for the values given:

a. a ϭ 3

b. a ϭ 5

c. a ϭ Ϫ6



Solution







a. 232 ϭ 19

ϭ3



b. 252 ϭ 125

ϭ5



c. 21Ϫ62 2 ϭ 136

ϭ6



Now try Exercises 7 and 8







The pattern seemed to indicate that 2a2 ϭ a and that our search for an inverse

operation was complete—until Example 1(c), where we found that 21Ϫ62 2 Ϫ6.

Using the absolute value concept, we can “repair” this apparent discrepancy and state a

general rule for simplifying these expressions: 2a2 ϭ ͿaͿ. For expressions like 249x2

and 2y6, the radicands can be rewritten as perfect squares and simplified in the same

manner: 249x2 ϭ 217x2 2 ϭ 7ͿxͿ and 2y6 ϭ 21y3 2 2 ϭ Ϳy3Ϳ.

The Square Root of a2: 2a2

For any real number a,

2a2 ‫ ͦ ؍‬aͦ.



EXAMPLE 2







Simplifying Square Root Expressions

Simplify each expression.

a. 2169x2



Solution







b. 2x2 Ϫ 10x ϩ 25



a. 2169x2 ϭ Ϳ13xͿ

ϭ 13ͿxͿ

b. 2x Ϫ 10x ϩ 25 ϭ 21x Ϫ 52

ϭ Ϳx Ϫ 5Ϳ

2



since x could be negative

2



since x ؊ 5 could be negative



Now try Exercises 9 and 10



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65



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CHAPTER R A Review of Basic Concepts and Skills







CAUTION



In Section R.2, we noted that 1A ϩ B2 2 A2 ϩ B2, indicating that you cannot square

the individual terms in a sum (the square of a binomial results in a perfect square

trinomial). In a similar way, 2A2 ϩ B2 A ϩ B, and you cannot take the square root

of individual terms. There is a big difference between the expressions 2A2 ϩ B2 and

21A ϩ B2 2 ϭ ͿA ϩ BͿ. Try evaluating each when A ϭ 3 and B ϭ 4.

3 3

3

3

To investigate expressions like 2

x , note the radicand in both 1

8 and 1

Ϫ64 can

be written as a perfect cube. From our earlier definition of cube roots we know

3

3

3

3

1

8ϭ 2

122 3 ϭ 2, 1

Ϫ64 ϭ 2

1Ϫ42 3 ϭ Ϫ4, and that every real number has only

one real cube root. For this reason, absolute value notation is not used or needed when

taking cube roots.



3 3

The Cube Root of a3: 2

a



For any real number a,

3 3

2

a ‫ ؍‬a.



EXAMPLE 3







Simplifying Cube Root Expressions

Simplify each expression.

3

3

a. 2

b. 2

Ϫ27x3

Ϫ64n6



Solution







3

3

a. 2

Ϫ27x3 ϭ 2

1Ϫ3x2 3

ϭ Ϫ3x



3

3

b. 2

Ϫ64n6 ϭ 2

1Ϫ4n2 2 3

ϭ Ϫ4n2



Now try Exercises 11 and 12







We can extend these ideas to fourth roots, fifth roots, and so on. For example, the

5

fifth root of a is b only if b5 ϭ a. In symbols, 1a ϭ b implies b5 ϭ a. Since an odd

number of negative factors is always negative: 1Ϫ22 5 ϭ Ϫ32, and an even number of

negative factors is always positive: 1Ϫ22 4 ϭ 16, we must take the index into account

n

when evaluating expressions like 1an. If n is even and the radicand is unknown,

absolute value notation must be used.

n



WORTHY OF NOTE

2

Just as 1

Ϫ16 is not a real number,

4

6

1 Ϫ16 and 1

Ϫ16 do not represent

real numbers. An even number of

repeated factors is always positive!



EXAMPLE 4







The nth Root of an: 2an

For any real number a,

n

1. 1an ϭ ͿaͿ when n is even.

Simplifying Radical Expressions

Simplify each expression.

4

4

a. 1

b. 1

81

Ϫ81

4

5

4

e. 216m

f. 232p5



Solution



A. You’ve just seen how

we can simplify radical

n

expressions of the form 1a n







n



2. 1an ϭ a when n is odd.



4

1

81 ϭ 3

5

132 ϭ 2

4

4

2

16m4 ϭ 2

12m2 4

ϭ Ϳ2mͿ or 2ͿmͿ

6

g. 2

1m ϩ 52 6 ϭ Ϳm ϩ 5Ϳ



a.

c.

e.



5

c. 1

32

6

g. 2 1m ϩ 52 6



5

d. 1

Ϫ32

7

h. 2 1x Ϫ 22 7



4

b. 1

Ϫ81 is not a real number

5

d.

1

Ϫ32 ϭ Ϫ2

5

5

f.

232p5 ϭ 2 12p2 5

ϭ 2p

7

h. 2

1x Ϫ 22 7 ϭ x Ϫ 2



Now try Exercises 13 and 14







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