Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models

Подождите немного. Документ загружается.

70 CHAPTER R A Review of Basic Concepts and Skills R–70

EXAMPLE 9

䊳

Adding and Subtracting Radical Expressions

Simplify and combine (if possible).

a. b.

Solution

䊳

a. simplify radicals:

like radicals

result

simplify radicals

result

Now try Exercises 31 through 34

䊳

E. Multiplication and Division of Radical Expressions;

Radical Expressions in Simplest Form

Multiplying radical expressions is simply an extension of our earlier work. The multi-

plication can take various forms, from the distributive property to any of the special

products reviewed in Section R.2. For instance, even if

A or B is a radical term.

EXAMPLE 10

䊳

Multiplying Radical Expressions

Compute each product and simplify.

a. b.

c. d.

Solution

䊳

a. distribute

simplify: ,

result

b. F-O-I-L

extract roots

and

simplify

result

simplify each term

result

Now try Exercises 35 through 38

䊳

One application of products and powers of radical expressions is to evaluate cer-

tain quadratic expressions, as illustrated in Example 11.

⫽ 11 ⫺ 612

⫽ 9 ⫺ 612 ⫹ 2

1A ⫺ B2

⫽ A

⫺ 2AB ⫹ B

13 ⫺ 122

⫽ 132

⫺ 21321122⫹ 1122

⫽ x

⫺ 7

1A ⫹ B21A ⫺ B2⫽ A

⫺ B

1x ⫹ 1721x ⫺ 172⫽ x

⫺ 1172

⫽ 3015 ⫹ 20130

⫽ 1215 ⫹ 20130 ⫹ 1815

1212 ⫹ 613213110 ⫹ 1152⫽ 6120 ⫹ 2130 ⫹ 18130 ⫹ 6145

⫽ 1512 ⫺ 60

1132

⫽ 31 18 ⫽ 31 2 ⫽ 51321 2 ⫺ 1202132

513

116 ⫺ 4132⫽ 5118 ⫺ 201132

13 ⫺ 122

1x ⫹ 1721x ⫺ 172

1212

⫹ 613213110 ⫹ 1152513116 ⫺ 4132

1A ⫾ B2

⫽ A

⫾ 2AB ⫹ B

⫽⫺x2

⫽ 2x2

⫺ 3x2

16x

⫺ x2

54x

⫽ 2

⫺ x2

⫽ 715

⫽ 315

⫹ 415

145 ⫽ 19

5; 120 ⫽ 14

5145 ⫹ 2120 ⫽ 315 ⫹ 212152

16x

⫺ x2

54x

145 ⫹ 2120

College Algebra Graphs & Models—

D. Addition and Subtraction of Radical Expressions

Since 3x and 5x are like terms, we know If the sum becomes

illustrating how like radical expressions can be combined. Like

radicals are those that have the same index and radicand. In some cases, we can iden-

tify like radicals only after radical terms have been simpliﬁed.

7 ⫹ 51

7 ⫽ 81

x ⫽ 1

7,3x ⫹ 5x ⫽ 8x.

D. You’ve just seen how

we can add and subtract

radical expressions

LOOKING AHEAD

Notice that the answer for

Example 10c contains no radical

terms, since the outer and inner

products sum to zero. This result

will be used to simplify certain

radical expressions in this section

and later in Chapter 2.

cob19545_chR_065-080.qxd 7/28/10 2:14 PM Page 70

R–71 Section R.6 Radicals, Rational Exponents, and Radical Equations 71

College Algebra Graphs & Models—

EXAMPLE 11

䊳

Evaluating a Quadratic Expression

Show that when is evaluated at the result is zero.

Solution

䊳

original expression

substitute for

square binomial; distribute

commutative and associative properties

0 ✓

A calculator check is shown in the ﬁgure.

Now try Exercises 39 through 42

䊳

When we applied the quotient property in Example 8A, we obtained a denomina-

tor free of radicals. Sometimes the denominator is not automatically free of radicals,

and the need to write radical expressions in simplest form comes into play. This process

is called rationalizing the denominator.

Radical Expressions in Simplest Form

A radical expression is in simplest form if:

1. The radicand has no perfect nth root factors.

2. The radicand contains no fractions.

3. No radicals occur in a denominator.

As with other types of simpliﬁcation, the desired form can be achieved in various

ways. If the denominator is a single radical term, we multiply the numerator and de-

nominator by the factors required to eliminate the radical in the denominator [see Ex-

amples 12(a) and 12(b)]. If the radicand is a rational expression, it is generally easier

to build an equivalent fraction within the radical having perfect nth root factors in the

denominator [see Example 12(c)].

EXAMPLE 12

䊳

Simplifying Radical Expressions

Simplify by rationalizing the denominator. Assume

a. b.

Solution

䊳

a. multiply numerator and denominator by

simplify—denominator is now rational

b. multiply using two additional factors of

product property

Now try Exercises 43 and 44

䊳

⫽ x ⫽

⫺72

⫽

⫺72

⫺7

⫽

⫺711

x211

x11

x211

⫽

213

51132

⫽

213

513

⫽

513

⫺7

523

a, x ⫽ 0.

14 ⫹ 3 ⫺ 8 ⫹ 12⫹ 1413

⫺ 4132

4 ⫹ 413

⫹ 3 ⫺ 8 ⫺ 413 ⫹ 1

2 ⫹ 13 12 ⫹ 132

⫺ 412 ⫹ 132⫹ 1

⫺ 4x ⫹ 1

x ⫽ 2 ⫹ 13

⫺ 4x ⫹ 1

cob19545_chR_065-080.qxd 7/28/10 2:14 PM Page 71

In some applications, the denominator may be a sum or difference containing a

radical term. In this case, the methods from Example 12 are ineffective, and instead we

multiply by a conjugate since If either A or B is a square

root, the result will be a denominator free of radicals.

EXAMPLE 13

䊳

Simplifying Radical Expressions Using a Conjugate

Simplify the expression by rationalizing the denominator.

Write the answer in exact form and approximate form

rounded to three decimal places.

Solution

䊳

multiply by the conjugate of the denominator

FOIL

difference of squares

simplify

exact form

approximate form

Now try Exercises 45 through 48

䊳

F. Equations and Formulas Involving Radicals

A radical equation is any equation that contains terms with a variable in the radicand.

To solve a radical equation, we attempt to isolate a radical term on one side, then

apply the appropriate nth power to free up the radicand and solve for the unknown.

This is an application of the power property of equality.

The Power Property of Equality

If and v are real-valued expressions and ,

then

for n an integer,

Raising both sides of an equation to an even power can also introduce a false solu-

tion (extraneous root). Note that by inspection, the equation has only the

solution But the equation (obtained by squaring both sides) has

both and as solutions, yet does not satisfy the original equation. This

means we should check all solutions of an equation where an even power is applied.

x ⫽ 1x ⫽ 1x ⫽ 4

1x ⫺ 22

⫽ xx ⫽ 4.

x ⫺ 2 ⫽ 1x

n ⱖ 2.

u ⫽ v

⫽ v

u ⫽ v1

⬇ 3.605

⫽

316

⫹ 512

⫽

316

⫹ 212 ⫹ 312

6 ⫺ 2

⫽

216

⫹ 212 ⫹ 118 ⫹ 16

1162

⫺ 1122

2 ⫹ 13

16 ⫺ 12

⫽

2 ⫹ 13

16 ⫺ 12

16 ⫹ 12

2 ⫹ 13

16 ⫺ 12

1A ⫹ B21A ⫺ B2⫽ A

⫺ B

72 CHAPTER R A Review of Basic Concepts and Skills R–72

College Algebra Graphs & Models—

E. You’ve just seen how we

can multiply and divide radical

expressions and write a radical

expression in simplest form

cob19545_chR_065-080.qxd 11/23/10 7:13 PM Page 72

R–73 Section R.6 Radicals, Rational Exponents, and Radical Equations 73

College Algebra Graphs & Models—

EXAMPLE 14

䊳

Solving Radical Equations

Solve each radical equation:

a. b.

Solution

䊳

a. original equation

isolate radical term (subtract 12)

apply power property, power is even

simplify, square binomial

set equal to zero

factor

apply zero product property

result, check for extraneous roots

Check

䊳

✓

Check

䊳

The only solution is is extraneous. A calculator check

is shown in the ﬁgures.

original equation

isolate radical term (subtract 4, divide by 2)

apply power property, power is odd

simplify:

solve

Substituting for x in the original equation veriﬁes it is a solution.

Now try Exercises 49 through 52

䊳

Sometimes squaring both sides of an equation still results in an equation with a

radical term, but often there is one fewer than before. In this case, we simply repeat the

process, as indicated by the ﬂowchart in Figure R.7.

EXAMPLE 15

䊳

Solving Radical Equations

Solve the equation:

Solution

䊳

original equation

isolate one radical

power property

;

simplify

isolate radical

divide by four

power property

possible solution

1 ⫽ x

4 ⫽ x ⫹ 3

2 ⫽ 1x ⫹ 3

8 ⫽ 41x ⫹ 3

x ⫹ 15 ⫽ x ⫹ 41x ⫹ 3 ⫹ 7

A ⫽ 1x ⫹ 3, B ⫽ 21A ⫹ B2

x ⫹ 15 ⫽ 1x ⫹ 32⫹ 41x ⫹ 3 ⫹ 4

11x ⫹ 15

⫽ 11x ⫹ 3 ⫹ 22

1x ⫹ 15 ⫽ 1x ⫹ 3 ⫹ 2

1x ⫹ 15

⫺ 1x ⫹ 3 ⫽ 2

1x ⫹ 15

⫺ 1x ⫹ 3 ⫽ 2.

⫺3

x ⫽⫺3

x ⫺ 52

⫽ x ⫺ 5 x ⫺ 5 ⫽⫺8

x ⫺ 52

⫽ 1⫺22

x ⫺ 5 ⫽⫺2

2 1

x ⫺ 5 ⫹ 4 ⫽ 0

x ⫽ 6; x ⫽ 1

13 ⫽ 11x

⫹ 12 ⫽ 11

13112⫺ 2

⫹ 12 ⫽ 112⫹ 10x ⫽ 1:

16 ⫽ 16

116

⫹ 12 ⫽ 16

13162⫺ 2

⫹ 12 ⫽ 162⫹ 10x ⫽ 6:

x ⫽ 6

x ⫽ 1

x ⫺ 6 ⫽ 0

x ⫺ 1 ⫽ 0

0 ⫽ 1x ⫺ 621x ⫺ 12

0 ⫽ x

⫺ 7x ⫹ 6

3 x ⫺ 2 ⫽ x

⫺ 4x ⫹ 4

113x ⫺ 2

⫽ 1x ⫺ 22

13x ⫺ 2 ⫽ x ⫺ 2

13x ⫺ 2

⫹ 12 ⫽ x ⫹ 10

x ⫺ 5 ⫹ 4 ⫽ 013x ⫺ 2 ⫹ 12 ⫽ x ⫹ 10

cob19545_chR_065-080.qxd 11/22/10 11:15 AM Page 73

Check

䊳

original equation

substitute 1 for

simplify

solution checks

✓

Now try Exercises 53 and 54

䊳

Since rational exponents are so closely related to radicals, the solution process for

each is very similar. The goal is still to “undo” the radical (rational exponent) and solve

for the unknown.

Power Property of Equality

For real-valued expressions u and v, with positive integers m, n, and

in lowest terms:

If m is odd If m is even

and and

then then

The power property of equality basically says that if certain conditions are satis-

ﬁed, both sides of an equation can be raised to any needed power.

EXAMPLE 16

䊳

Solving Equations with Rational Exponents

Solve each equation:

a. b.

Solution

䊳

a. original equation;

isolate variable term (add 9, divide by 3)

apply power property, note

is odd

simplify

result

Check

䊳

substitute 15 for

in the original equation

simplify, rewrite exponent

✓ solution checks

b. original equation;

apply power property, note

is even

simplify

result

The solutions are and

Verify by checking both in the original equation.

Now try Exercises 55 through 58

䊳

3 ⫺ 8 ⫽⫺5.3 ⫹ 8 ⫽ 11

x ⫽ 3 ⫾ 8

⫽ 14

⫽ 84 x ⫺ 3 ⫽⫾8

31x ⫺ 32

⫽⫾4

⫽

1x ⫺ 32

⫽ 4

15 ⫽ 15

⫽ 8 3 182⫺ 9 ⫽ 15

16 ⫽ 2 3 122

⫺ 9 ⫽ 15

3 116

⫺ 9 ⫽ 15

3 115 ⫹ 12

⫺ 9 ⫽ 15

x ⫽ 15

⫽ 18

⫽ 164 x ⫹ 1 ⫽ 16

31x ⫹ 12

⫽ 8

1x ⫹ 12

⫽ 8

⫽

3 1x ⫹ 12

⫺ 9 ⫽ 15

1x ⫺ 32

⫽ 431x ⫹ 12

⫺ 9 ⫽ 15

u ⫽⫾v

u ⫽ v

⫽⫾v

⫽ v

⫽ v 1v 7 02,u

⫽ v,

2 ⫽ 2

4 ⫺ 2 ⫽ 2

116

⫺ 14 ⫽ 2

1112⫹ 15

⫺ 1112⫹ 3 ⫽ 2

1x ⫹ 15

⫺ 1x ⫹ 3 ⫽ 2

74 CHAPTER R A Review of Basic Concepts and Skills R–74

College Algebra Graphs & Models—

Radical Equations

Isolate

radical term

Apply

power property

Does the result

contain a radical?

Solve using

properties of equality

Check results in

original equation

YES

Figure R.7

cob19545_chR_065-080.qxd 7/28/10 2:15 PM Page 74

R–75 Section R.6 Radicals, Rational Exponents, and Radical Equations 75

College Algebra Graphs & Models—

CAUTION

䊳

As you continue solving equations with radicals and rational exponents, be careful not to

arbitrarily place the “ ” sign in front of terms given in radical form. The expression

indicates the positive square root of 18, where The equation be-

comes after applying the power property, with solutions

since the square of either number produces 18.

In Section R.4, we used a technique called u-substitution to factor expressions

in quadratic form. The following equations are also in quadratic form since the de-

gree of the leading term is twice the degree of the middle term:

and . The ﬁrst equation and its solution appear in

Example 17.

EXAMPLE 17

䊳

Solving Equations in Quadratic Form

Solve using a u-substitution: .

Solution

䊳

This equation is in quadratic form since it can be rewritten as:

where the degree of leading term is twice that of second

term. If we let then and the equation becomes ,

which is factorable.

factor

or solution in terms of

or resubstitute for

or cube both sides:

or solve for

Both solutions check.

Now try Exercises 59 and 60

䊳

A right triangle is one that has a angle (see Figure R.8). The longest side (opposite

the right angle) is called the hypotenuse, while the other two sides are simply called

“legs.” The Pythagorean theorem is a formula that says if you add the square of each

leg, the result will be equal to the square of the hypotenuse. Furthermore, we note the

converse of this theorem is also true.

Pythagorean Theorem

1. For any right triangle with legs a and b and hypotenuse c,

2. For any triangle with sides a, b, and c, if then the triangle

is a right triangle.

A geometric interpretation of the theorem is given in Figure R.9, which shows

⫹ 4

⫽ 5

⫹ b

⫽ c

⫹ b

⫽ c

90°

x ⫽⫺8 x ⫽ 125

132⫽ 1 1x

⫽ 1⫺22

⫽ 5

⫽⫺2 x

⫽ 5

u ⫽⫺2 u ⫽ 5

1u ⫺ 521u ⫹ 22⫽ 0

⫺ 3u

⫺ 10 ⫽ 0u

⫽ x

u ⫽ x

⫺ 31x

⫺ 10 ⫽ 0,

⫺ 3x

⫺ 10 ⫽ 0

⫹ x2

⫺ 81x

⫹ x2⫹ 12 ⫽ 0

⫺ 3x

⫺ 10 ⫽ 0

1x ⫽⫺312, x ⫽ 3122,

x ⫽⫾312

x ⫽⫾118

⫽ 18118 ⫽ 312.

118

⫾

Hypotenuse

Leg

90⬚

Figure R.8

Figure R.9

Area

16 in

Area

9 in

Area

25 in

⫹ 12

⫽ 13

25 ⫹ 144 ⫽ 169 

⫹ 24

⫽ 25

49 ⫹ 576 ⫽ 625

⫹ b

⫽ c

eneral case

cob19545_chR_065-080.qxd 11/22/10 11:16 AM Page 75

76 CHAPTER R A Review of Basic Concepts and Skills R–76

College Algebra Graphs & Models—

EXAMPLE 18

䊳

Applying the Pythagorean Theorem

An extension ladder is placed 9 ft from the base of a building in an effort to reach a

third-story window that is 27 ft high. What is the minimum length of the ladder

required? Answer in exact form using radicals, and in approximate form by

rounding to one decimal place.

Solution

䊳

We can assume the building makes a angle with the

ground, and use the Pythagorean theorem to ﬁnd the

required length. Let c represent this length.

Pythagorean theorem

substitute 9 for

and 27 for

add

deﬁnition of square root;

exact form:

approximate form

The ladder must be at least 28.5 ft tall.

Now try Exercises 63 and 64

䊳

c ⬇ 28.5 ft

1810 ⫽ 181

10 ⫽ 9110 c ⫽ 9110

c 7 0 c ⫽ 1810

⫽ 810

⫽ 81, 27

⫽ 729 c

⫽ 81 ⫹ 729

⫽ 192

⫹ 1272

⫽ a

⫹ b

90°

9 ft

27 ft

F. You’ve just seen how

we can solve equations and

use formulas involving radicals

2. The conjugate of is .2 ⫺ 13

䊳

CONCEPTS AND VOCABULARY

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

R.6 EXERCISES

1. if is a(n) integer.n 7 01

⫽

冟

3. By decomposing the rational exponent, we can

rewrite as 116

.16

4. is an example of the

property of exponents.

⫽ x

5. Discuss/Explain what it means when we say an

expression like has been written in simplest

form.

6. Discuss/Explain why it would be easier

to simplify the expression given using

rational exponents rather than radicals.

䊳

DEVELOPING YOUR SKILLS

Evaluate the expression for the values given.

7. a. b.

8. a. b.

Simplify each expression, assuming that variables can

represent any real number.

9. a. b.

c. d. 2x

⫺ 6x ⫹ 9281m

21x ⫺ 32

249p

x ⫽⫺8x ⫽ 7

x ⫽⫺10x ⫽ 9

10. a. b.

c. d.

11. a. b.

c. d.

12. a. b.

c. d.

⫺64

27q

⫺125p

⫺8

216z

⫺216x

24a

⫹ 12a ⫹ 92v

21y ⫹ 22

225n

cob19545_chR_065-080.qxd 11/23/10 7:18 PM Page 76

R–77 Section R.6 Radicals, Rational Exponents, and Radical Equations 77

College Algebra Graphs & Models—

24. a. b.

c. d.

e. f.

25. a. b.

c. d.

26. a. b.

c. d.

27. a. b.

c. d.

28. a. b.

c. d.

29. a. b.

c. d.

30. a. b.

c. d.

Simplify and add (if possible).

31. a.

32. a.

33. a.

34. a.

c. 275r

⫹ 132 ⫺ 127r ⫹ 138

110b ⫹ 1200b ⫺ 120 ⫹ 140

54m

⫺ 2m2

16m

272x

⫹ 150 ⫺ 17x ⫹ 127

14 ⫹ 13x ⫺ 112x ⫹ 145

3x1

54x ⫺ 52

16x

3240q ⫹ 9210q

3212x ⫺ 5275x

5212 ⫹ 2227

⫺3280 ⫹ 22125

2228p ⫺ 3263p

7218m ⫺ 250m

8248 ⫺ 32108

12272 ⫺ 9298

81a

2b1

32x

⫺9

125

27x

72b

227y

23y

16x

108n

28m

22m

5cd

25cd

3 B

25ab

⫺

25q220q

5.122p232p

3 B

12y

⫺

23b212b

2.5218a22a

⫺20 ⫹ 232

12 ⫺ 248

254m

27a

128a

28x

13. a. b.

c. d.

e. f.

14. a. b.

c. d.

e. f.

15. a. b.

c. d.

16. a. b.

c. d.

17. a. b.

c. d.

18. a. b.

c. d.

19. a. b.

c. d.

20. a. b.

c. d.

Use properties of exponents to simplify. Answer in

exponential form without negative exponents.

21. a. b.

22. a. b.

Simplify each expression. Assume all variables represent

nonnegative real numbers.

23. a. b.

c. d.

e. f.

27 ⫺ 272

⫺6 ⫹ 228

232p

64m

⫺22

⫺125p

218m

12x

⫺

24x

64y

12n

⫺

⫺a

⫺

1⫺1252

⫺

a⫺

⫺100

⫺a

27x

⫺

1⫺272

⫺

a⫺

⫺144

⫺125v

27w

⫺

⫺27p

⫺

25x

2⫺121

⫺2

16m

⫺216

49v

2⫺36

⫺2

81n

⫺125

1p ⫹ 42

1q ⫺ 92

⫺1024z

1024z

⫺811

1h ⫹ 22

1k ⫺ 32

⫺243x

243x

⫺641

cob19545_chR_065-080.qxd 7/28/10 2:15 PM Page 77

78 CHAPTER R A Review of Basic Concepts and Skills R–78

College Algebra Graphs & Models—

46. a. b.

47. a. b.

48. a. b.

Solve each equation and check your solutions by

substitution. Identify any extraneous roots.

49. a.

50. a.

51. a.

52. a.

53. a.

54. a.

Write the equation in simpliﬁed form, then solve. Check

all answers by substitution.

55. a.

56. a.

57. a.

58. a.

59.

60. x

⫺ 9x

⫽⫺8

⫺ 2x

⫺ 15 ⫽ 0

⫺ 4 ⫽ x

⫺ 10 ⫽ 1

⫺31x ⫺ 22

⫹ 29 ⫽⫺19

21x ⫹ 52

⫺ 11 ⫽ 7

0.5

⫹ 92 ⫽⫺43

0.3

⫺ 39 ⫽ 42

⫺2x

⫹ 47 ⫽⫺7

⫹ 17 ⫽ 9

13x ⫹ 4

⫺ 17x ⫽⫺2

13x

⫽ 1x ⫺ 3 ⫹ 3

12x ⫹ 31

⫹ x ⫽ 2

1x ⫹ 7

⫺ 1x ⫽ 1

112x ⫹ 9

⫺ 124x ⫽⫺3

1x ⫺ 2

⫺ 12x ⫽⫺2

x ⫽ 3 ⫹ 223 ⫺ x

1x ⫺ 9 ⫹ 1x ⫽ 9

x ⫹ 3 ⫽ 21

2x ⫹ 17

6x ⫺ 7

⫺ 5 ⫽⫺6

3 ⫺ 4x ⫺ 7 ⫽⫺4

⫺3 ⫽ 1

5p ⫹ 2

2x ⫺ 9 ⫽ 1

3x ⫹ 7

2m ⫹ 3

⫺5

⫹ 2 ⫽ 3

7 ⫺ 3x ⫺ 3 ⫽⫺7

2 ⫽ 1

3m ⫺ 1

⫺5 ⫽ 15x ⫺ 1 ⫺ x

⫺214x ⫺ 1

⫽⫺10

x ⫽ 13x ⫹ 1

⫹ 3

⫺313x ⫺ 5

⫽⫺9

1 ⫹ 16

5 ⫹ 213

1 ⫹ 12

16 ⫹ 114

7 ⫹ 16

3 ⫺ 312

110 ⫺ 3

13 ⫹ 12

1x ⫹ 13

17 ⫹ 3

Compute each product and simplify the result.

35. a. b.

c. d.

36. a. b.

c. d.

37. a.

38. a.

Use a substitution to verify the solutions to the

quadratic equation given. Verify results using a

calculator.

39.

a. b.

40.

a. b.

41.

a. b.

42.

a. b.

Rationalize each expression by building perfect nth root

factors for each denominator. Assume all variables

represent positive quantities.

43. a. b.

c. d.

44. a. b.

c. d.

Simplify the following expressions by rationalizing the

denominators. Where possible, state results in exact

form and approximate form, rounded to hundredths.

45. a. b.

1x ⫺ 12

3 ⫹ 111

⫺8

12x

125

12n

⫺4

120

4pB

50b

27x

112

x ⫽ 7 ⫹ 215x ⫽ 7 ⫺ 215

⫺ 14x ⫹ 29 ⫽ 0

x ⫽⫺1 ⫺ 110

x ⫽⫺1 ⫹ 110

⫹ 2x ⫺ 9 ⫽ 0

x ⫽ 5 ⫹ 17

x ⫽ 5 ⫺ 17

⫺ 10x ⫹ 18 ⫽ 0

x ⫽ 2 ⫺ 13

x ⫽ 2 ⫹ 13

⫺ 4x ⫹ 1 ⫽ 0

1315

⫹ 41221115 ⫹ 162

113

⫹ 1221110 ⫹ 1112

15 ⫹ 4110

211 ⫺ 21102

1212

⫹ 616213110 ⫹ 172

115

⫺ 1142112 ⫹ 1132

13 ⫹ 217

213 ⫺ 2172

12 ⫹ 15

14 ⫹ 13214 ⫺ 132

116 ⫺ 12210.3152

16 ⫺ 132

1n ⫹ 1521n ⫺ 152

115 ⫹ 17217122

cob19545_chR_065-080.qxd 11/22/10 11:17 AM Page 78

R–79 Section R.6 Radicals, Rational Exponents, and Radical Equations 79

College Algebra Graphs & Models—

䊳

WORKING WITH FORMULAS

61. Fish length to weight relationship:

The length to weight relationship of a female

Paciﬁc halibut can be approximated by the formula

shown, where W is the weight in pounds and L is

the length in feet. A ﬁsherman lands a halibut that

weighs 400 lb. Approximate the length of the ﬁsh

(round to two decimal places).

L  1.131W2

62. Timing a falling object:

The time it takes an object to fall a certain distance is

given by the formula shown, where t is the time in

seconds and s is the distance the object has fallen.

Approximate the time it takes an object to hit the

ground, if it is dropped from the top of a building

that is 80 ft in height (round to hundredths).

t 

䊳

APPLICATIONS

63. Length of a cable: A radio

tower is secured by cables that

are anchored in the ground 8 m

from its base. If the cables are

attached to the tower 24 m above

the ground, what is the length of

each cable? Answer in (a) exact

form using radicals, and (b)

approximate form by rounding

to one decimal place.

64. Height of a kite: Benjamin

Franklin is ﬂying his kite in a storm once again.

John Adams has walked to a position directly under

the kite and is 75 ft from Ben. If the kite is 50 ft

above John Adams’head, how much string S has

Ben let out? Answer in (a) exact form using

radicals, and (b) approximate form by rounding to

one decimal place.

The time T (in days) required for a planet to

make one revolution around the sun is modeled

by the function where R is the

maximum radius of the planet’s orbit (in

millions of miles). This is known as Kepler’s

third law of planetary motion. Use the equation

given to approximate the number of days

required for one complete orbit of each planet,

given its maximum orbital radius.

T  0.407R

75 ft

50 f

65. a. Earth: 93 million mi

b. Mars: 142 million mi

c. Mercury: 36 million mi

66. a. Venus: 67 million mi

b. Jupiter: 480 million mi

c. Saturn: 890 million mi

67. Accident investigation:After an accident, police

ofﬁcers will try to determine the approximate

velocity V that a car was traveling using the formula

where L is the length of the skid marks

in feet and V is the velocity in miles per hour. (a) If

the skid marks were 54 ft long, how fast was the car

traveling? (b) Approximate the speed of the car if

the skid marks were 90 ft long.

68. Wind-powered energy: If a wind-powered

generator is delivering P units of power, the

velocity V of the wind (in miles per hour) can be

determined using where k is a constant

that depends on the size and efﬁciency of the

generator. Rationalize the radical expression and

use the new version to ﬁnd the velocity of the wind

if and the generator is putting out 13.5

units of power.

k ⫽ 0.004

V ⫽

V ⫽ 226L

24 m

8 m

cob19545_chR_065-080.qxd 7/28/10 2:16 PM Page 79