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F. Equations and Formulas Involving Radicals

F. Equations and Formulas Involving Radicals

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EXAMPLE 14

a. 13x Ϫ 2 ϩ 12 ϭ x ϩ 10

Solution

73

3

b. 2 1

xϪ5ϩ4ϭ0

a. 13x Ϫ 2 ϩ 12 ϭ x ϩ 10

13x Ϫ 2 ϭ x Ϫ 2

1 13x Ϫ 22 2 ϭ 1x Ϫ 22 2

3x Ϫ 2 ϭ x2 Ϫ 4x ϩ 4

0 ϭ x2 Ϫ 7x ϩ 6

0 ϭ 1x Ϫ 62 1x Ϫ 12

x Ϫ 6 ϭ 0 or x Ϫ 1 ϭ 0

x ϭ 6 or x ϭ 1

Check

x ϭ 6:

Check

x ϭ 1:

original equation

apply power property, power is even

simplify, square binomial

set equal to zero

factor

apply zero product property

result, check for extraneous roots

13162 Ϫ 2 ϩ 12 ϭ 162 ϩ 10

116 ϩ 12 ϭ 16

16 ϭ 16 ✓

13112 Ϫ 2 ϩ 12 ϭ 112 ϩ 10

11 ϩ 12 ϭ 11

13 ϭ 11x

The only solution is x ϭ 6; x ϭ 1 is extraneous. A calculator check

is shown in the figures.

3

b. 2 1

xϪ5ϩ4ϭ0

3

1

x Ϫ 5 ϭ Ϫ2

3

1 1x Ϫ 52 3 ϭ 1Ϫ22 3

x Ϫ 5 ϭ Ϫ8

x ϭ Ϫ3

original equation

isolate radical term (subtract 4, divide by 2)

apply power property, power is odd

3

simplify: 1

x Ϫ 52 3 ϭ x Ϫ 5

solve

Substituting Ϫ3 for x in the original equation verifies it is a solution.

Now try Exercises 49 through 52

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

Solve the equation: 1x ϩ 15 Ϫ 1x ϩ 3 ϭ 2.

Solution

1x ϩ 15 Ϫ 1x ϩ 3 ϭ 2

1x ϩ 15 ϭ 1x ϩ 3 ϩ 2

1 1x ϩ 152 2 ϭ 1 1x ϩ 3 ϩ 22 2

x ϩ 15 ϭ 1x ϩ 32 ϩ 4 1x ϩ 3 ϩ 4

x ϩ 15 ϭ x ϩ 4 1x ϩ 3 ϩ 7

8 ϭ 4 1x ϩ 3

2 ϭ 1x ϩ 3

4ϭxϩ3

1ϭx

original equation

power property

1A ϩ B2 2; A ϭ 1x ϩ 3, B ϭ 2

simplify

divide by four

power property

possible solution

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Figure R.7

Check

1x ϩ 15 Ϫ 1x ϩ 3 ϭ 2

1112 ϩ 15 Ϫ 1112 ϩ 3 ϭ 2

116 Ϫ 14 ϭ 2

4Ϫ2ϭ2

2 ϭ 2✓

Isolate

original equation

substitute 1 for x

simplify

solution checks

Now try Exercises 53 and 54

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.

Apply

power property

Does the result

Power Property of Equality

YES

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

in lowest terms:

If m is odd

m

and u n ϭ v,

NO

If m is even

m

and u n ϭ v 1v 7 02,

then 1u n 2 m ϭ vm

m n

Solve using

properties of equality

then 1u n 2 m ϭ Ϯv m

n

m n

n

n

n

u ϭ vm

u ϭ Ϯvm

The power property of equality basically says that if certain conditions are satisfied, both sides of an equation can be raised to any needed power.

Check results in

original equation

EXAMPLE 16

Solving Equations with Rational Exponents

Solve each equation:

3

a. 31x ϩ 12 4 Ϫ 9 ϭ 15

Solution

3

a. 31x ϩ 12 4 Ϫ 9 ϭ 15

1x ϩ 12 ϭ 8

3

4

3 1x ϩ 12 4 ϭ 8

x ϩ 1 ϭ 16

x ϭ 15

3 4

4 3

Check

4

3

3

4

3115 ϩ 12 Ϫ 9 ϭ 15

3116 2 Ϫ 9 ϭ 15

3122 3 Ϫ 9 ϭ 15

3182 Ϫ 9 ϭ 15

15 ϭ 15 ✓

1

4

b.

3

1x Ϫ 32 ϭ 4

2

3

3 1x Ϫ 32 4 ϭ Ϯ 4

xϪ3ϭ Ϯ8

xϭ3Ϯ8

2 3

3 2

3

2

b. 1x Ϫ 32 3 ϭ 4

2

original equation; mn ϭ 34

isolate variable term (add 9, divide by 3)

apply power property, note m is odd

simplify 383 ϭ 183 2 4 ϭ 16 4

4

1

result

substitute 15 for x in the original equation

simplify, rewrite exponent

4

1

16 ϭ 2

23 ϭ 8

solution checks

original equation; mn ϭ 23

apply power property, note m is even

simplify 342 ϭ 142 2 3 ϭ 8 4

3

1

result

The solutions are 3 ϩ 8 ϭ 11 and 3 Ϫ 8 ϭ Ϫ5.

Verify by checking both in the original equation.

Now try Exercises 55 through 58

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CAUTION

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

indicates the positive square root of 18, where 118 ϭ 312. The equation x2 ϭ 18 becomes x ϭ Ϯ 118 after applying the power property, with solutions x ϭ Ϯ312

1x ϭ Ϫ312, x ϭ 3122, since the square of either number produces 18.

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

since

the de2

1

gree of the leading term is twice the degree of the middle term: x3 Ϫ 3x3 Ϫ 10 ϭ 0

and 1x2 ϩ x2 2 Ϫ 81x2 ϩ x2 ϩ 12 ϭ 0. The first equation and its solution appear in

Example 17.

EXAMPLE 17

2

1

Solve using a u-substitution: x3 Ϫ 3x3 Ϫ 10 ϭ 0.

Solution

This

equation

is in quadratic form since it can be rewritten as:

1

1

1x3 2 2 Ϫ 31x3 2 1 Ϫ 10 ϭ

0, where the2 degree of leading term is twice that of second

1

term. If we let u ϭ x3, then u2 ϭ x3 and the equation becomes u2 Ϫ 3u1 Ϫ 10 ϭ 0,

which is factorable.

1u Ϫ 521u ϩ 22

uϭ5

1

x3 ϭ 5

1

1x3 2 3 ϭ 53

x ϭ 125

ϭ0

or

u ϭ Ϫ2

1

or

x3 ϭ Ϫ2

1

or 1x3 2 3 ϭ 1Ϫ22 3

or

x ϭ Ϫ8

factor

solution in terms of u

1

resubstitute x 3 for u

cube both sides: 13 132 ϭ 1

solve for x

Both solutions check.

Now try Exercises 59 and 60

Figure R.8

Hypotenuse

Leg

90Њ

Leg

A right triangle is one that has a 90° 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, a2 ϩ b2 ϭ c2

2. For any triangle with sides a, b, and c, if a2 ϩ b2 ϭ c2, then the triangle

is a right triangle.

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

Figure R.9

32 ϩ 42 ϭ 52.

Area

16 in2

4

5

ea

Ar in2

25

3

Area

9 in2

25

13

12

24

ϩ

ϭ

25 ϩ 144 ϭ 169

52

122

c

7

5

132

b

ϩ

ϭ

49 ϩ 576 ϭ 625

72

242

252

ϩ b2 ϭ c2

general case

a2

a

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CHAPTER R A Review of Basic Concepts and Skills

EXAMPLE 18

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

We can assume the building makes a 90° angle with the

ground, and use the Pythagorean theorem to find the

required length. Let c represent this length.

c2 ϭ a2 ϩ b2

c2 ϭ 192 2 ϩ 1272 2

c2 ϭ 81 ϩ 729

c2 ϭ 810

c ϭ 1810

c ϭ 9 110

c Ϸ 28.5 ft

Pythagorean theorem

substitute 9 for a and 27 for b

92 ϭ 81, 272 ϭ 729

c

27 ft

definition of square root; c 7 0

exact form: 1810 ϭ 181 # 10 ϭ 9 110

approximate form

The ladder must be at least 28.5 ft tall.

9 ft

F. You’ve just seen how

we can solve equations and

Now try Exercises 63 and 64

R.6 EXERCISES

CONCEPTS AND VOCABULARY

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

n

1. 1an ϭ ͿaͿ if n 7 0 is a(n)

integer.

3. By decomposing the rational exponent, we can

3

?

rewrite 16 4 as 116 ? 2 ?.

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

expression like 1A has been written in simplest

form.

2. The conjugate of 2 Ϫ 13 is

.

4. 1x2 2 3 ϭ x2 3 ϭ x1 is an example of the

property of exponents.

3 2

#

3 2

6. Discuss/Explain why it would be easier x12

to simplify the expression given using

1

rational exponents rather than radicals. x3

Evaluate the expression 2x2 for the values given.

7. a. x ϭ 9

b. x ϭ Ϫ10

8. a. x ϭ 7

b. x ϭ Ϫ8

Simplify each expression, assuming that variables can

represent any real number.

9. a. 249p2

c. 281m4

b. 21x Ϫ 32 2

d. 2x2 Ϫ 6x ϩ 9

10. a. 225n2

c. 2v10

3

11. a. 1

64

3

c. 2

216z12

3

12. a. 1

Ϫ8

3

c. 2

27q9

b. 21y ϩ 22 2

d. 24a2 ϩ 12a ϩ 9

3

b. 2

Ϫ216x3

d.

v3

B Ϫ8

3

3

b. 2

Ϫ125p3

d.

w3

B Ϫ64

3

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6

13. a. 1

64

5

c. 2243x10

5

e. 2 1k Ϫ 32 5

4

14. a. 1 81

5

c. 21024z15

5

e. 2 1q Ϫ 92 5

3

15. a. 1 Ϫ125

6

b. 1

Ϫ64

5

d. 2Ϫ243x5

6

f. 2 1h ϩ 22 6

4

b. 1 Ϫ81

5

d. 2Ϫ1024z20

6

f. 2 1p ϩ 42 6

4

b. Ϫ 281n12

10

c. 2Ϫ36

49v

B 36

4

b. Ϫ 216m24

d.

16. a. 1 Ϫ216

3

c. 2Ϫ121

25x6

d.

B 4

24. a. 28x6

2 3

c. 227a2b6

9

12 Ϫ 248

e.

8

25. a. 2.5 218a22a3

c.

26. a. 5.1 22p232p5

c.

3

16 2

b. a b

25

2

3

17. a. 8

4 Ϫ2

c. a b

25

3

d. a

27. a.

Ϫ27p6

8q3

2

3

b

28. a.

4

b. a b

9

3

16 Ϫ4

c. a b

81

3

2

Ϫ125v9 3

b

d. a

27w6

b. aϪ

3

19. a. Ϫ1442

c. 1Ϫ272 Ϫ3

2

b. aϪ

3

20. a. Ϫ1002

c. 1Ϫ1252

d. Ϫa

4

b

25

21. a. 12n p 2

22. a. a

1

4x2

b

23y

20

B 4x4

5

29. a. 2

32x10y15

Ϫ20 ϩ 232

4

2

b. Ϫ 23b212b2

3

f.

3

3

d. 29v2u23u5v2

4

b. Ϫ 25q220q3

5

d. 25cd2 125cd

3

b.

c. 31 b

4

e. 2b 1

b

27x

b

64

3

4

30. a. 2

81a12b16

4

c. 32a

3

4

e. 1

c1

c

3

2

x9 Ϫ3

d. Ϫa b

8

3

3

2

108n4

3

2

4n

81

d. 12 3 9

A 8z

b.

3

2

72b5

3

2

3b2

125

d. Ϫ9 3

A 27x6

4 5

b. x 2

x

d.

3

1

6

26

5 6

b. a 2

a

d.

3

1

3

4

2

3

4

Ϫ23

2 Ϫ25 5

3

c.

227y7

4

Use properties of exponents to simplify. Answer in

exponential form without negative exponents.

24x8

28m5

d. 254m6n8

3

2

3 Ϫ43

49

b

36

ab2 25ab4

B 3 B 27

22m

45

c.

B 16x2

3

2

18. a. 92

x3y 4x5y

B 3 B 12y

3

b. 3 2

128a4b2

2

b. a

3

8y4

3

64y2

b

1

3

b. 12x y 2

Ϫ14

3

4

4

Simplify each expression. Assume all variables represent

nonnegative real numbers.

23. a. 218m2

3 3

c. 264m3n5

8

Ϫ6 ϩ 228

e.

2

3

b. Ϫ2 2

Ϫ125p3q7

d. 232p3q6

f.

27 Ϫ 272

6

31. a.

b.

c.

d.

12 272 Ϫ 9 298

8 248 Ϫ 3 2108

7 218m Ϫ 250m

2 228p Ϫ 3 263p

32. a.

b.

c.

d.

Ϫ3280 ϩ 2 2125

5 212 ϩ 2 227

3 212x Ϫ 5 275x

3 240q ϩ 9 210q

3

3

33. a. 3x 1 54x Ϫ 5 216x4

b. 14 ϩ 13x Ϫ 112x ϩ 145

c. 272x3 ϩ 150 Ϫ 17x ϩ 127

3

3

34. a. 5 254m3 Ϫ 2m216m3

b. 110b ϩ 1200b Ϫ 120 ϩ 140

c. 275r3 ϩ 132 Ϫ 127r ϩ 138

77

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Compute each product and simplify the result.

35. a. 17 122

b. 131 15 ϩ 172

c. 1n ϩ 1521n Ϫ 152 d. 16 Ϫ 132 2

46. a.

7

17 ϩ 3

b.

12

1x ϩ 13

47. a.

110 Ϫ 3

13 ϩ 12

b.

7 ϩ 16

3 Ϫ 3 12

37. a. 13 ϩ 217213 Ϫ 2 172

b. 1 15 Ϫ 1142 1 12 ϩ 1132

c. 12 12 ϩ 6 162 13110 ϩ 172

48. a.

1 ϩ 12

16 ϩ 114

b.

1 ϩ 16

5 ϩ 2 13

2

36. a. 10.3 152 2

b. 151 16 Ϫ 122

c. 14 ϩ 13214 Ϫ 132 d. 12 ϩ 152 2

38. a. 15 ϩ 4110211 Ϫ 2 1102

b. 1 13 ϩ 122 1 110 ϩ 1112

c. 1315 ϩ 4 122 1 115 ϩ 162

Use a substitution to verify the solutions to the

quadratic equation given. Verify results using a

calculator.

39. x2 Ϫ 4x ϩ 1 ϭ 0

a. x ϭ 2 ϩ 13

b. x ϭ 2 Ϫ 13

40. x Ϫ 10x ϩ 18 ϭ 0

a. x ϭ 5 Ϫ 17

b. x ϭ 5 ϩ 17

41. x2 ϩ 2x Ϫ 9 ϭ 0

a. x ϭ Ϫ1 ϩ 110

b. x ϭ Ϫ1 Ϫ 110

42. x2 Ϫ 14x ϩ 29 ϭ 0

a. x ϭ 7 Ϫ 2 15

b. x ϭ 7 ϩ 2 15

2

Rationalize each expression by building perfect nth root

factors for each denominator. Assume all variables

represent positive quantities.

3

112

27

c.

B 50b

5

e. 3

1a

43. a.

20

B 27x3

1

d. 3

A 4p

b.

Ϫ4

125

44. a.

b.

B 12n3

120

5

3

c.

d. 3

B 12x

A 2m2

Ϫ8

e. 3

3 15

Simplify the following expressions by rationalizing the

denominators. Where possible, state results in exact

form and approximate form, rounded to hundredths.

45. a.

8

3 ϩ 111

b.

6

1x Ϫ 12

Solve each equation and check your solutions by

substitution. Identify any extraneous roots.

49. a. Ϫ313x Ϫ 5 ϭ Ϫ9

b. x ϭ 13x ϩ 1 ϩ 3

50. a. Ϫ2 14x Ϫ 1 ϭ Ϫ10

b. Ϫ5 ϭ 15x Ϫ 1 Ϫ x

3

51. a. 2 ϭ 13m Ϫ 1

3

b. 2 1 7 Ϫ 3x Ϫ 3 ϭ Ϫ7

3

1 2m ϩ 3

ϩ2ϭ3

c.

Ϫ5

3

3

d. 1 2x Ϫ 9 ϭ 1 3x ϩ 7

3

52. a. Ϫ3 ϭ 1 5p ϩ 2

3

b. 3 1 3 Ϫ 4x Ϫ 7 ϭ Ϫ4

3

1

6x Ϫ 7

Ϫ 5 ϭ Ϫ6

c.

4

3

3

d. 3 1 x ϩ 3 ϭ 2 1 2x ϩ 17

53. a.

b.

c.

d.

1x Ϫ 9 ϩ 1x ϭ 9

x ϭ 3 ϩ 223 Ϫ x

1x Ϫ 2 Ϫ 12x ϭ Ϫ2

112x ϩ 9 Ϫ 124x ϭ Ϫ3

54. a.

b.

c.

d.

1x ϩ 7 Ϫ 1x ϭ 1

12x ϩ 31 ϩ x ϭ 2

13x ϭ 1x Ϫ 3 ϩ 3

13x ϩ 4 Ϫ 17x ϭ Ϫ2

Write the equation in simplified form, then solve. Check

3

55. a. x5 ϩ 17 ϭ 9

3

b. Ϫ2x4 ϩ 47 ϭ Ϫ7

5

56. a. 0.3x2 Ϫ 39 ϭ 42

5

b. 0.5x3 ϩ 92 ϭ Ϫ43

2

57. a. 21x ϩ 52 3 Ϫ 11 ϭ 7

4

b. Ϫ31x Ϫ 22 5 ϩ 29 ϭ Ϫ19

1

3

58. a. 3x3 Ϫ 10 ϭ 1

x

2

5 2

b. 2 1x Ϫ 4 ϭ x5

2

1

59. x3 Ϫ 2x3 Ϫ 15 ϭ 0

3

60. x3 Ϫ 9x2 ϭ Ϫ8

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WORKING WITH FORMULAS

1

61. Fish length to weight relationship: L ‫ ؍‬1.131W2 3

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).

79

1s

4

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).

62. Timing a falling object: t ‫؍‬

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

approximate form by rounding

to one decimal place.

24 m

c

8m

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

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

radicals, and (b) approximate form by rounding to

one decimal place.

S

50 ft

75 ft

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

make one revolution around3 the sun is modeled

by the function T ‫ ؍‬0.407R2, 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,

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

V ϭ 2 26L, 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

3 P

, where k is a constant

determined using V ϭ

Ak

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 k ϭ 0.004 and the generator is putting out 13.5

units of power.

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69. Surface area: The lateral surface

area (surface area excluding the base)

h

S of a cone is given by the formula

2

2

S ϭ ␲r 2r ϩ h , where r is the

r

radius of the base and h is the height

of the cone. Find the lateral surface

area of a cone that has a radius of 6 m and a height of

10 m. Answer in simplest form.

70. Surface area: The lateral surface

a

area S of a frustum (a truncated

cone) is given by the formula

h

S ϭ ␲1a ϩ b2 2h2 ϩ 1b Ϫ a2 2,

b

where a is the radius of the upper

base, b is the radius of the lower

base, and h is the height. Find the surface area of a

frustum where a ϭ 6 m, b ϭ 8 m, and h ϭ 10 m.

71. Planetary motion: The time T (in days) for a

planet to make one revolution (elliptical orbit)

3

around the sun is modeled by T ϭ 0.407R2, where

R is the maximum radius of the planet’s orbit in

millions of miles (Kepler’s third law of planetary

motion). Use the equation to approximate the

maximum radius of each orbit, given the number of

days it takes for one revolution. (See Exercises 65

and 66.)

a. Mercury: 88 days

b. Venus: 225 days

c. Earth: 365 days

d. Mars: 687 days

e. Jupiter: 4333 days

f. Saturn: 10,759 days

72. 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

3 P

, where k is a constant

determined using V ϭ

Ak

that depends on the size and efficiency of the

generator. Given k ϭ 0.004, approximately how

many units of power are being delivered if the

wind is blowing at 27 miles per hour? (See

Exercise 68.)

EXTENDING THE CONCEPT

The expression x2 ؊ 7 is not factorable using integer

values. But the expression can be written in the form

x2 ؊ 1 272 2, enabling us to factor it as a “binomial”

and its conjugate: 1x ؉ 272 1x ؊ 272. Use this idea to

factor the following expressions.

73. a. x2 Ϫ 5

b. n2 Ϫ 19

74. a. 4v2 Ϫ 11

b. 9w2 Ϫ 17

75. The following terms form a pattern that continues

until the sixth term is found:

23x ϩ 29x ϩ 227x ϩ p (a) Compute the sum

of all six terms; (b) develop a system (investigate

the pattern further) that will enable you to find the

sum of 12 such terms without actually writing out

the terms.

76. Find a quick way to simplify the expression

without the aid of a calculator.

a a a a a3 b

5

6

3

2

4

5

3

4

2

5

aaaa

80

10

3

1

1

1

1

9

77. If 1x2 ϩ xϪ2 2 2 ϭ , find the value of x2 ϩ xϪ2.

2

78. Rewrite by rationalizing the numerator:

1x ϩ h Ϫ 1x

h

Determine the values of x for which each expression

represents a real number.

79.

1x Ϫ 1

x2 Ϫ 4

80.

x2 Ϫ 4

1x Ϫ 1

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Overview of Chapter R

OVERVIEW OF CHAPTER R

Prerequisite Definitions, Properties, Formulas, and Relationships

Notation and Relations

concept

• Set notation:

notation

{members}

ʦ

• Is an element of

л or { }

• Empty set

Is

a

proper

subset

of

(

5x | x p6

• Defining a set

description

braces enclose the members of a set

indicates membership in a set

a set having no elements

indicates the elements of one set

are entirely contained in another

the set of all x, such that x . . .

Sets of Numbers

• Natural: ‫ ގ‬ϭ 51, 2, 3, 4, p6

• Integers: ‫ ޚ‬ϭ 5p , Ϫ3, Ϫ2, Ϫ1, 0, 1, 2, 3, p6

• Irrational: ‫ ވ‬ϭ {numbers with a nonterminating,

example

set of even whole numbers

A ϭ 50, 2, 4, 6, 8, p6

14 ʦ A

odd numbers in A

S ϭ 50, 6, 12, 18, 24, p6

S ( A

S ϭ 5x | x ϭ 6n for n ʦ ‫ޗ‬6

• Whole: ‫ ޗ‬ϭ 50, 1, 2, 3, p6

• Rational: ‫ ޑ‬ϭ e q , where p, q ʦ ‫ ;ޚ‬q 0 f

• Real: ‫ ޒ‬ϭ {all rational and irrational numbers}

p

nonrepeating decimal form}

Absolute Value of a Number

Distance between numbers a and b

on the number line

• The distance between a number n

Ϳa Ϫ bͿ or Ϳb Ϫ aͿ

and zero (always positive)

n

if n Ն 0

0n 0 ϭ e

Ϫn if n 6 0

For a complete review of these ideas, go to www.mhhe.com/coburn.

R.1 Properties of Real Numbers: For real numbers a, b, and c,

Commutative Property

• Addition: a ϩ b ϭ b ϩ a

• Multiplication: a # b ϭ b # a

Identities

Associative Property

• Addition: 1a ϩ b2 ϩ c ϭ a ϩ 1b ϩ c2

• Multiplication: 1a # b2 # c ϭ a # 1b # c2

Inverses

• Additive: a ϩ 1Ϫa2 ϭ 0

• Additive: 0 ϩ a ϭ a

• Multiplicative: 1 # a ϭ a

p q

• Multiplicative: q # p ϭ 1; p, q

0

R.2 Properties of Exponents: For real numbers a and b, and integers m, n, and p

(excluding 0 raised to a nonpositive power),

Product property: b

• Power property: 1bm 2 n ϭ bmn

am p amp

Product to a power: 1ambn 2 p ϭ amp # bnp

Quotient

to

a

power:

b ϭ np 1b 02

a

bm

bn

b

mϪn

Quotient property: n ϭ b

1b 02

0

Zero

exponents:

ϭ

1

1b

02

b

b

Ϫn

n

1

a

b

• Scientific notation: N ϫ 10k; 1 Յ ͿNͿ 6 10, k ʦ ‫ޚ‬

Negative exponents: bϪn ϭ n ; a b ϭ a b

a

b

b

1a, b 02

m

# bn ϭ bmϩn

81

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CHAPTER R A Review of Basic Concepts and Skills

R–82

R.2 Polynomials

• A polynomial is a sum or difference of monomial terms

• Polynomials are classified as a monomial, binomial, trinomial, or polynomial, depending on the number of terms

• The degree of a polynomial in one variable is the same as the largest exponent occuring on the variable in any term

• A polynomial expression is in standard form when written with the terms in descending order of degree

R.3 Solving Linear Equations and Inequalities

• Properties of Equality:

If A and B are algebraic expressions, where A ϭ B,

then A ϩ C ϭ B ϩ C.

multiplicative property

If A and B are algebraic expressions, where A ϭ B,

then A # C ϭ B # C

B

A

[C can be positive or negative]

and ϭ ;C 0

C

C

A linear equation in one variable is one that can be written in the form ax ϩ b ϭ c, where the exponent on the

variable is a 1.

To solve a linear equation, we attempt to isolate the term containing the variable using the additive property, then

solve for the variable using the multiplicative property.

An equation can be an identity (always true), a contradiction (never true) or conditional (true or false depending

on the input value[s]).

If an equation contains fractions, multiplying both sides by the least common denominator of all fractions will

“clear the denominators” and reduce the amount of work required to solve.

Inequalities are solved using properties similar to those used for solving equations. The one exception is when

multiplying or dividing by a negative quantity, as the inequality symbol must then be reversed to maintain the

truth of the resulting statement.

Solutions to an inequality can be given using a simple inequality, graphed on a number line, stated in set notation,

or stated using interval notation.

Given two sets A and B: A intersect B 1A ă B2 is the set of elements shared by both A and B (elements common to

both sets). A union B 1A ´ B2 is the set of elements in either A or B (elements are combined to form a larger set).

Compound inequalities are formed using the conjunction “and” or the conjunction “or.” The result can be either a

joint inequality as in Ϫ3 6 x Յ 5, or a disjoint inequality, x 6 Ϫ2 or x 7 7.

R.4 Special Factorizations

• A2 Ϫ B2 ϭ 1A ϩ B21A Ϫ B2

• A2 Ϯ 2AB ϩ B2 ϭ 1A Ϯ B2 2

3

3

2

2

• A Ϫ B ϭ 1A Ϫ B21A ϩ AB ϩ B 2

• A3 ϩ B3 ϭ 1A ϩ B21A2 Ϫ AB ϩ B2 2

• Certain equations of higher degree can be solved using factoring skills and the zero product property.

R.5 Rational Expressions: For polynomials P, Q, R, and S with no denominator of zero,

P#R

P

P#R

P

ϭ

Equivalence:

ϭ

Q#R

Q

Q

Q#R

P R

P#R

PR

R

P S

PS

P

Multiplication: # ϭ # ϭ

• Division: Ϭ ϭ # ϭ

Q S

Q S

QS

Q

S

Q R

QR

Q

PϩQ

Q

PϪQ

P

P

• Subtraction: Ϫ ϭ

R

R

R

R

R

R

1. Find the LCD of all rational expressions.

2. Build equivalent expressions using LCD.

4. Write the result in lowest terms.

To solve rational equations, first clear denominators using the LCD, noting values that must be excluded.

Multiplying an equation by a variable quantity sometimes introduces extraneous solutions. Check all results in the

original equation.

• Lowest terms:

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Practice Test

• 1a is a real number only for a Ն 0

n

• 1a ϭ b, only if bn ϭ a

83

• 1a ϭ b, only if b2 ϭ a

n

• If n is even, 1a represents a real number only

if a Ն 0

n

• For any real number a, 1an ϭ a when n is odd

m

If a is a real number and

n is an integer greater

• If n is a rational number written in lowest terms

1

n

n

m

m

than 1, then 1 a ϭ an provided 1 a represents a

n

n

with n Ն 2, then a n ϭ 11a2 m and a n ϭ 1am

real number

n

provided 2a represents a real number.

n

n

n

n

If 1A and 1B represent real numbers and B 0,

If 1A and 1B represent real numbers,

n

n

n

n

1AB ϭ 1 A # 1B

1A

n A

ϭ n

BB

1B

A radical expression is in simplest form when:

1. the radicand has no factors that are perfect nth roots,

2. the radicand contains no fractions, and

3. no radicals occur in a denominator.

To solve radical equations, isolate the radical on one side, then apply the appropriate “nth power” to free up the

radicand. Repeat the process if needed. See flowchart on page 74.

For equations with a rational exponent mn, isolate the variable term and raise both sides to the mn power. If m is even,

there will be two real solutions.

n

• For any real number a, 1an ϭ ͿaͿ when n is even

R.6 Pythagorean Theorem

• For any right triangle with legs a and b and

hypotenuse c: a2 ϩ b2 ϭ c2.

• For any triangle with sides a, b, and c, if

a2 ϩ b2 ϭ c2, then the triangle is a right triangle.

PRACTICE TEST

1. State true or false. If false, state why.

7

a. 13 ϩ 42 2 ϭ 25

b. ϭ 0

0

c. x Ϫ 3 ϭ Ϫ3 ϩ x

d. Ϫ21x Ϫ 32 ϭ Ϫ2x Ϫ 3

2. State the value of each expression.

3

a. 1 121

b. 1

Ϫ125

c. 1Ϫ36

d. 1400

3. Evaluate each expression:

7

1

5

1

a. Ϫ aϪ b

b. Ϫ Ϫ

8

4

3

6

c. Ϫ0.7 ϩ 1.2

d. 1.3 ϩ 1Ϫ5.92

4. Evaluate each expression:

1

a. 1Ϫ42aϪ2 b

b. 1Ϫ0.621Ϫ1.52

3

Ϫ2.8

c.

d. 4.2 Ϭ 1Ϫ0.62

Ϫ0.7

#

0.08 12 10

5. Evaluate using a calculator: 2000 a1 ϩ

b

12

6. State the value of each expression, if possible.

a. 0 Ϭ 6

b. 6 Ϭ 0

7. State the number of terms in each expression and

identify the coefficient of each.

cϩ2

ϩc

a. Ϫ2v2 ϩ 6v ϩ 5

b.

3

8. Evaluate each expression given x ϭ Ϫ0.5 and

y ϭ Ϫ2. Round to hundredths as needed.

y

a. 2x Ϫ 3y2

b. 12 Ϫ x14 Ϫ x2 2 ϩ

x

9. Translate each phrase into an algebraic expression.

a. Nine less than twice a number is subtracted from

the number cubed.

b. Three times the square of half a number is

subtracted from twice the number.

10. Create a mathematical model using descriptive

variables.

a. The radius of the planet Jupiter is approximately

119 mi less than 11 times the radius of the Earth.

Express the radius of Jupiter in terms of the

b. Last year, Video Venue Inc. earned \$1.2 million

more than four times what it earned this year.

Express last year’s earnings of Video Venue Inc.

in terms of this year’s earnings.

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