Page 16SECTION R.2Exponents and RadicalsSECTION R.2 Exponents and RadicalsOBJECTIVES

1. Apply Properties of Exponents

2. Apply Scientific Notation

3. Evaluate nth-Roots

4. Simplify Expressions of the Forms a1/n and am/n

5. Simplify Radicals

6. Apply Operations on Radicals

1. Apply Properties of Exponents

In Section R.1, we learned that exponents are used to represent repeated multiplication. Applications ofexponents appear in many fields of study, including computer science. Computer engineers define a bitas a fundamental unit of information having just two possible values. These values are represented byeither 0 or 1. A byte is usually taken as 8 bits. Computer programmers know that there are 2n possiblevalues for an n-bit variable. So 1 byte has 28 = 256 possible values.

Three bytes are often used to represent color on a computer screen. The intensity of each of the colorsred, green, and blue ranges from 0 to 255 (a total of 256 possible values each). So the number of colorsthat can be represented by this system is

There are over 16 million possible colors available using this system. For example, the color given by red137, green 21, blue 131, is a deep pink. See Figure R-2.

Figure R-2

A similar approach can be used to show that bmbn = bm+n for natural numbers m and n.

We would like to extend this result to expressions where m and n are negative integers or zero. Forexample,

Now consider an example involving a negative exponent.

These observations lead to two important definitions.

Definition of b0 and b−n

If b is a nonzero real number and n is a positive integer, then

b0 = 1 Examples: (5000)0 = 1 and (−3x)0 = 1 for x ≠ 0

Examples: and for x ≠ 0

The definitions given here have two important restrictions.

By definition, b0 = 1 provided that b ≠ 0. Therefore,

The value of 00 is not defined here. The value of 00 is said to be indeterminate and is examined incalculus.

Page 17

For a positive integer n, by definition, provided that b ≠ 0. Therefore,

The value of 0−n is not defined here.

All examples and exercises in the text will be given under the assumption that the variable expressionsavoid these restrictions. For example, the expression x0 will be stated with the implied restriction that x≠ 0.

EXAMPLE 1 Simplifying Expressions with a Zero or Negative Exponent

Simplify.

Lecture: Definition of a Nonzero Base b to the Zero Power

PDF Transcript for Lecture: Definition of a Nonzero Base b to the Zero Power

a. 6y0

b. −50

Lecture: Definition of a Base b Raised to a Negative Integer Exponent

PDF Transcript for Lecture: Definition of a Base b Raised to a Negative Integer Exponent

c. 5−2

d.

e. 5x−8y2

Solution:

a.

Table R-5

b.

c.

d.

e.

Skill Practice 1

Simplify.

a. 4z0

b. −40

c. 2−3

d.

e. −3a4b−9

Answers

1. a. 4

b. −1

c. or

d. n5

e.

The property bmbn = bm+n is one of several important properties of exponents that can be used tosimplify an algebraic expression (Table R-5).

Lecture: Summarizing Properties of Exponents

PDF Transcript for Lecture: Summarizing Properties of Exponents

Properties of ExponentsLet a and b be real numbers and m and n be integers.*

Property Example Expanded Form

bm · bn = bm+nx4 · x3 = x4+3 = x7x4 · x3 = (x · x · x · x) (x · x · x) = x7

(bm)n = bm·n (x2)3 = x2·3 = x6 (x2)3 = (x2)(x2)(x2) = x6

(ab)m = ambm (4x)3 = 43x3

*The properties are stated under the assumption that thevariables are restricted to avoid the expressions 00 and .Expressions of the form 00 and are said to be indeterminateand are examined in calculus.

Lecture: Simplifying an Exponential Expression by Using thePower Properties of Exponents

PDF Transcript for Lecture: Simplifying an ExponentialExpression by Using the Power Properties of Exponents

Page 18EXAMPLE 2 Simplifying Expressions Containing Exponents

Simplify. Write the answers with positive exponents only.

a.

b. (−3x)−4(4x−2y3)3

Solution:

a.

b.

Skill Practice 2

Simplify. Write the answers with positive exponents only.

a. (3w)−3(5wt5)2

b.

Answers

2. a.

b.

Animation: Introduction to scientific notation

2. Apply Scientific Notation

In many applications of science, technology, and business we encounter very large or very smallnumbers. For example:

eBay Inc. purchased the Internet communications company, Skype Technologies, for approximately$2,600,000,000. (Source: www.ebay.com)

The diameter of a capillary is measured as 0.000 005 m.

The mean surface temperature of the planet Saturn is −300°F.

Very large and very small numbers are sometimes cumbersome to write because they contain numerouszeros. Furthermore, it is difficult to determine the location of the decimal point when performingcalculations with such numbers. For these reasons, scientists will often write numbers using scientificnotation.

Lecture: Introduction to Scientific Notation

PDF Transcript for Lecture: Introduction to Scientific Notation

Page 19Scientific Notation

A number expressed in the form a × 10n, where 1 ≤ |a| < 10 and n is an integer is said to be inscientific notation.

Examples

Skype Purchase $2,600,000,000 = $2.6 × 1,000,000,000 = $2.6 × 109

Capillary Size 0.000 005 m = 5.0 × 0.000001 m = 5.0 × 10−6 m

Saturn Temp. −300°F = −3 × 100°F = −3 × 102 °F

To write a number in scientific notation, the number of positions that the decimal point must be moveddetermines the power of 10. Numbers 10 or greater require a positive exponent on 10. Numbers

between 0 and 1 require a negative exponent on 10.

Lecture: Converting Numbers from Scientific Notation to Standard Decimal Form

PDF Transcript for Lecture: Converting Numbers from Scientific Notation to Standard Decimal Form

EXAMPLE 3 Writing Numbers in Scientific Notation and Standard Decimal Notation

Write the numbers in scientific notation or standard decimal notation.

a. 0.0000002

b. 230 billion

c. 1.36 × 107

d. 3.9 × 10−3

Solution:

a.

b.

c.

d.

Skill Practice 3

Write the numbers in scientific notation or standard decimal notation.

a. 0.0000035

b. 49,500

c. 5.86 × 102

d. 5.0 × 10−2

Answers

3. a. 3.5 × 10−6

b. 4.95 × 104

c. 586

d. 0.05

Example 4 demonstrates the process to multiply numbers written in scientific notation.

EXAMPLE 4 Performing Calculations with Scientific Notation

A light-year is the distance that light travels in 1 yr. If light travels at a speed of 6.7 × 108 mph, how farwill it travel in 1 yr (8.76 × 103 hr)?

Lecture: Multiplying and Dividing Numbers in Scientific Notation

PDF Transcript for Lecture: Multiplying and Dividing Numbers in Scientific Notation

Page 20Solution:

Skill Practice 4

A satellite travels 1.72 × 104 mph. How far does it travel in 24 hr (2.4 × 101 hr)?

Answer

4. 4.128 × 105 mi

3. Evaluate nth-Roots

We now want to extend the definition of bn to expressions in which the exponent, n, is a rational number.

First we need to understand the relationship between nth-powers and nth-roots. From Section R.1, weknow that for a ≥ 0, if b2 = a and b ≥ 0. Square roots are a special case of nth-roots.

Definition of an nth-Root

For a positive integer n > 1, the principal nth-root of a, denoted by , is a number b such that

If n is even, then we require that a ≥ 0 and b ≥ 0.

For the expression , the symbol is called a radical sign, the value a is called the radicand, andn is called the index.

Lecture: Introduction to nth Roots

PDF Transcript for Lecture: Introduction to nth Roots

EXAMPLE 5 Simplifying nth-Roots

Simplify.

a.

b.

c.

d.

e.

Solution:

a.

b.

c.

d.

e.

Page 21Skill Practice 5

Simplify.

a.

b.

c.

d.

e.

Answers

5. a. −5

b.

c. 0.1

d. Not a real number

e. −2

4. Simplify Expressions of the Forms a1/n and am/n

Next, we want to define an expression of the form an, where n is a rational number. Furthermore, wewant a definition for which the properties of integer exponents can be extended to rational exponents.For example, we want

Definition of an a1/n and am/n

Let m and n be integers such that m/n is a rational number in lowest terms and n > 1. Then,

If n is even, we require that a ≥ 0.

The definition of am/n indicates that am/n can be written as a radical whose index is the denominator ofthe rational exponent. The order in which the nth-root and exponent m are performed within the radicaldoes not affect the outcome. For example:

EXAMPLE 6 Simplifying Expressions of the Form a1/n and am/n

Write the expressions using radical notation and simplify if possible.

Lecture: Definition of "a" to the 1/n Power

PDF Transcript for Lecture: Definition of "a" to the 1/n Power

a. 251/2

b.

c. (−81)1/4

Lecture: Definition of "a" to the m/n Power

PDF Transcript for Lecture: Definition of "a" to the m/n Power

d. 323/5

e. 81−3/4

Solution:

a.

b.

c. (−81)1/4 is undefined because is not a real number.

d.

e.

Page 22Skill Practice 6

Simplify the expressions if possible.

a. 361/2

b.

Table R-6

c. (−9)1/2

d. (−1)4/3

e. (−125)−2/3

Answers

6. a. 6

b.

c. Undefined (not a real number)

d. 1

e.

5. Simplify Radicals

In Example 6, we simplified several expressions with rational exponents. Next, we want to simplifyradical expressions. First consider expressions of the form . The value of is not necessarily a.Since represents the principal nth-root of a, then must be nonnegative for even values of n. Forexample:

In Table R-6, we generalize this result and give three other important properties of radicals.

Lecture: The Product and Quotient Properties of Radicals

PDF Transcript for Lecture: The Product and Quotient Properties of Radicals

Properties of Radicals

Let n > 1 be an integer and a and b be real numbers. The following properties are true provided that thegiven radicals are real numbers.

Property Examples

1. If n is even, .

2. If n is odd, .

3. Product Property

4. Quotient Property

5. Nested Radical Property

Properties 1–5 follow from the definition of a1/n and the properties of exponents. We use theseproperties to simplify radical expressions and must address four specific criteria.

Lecture: Simplified Form of a Radical

PDF Transcript for Lecture: Simplified Form of a Radical

Simplified Form of a Radical

Suppose that the radicand of a radical is written as a product of prime factors. Then the radical issimplified if all of the following conditions are met.

1. The radicand has no factor other than 1 that is a perfect nth-power. This means that all exponentsin the radicand must be less than the index.

2. No fractions may appear in the radicand.

3. No denominator of a fraction may contain a radical.

4. The exponents in the radicand may not all share a common factor with the index.

In Example 7, we simplify expressions that fail condition 1. Also notice that the expressions in Example 7are assumed to have positive radicands. This eliminates the need to insert absolute value bars aroundthe simplified form of .

Lecture: Simplifying Radicals by Using the Product Property of Radicals

PDF Transcript for Lecture: Simplifying Radicals by Using the Product Property of Radicals

Page 23EXAMPLE 7 Simplifying Radicals Using the Product Property

Simplify each expression. Assume that all variables represent positive real numbers.

a.

b.

c.

Solution:

a.

b.

c.

Skill Practice 7

Simplify each expression. Assume that all variables represent positive real numbers.

a.

b.

c.

Answers

7. a.

b.

c.

6. Apply Operations on Radicals

In Example 8(a), we will use the product property of radicals to multiply two radical expressions. InExample 8(b), we will use the quotient property of radicals to divide radical expressions.

EXAMPLE 8 Multiplying and Dividing Radical Expressions

Multiply or divide as indicated. Assume that x and y represent positive real numbers.

a.

b.

Solution:

a.

Page 24b.

Skill Practice 8

Multiply or divide as indicated. Assume that c, d, and y represent positive real numbers.

a.

b.

TIP

In Example 8(b), the purpose of writing the quotient of two radicals as a single radical is to simplify theresulting fraction in the radicand.

In Example 8(b), we removed the radical from the denominator of the fraction. This is calledrationalizing the denominator.

We can use the distributive property to add or subtract radical expressions. However, the radicals mustbe like radicals. This means that the radicands must be the same and the indices must be the same. Forexample:

and are like radicals.

and are not like radicals because the indices are different.

and are not like radicals because the radicands are different.

Lecture: Adding and Subtracting Radicals

PDF Transcript for Lecture: Adding and Subtracting Radicals

EXAMPLE 9 Adding and Subtracting Radicals

Add or subtract as indicated. Assume that all variables represent positive real numbers.

a.

b.

Solution:

a.

b.

Skill Practice 9

Add or subtract as indicated. Assume that all variables represent positive real numbers.

a.

b.

Answers

8. a.

b.

9. a.

b.

Page 25SECTION R.2 Practice ExercisesConcept Connections

1. For a nonzero real number b, the value of b0 = .

Answers

2. For a nonzero real number b, the value .

3. From the properties of exponents, .

Answers

4. If b ≠ 0, .

5. From the properties of exponents, .

Answers

6. Given the expression , the value a is called the and n is called the .

7. The expression am/n can be written in radical notation as , provided that is areal number.

Answers

8. The expression a1/n can be written in radical notation as , provided that is a realnumber.

Objective 1: Apply Properties of Exponents

For Exercises 9–14, simplify each expression. (See Example 1)

Exercise: Definition of a Nonzero Base b to the Zero Power

PDF Transcript for Exercise: Definition of a Nonzero Base b to the Zero Power

9. a. 80

b. −80

c. 8x0

d. (8x)0

Answers

10. a. 70

b. −70

c. 7y0

d. (7y)0

11. a. 8−2

b. 8x−2

c. (8x)−2

d. −8−2

Answers

12. a. 7−2

b. 7y−2

c. (7y)−2

d. −7−2

Exercise: Definition of a Base b Raised to a Negative Integer Exponent

PDF Transcript for Exercise: Definition of a Base b Raised to a Negative Integer Exponent

13. a.

b. q−2

c. 5p3q−2

d. 5p−3q2

Answers

14. a.

b. t−4

c. 11t−4u2

d. 11t4u−2

For Exercises 15–46, use the properties of exponents to simplify each expression. (SeeExample 2)

15. x7 · x6 · x−2

Answers

16. y−3 · y7 · y4

Exercise: Simplifying an Exponential Expression by Using the Product Property

PDF Transcript for Exercise: Simplifying an Exponential Expression by Using the Product Property

17. (−3c2d7)(4c−5d)

Answers

18. (−7m−3n−8)(3m−5n)

19.

Answers

20.

21.

Answers

22.

Exercise: Simplifying an Exponential Expression by Using the Quotient Property

PDF Transcript for Exercise: Simplifying an Exponential Expression by Using the Quotient Property

23.

Answers

24.

25. (p−2)7

Answers

26. (q−4)2

27.

Answers

28.

29. (4x2y−3)2

Answers

30. (−3w−3z5)2

31.

Answers

32.

33.

Answers

34.

35. (−2y)−3(6y−2z8)2

Answers

36. (−15z)−2(5z4w−6)3

37.

Answers

38.

39.

Answers

40.

41.

Answers

42.

43. (3x + 5)14(3x + 5)−2

Answers

44. (2y − 7z)−4(2y − 7z)13

45. 2−2 + 2−1 + 20 + 21 + 22

Answers

46. 3−2 + 3−1 + 30 + 31 + 32

Page 26Objective 2: Apply Scientific Notation

For Exercises 47–52, write the numbers in scientific notation. (See Example 3)

47. a. 350,000

b. 0.000035

c. 3.5

Answers

48. a. 2710

b. 0.00271

c. 2.71

Exercise: Introduction to Scientific Notation

PDF Transcript for Exercise: Introduction to Scientific Notation

49. The speed of light is approximately 29,980,000,000 cm/sec.

Answers

50. The mean distance between the Earth and the Sun is approximately 149,000,000 km.

Exercise: Introduction to Scientific Notation

PDF Transcript for Exercise: Introduction to Scientific Notation

51. The size of an HIV particle is approximately 0.00001 cm.

Answers

52. One picosecond is 0.000 000 000 001 sec.

For Exercises 53–58, write the number in standard decimal notation. (See Example 3)

Exercise: Converting Numbers from Scientific Notation to Standard Decimal Form

PDF Transcript for Exercise: Converting Numbers from Scientific Notation to Standard Decimal Form

53. a. 2.61 × 10−6

b. 2.61 × 106

c. 2.61 × 100

Answers

54. a. 3.52 × 10−2

b. 3.52 × 102

c. 3.52 × 100

55. A drop of water has approximately 1.67 × 1021 molecules of H2O.

Answers

56. A computer with a 3-terabyte hard drive can store approximately 3.0 × 1012 bytes.

57. A typical red blood cell is 7.0 × 10−6 m.

Answers

58. The blue light used to read a laser disc has a wavelength of 4.7 × 10−7 m.

For Exercises 59–66, perform the indicated operation. Write the answer in scientific notation.(See Example 4)

59.

Answers

60.

61. (6.2 × 1011)(3 × 104)

Answers

62. (8.1 × 106)(2 × 105)

63.

Answers

64.

65.

Answers

66.

67. Jonas has a personal music player with 80 gigabytes of memory (80 gigabytes is approximately 8× 1010 bytes). If each song requires an average of 4 megabytes of memory (approximately 4 ×106 bytes), how many songs can Jonas store on the device?

Answers

68. Joelle has a personal web page with 60 gigabytes of memory (approximately 6 × 1010 bytes). Shestores math videos on the site for her students to watch outside of class. If each video requires anaverage of 5 megabytes of memory (approximately 5 × 106 bytes), how many videos can shestore on her website?

69. A typical adult human has approximately 5 L of blood in the body. If 1 μL (1 microliter) contains 5× 106 red blood cells, how many red blood cells does a typical adult have? (Hint: 1 L = 106 μL.)

Answers

70. The star Proxima Centauri is the closest star (other than the Sun) to the Earth. It is approximately4.3 light-years away. If 1 light-year is approximately 5.9 × 1012 mi, how many miles is ProximaCentauri from the Earth?

Objective 3: Evaluate nth-Roots

For Exercises 71–78, simplify the expression. (See Example 5)

71.

Answers

72.

Exercise: Introduction to nth Roots

PDF Transcript for Exercise: Introduction to nth Roots

73.

Answers

74.

Exercise: Introduction to nth Roots

PDF Transcript for Exercise: Introduction to nth Roots

75.

Answers

76.

Exercise: Introduction to nth Roots

PDF Transcript for Exercise: Introduction to nth Roots

77.

Answers

78.

Objective 4: Simplify Expressions of the Forms a1/n and am/n

For Exercises 79–88, simplify each expression. (See Example 6)

Exercise: Definition of "a" to the 1/n Power

PDF Transcript for Exercise: Definition of "a" to the 1/n Power

79. a. 251/2

b. (−25)1/2

c. −251/2

Answers

80. a. 361/2

b. (−36)1/2

c. −361/2

81. a. 271/3

b. (−27)1/3

c. −271/3

Answers Page 27

82. a. 1251/3

b. (−125)1/3

c. −1251/3

83. a.

b.

Answers

84. a.

b.

Exercise: Definition of "a" to the m/n Power

PDF Transcript for Exercise: Definition of "a" to the m/n Power

85. a. 163/4

b. 16−3/4

c. −163/4

d. −16−3/4

e. (−16)3/4

f. (−16)−3/4

Answers

86. a. 813/4

b. 81−3/4

c. −813/4

d. −81−3/4

e. (−81)3/4

f. (−81)−3/4

87. a. 642/3

b. 64−2/3

c. −642/3

d. −64−2/3

e. (−64)2/3

f. (−64)−2/3

Answers

88. a. 82/3

b. 8−2/3

c. −82/3

d. −8−2/3

e. (−8)2/3

f. (−8)−2/3

Objective 5: Simplify Radicals

89. a. For what values of t will the statement be true?

b. For what value of t will the statement be true?

Answers

90. a. For what values of c will the statement be true?

b. For what value of c will the statement be true?

For Exercises 91–100, simplify each expression. Assume that all variable expressionsrepresent positive real numbers. (See Example 7)

Exercise: Simplified Form of a Radical

PDF Transcript for Exercise: Simplified Form of a Radical

91. a.

b.

c.

d.

Answers

92. a.

b.

c.

d.

93. a.

b.

Answers

94. a.

b.

95.

Answers

96.

97.

Answers

98.

99.

Answers

100.

6. Apply Operations on Radicals

For Exercises 101–112, multiply or divide as indicated. Assume that all variable expressionsrepresent positive real numbers. (See Example 8)

101.

Answers

102.

103.

Answers

104.

105.

Answers

106.

107.

Answers

108.

109.

Answers

110.

111.

Answers

112.

For Exercises 113–120, add or subtract as indicated. Assume that all variables representpositive real numbers. (See Example 9)

Exercise: Adding and Subtracting Radicals

PDF Transcript for Exercise: Adding and Subtracting Radicals

113.

Answers

114.

115.

Answers

116.

Exercise: Adding and Subtracting Radicals

PDF Transcript for Exercise: Adding and Subtracting Radicals

117.

Answers

118.

119.

Answers

120.

Page 28Mixed Exercises

For Exercises 121–122, use the Pythagorean theorem to determine the length of the missingside. Write the answer as a simplified radical.

121. Answers

122.

123. The slant length L for a right circular cone is given by , where r and h are the radiusand height of the cone. Find the slant length of a cone with radius 4 in. and height 10 in.Determine the exact value and a decimal approximation to the nearest tenth of an inch.

Answers

124. The lateral surface area A of a right circular cone is given by , where r and h arethe radius and height of the cone. Determine the exact value (in terms of π) of the lateral surfacearea of a cone with radius 6 m and height 4 m. Then give a decimal approximation to the nearestsquare meter.

125. The depreciation rate for a car is given by , where S is the value of the car after nyears, and C is the initial cost. Determine the depreciation rate for a car that originally cost$22,990 and was valued at $11,500 after 4 yr. Round to the nearest tenth of a percent.

Answers

126. For a certain oven, the baking time t (in hr) for a turkey that weighs x pounds can beapproximated by the model t = 0.84x3/5. Determine the baking time for a 15-lb turkey. Round to 1decimal place.

Write About It

127. Explain the difference between the expressions 6x0 and (6x)0

Answers

128. Explain why scientific notation is used.

129. Explain the similarity in simplifying the given expressions.

a. 2x + 3x

b.

c.

Answers

130. Explain why the given expressions cannot be simplified further.

a. 2x + 3y

b.

c.

Expanding Your Skills

For Exercises 131–132, refer to the formula . This gives the gravitational force F (in

Newtons, N) between two masses m1 and m2 (each measured in kg) that are a distance of dmeters apart. In the formula, G = 6.6726 × 10−11 N-m2/kg2.

131. Determine the gravitational force between the Earth (mass = 5.98 × 1024 kg) and Jupiter (mass =1.901 × 1027 kg) if at one point in their orbits, the distance between them is 7.0 × 1011 m.

Answers

132. Determine the gravitational force between the Earth (mass = 5.98 × 1024 kg) and an 80-kghuman standing at sea level. The mean radius of the Earth is approximately 6.371 × 106 m.

For Exercises 133–136, without the assistance of a calculator, fill in the blank with theappropriate symbol <, >, or =.

133. a. 515 517

b. 5−15 5−17

Answers

134. a.

b.

135. a. (−1)86 (−1)87

b. (1)86 (1)87

Answers

136. a. (−1)0 −141

b. (−1)42 (−1)0

Page 29

For Exercises 137–142, write each expression as a single radical for positive values of thevariable. (Hint: Write the radicals as expressions with rational exponents and simplify. Thenconvert back to radical form.)

137.

Answers

138.

139.

Answers

140.

141.

Answers

142.

For Exercises 143–144, evaluate the expression without the use of a calculator.

143.

Answers

144.

For Exercises 145–146, simplify the expression.

145.

Answers

146.