1 roots & radicals intermediate algebra. 2 roots and radicals radicals rational exponents...

1

Roots & Radicals

Intermediate Algebra

2

Roots and Radicals

Radicals

Rational Exponents

Operations with Radicals

Quotients, Powers, etc.

Solving Equations

Complex Numbers

3

Radicals

7.1

4

Square Roots

Finding Square Roots

32 = 9(-3)2 = 9 N.B. -32 = -9(½)2= (¼)

The square root of 9 is 3The square root of 9 is also –3The square root of (¼) is (½)

5

Square RootsThe square root symbol

Radical sign

The expression within is the radicand

Square Root

If a is a positive number, then

is the positive square root of a

is the negative square root of a

Also, 00 a

a

6

Approximating Square Roots


Perfect squares are numbers whose square roots are integers, for example 81 = 92.

Square roots of other numbers are irrational numbers, for example

We can approximate square roots with a calculator.

3,2

7


3.162 (Calculator)

We can determine that it is greater than 3 and less then 4 because 32 = 9 and 42 =16.

10

8

Cube Roots

2 is the cube root of 8 because 23 = 8.

8 and 23 above are radicands

3 is called the index (index 2 is omitted)

.

3 33 8 2 2

?27

?8

27

?27

3

3

3

9

Cube Roots Evaluated

2 is the cube root of 8 because 23 = 8.

8 and 23 above are radicands

3 is called the index (index 2 is omitted)

3 33 8 2 2

3

3

3

27 3

27 3

8 2

27 3

10

nth Roots

The number b is an nth root of a, , if

bn = a.

n a

11

nth Rootsnth Roots

An nth root of number a is a number whose nth power is a.

a number whose nth power is a

If the index n is even, then the radicand a must be nonnegative.

is not a real number

n a

4 4

5

16 2, 16

32 2

but

12

Radicals

7-8Page 397

13

Square Root of x2

7-7Page 393

xx 2

14

Product Rule for Radicals

7-9Page 398

15

Simplifying Radical Expressions

Product Rule –

nnn baba

5 425 25 2

54

7289

22

30310

12144436

1226436

yxxyxy

XXXXXX

kyky

16

Quotient Rule for Radicals

7-10Page 399

17


7-10Page 399

39

6

64

27

7

8

49

64

49

64

y

x

18


7-10Page 399

3

2

3 33

3 32

3 93

3 63

3 9

3 6

39

6

4

3

4

3

64

27

64

27

64

27

7

8

49

64

49

64

y

x

y

x

y

x

y

x

y

x

19

Radical Functions

Finding the domain of a square root function.

122)( xxf

20

Radical Functions

Finding the domain of a square root function.

( ) 2 12

' 2 12 0.

, | 6 .

f x x

Domain is all x s for which x

That is x x

21

Warm-Ups 7.1

22

7.1 T or F

1. T 6. F

2. F 7. F

3. T 8. F

4. F 9. T

5. T 10. T

23

Wind Chill

24

Wind Chill

25

Wind Chill

10.5 6.7 20 0.45 20 457 5 2591.4

110W

26

Wind Chill

10.5 6.7 20 0.45 20 457 5 2591.4

1103.56

W

27

Rational Exponents

7.2

28

Exponent 1/n When n Is Even

7-1Page 388

29

When n Is Even

definedyetnotis44

26464

5625625

10100100

2

1

66

1

44

1

2

1

30

Exponent 1/n When n Is Odd

7-2Page 389

31

Exponent 1/n When n Is Odd

2

1

32

1

32

1

32727

32727

55

1

33

1

33

1

32

nth Root of Zero

Page 389

0000 nn

33

Rational Exponents

7-4Page 390

34

Evaluating in Either Order

46488

4288

33 23

2

223

3

2

or

35

Negative Rational Exponents

7-5Page 391

36

Evaluating a - m/n

4

1

2

1

8

1

8

18 22

33

23

2

37

Rules for Rational Exponents

7-6Page 392

38

Simplifying

abba

yyy

3

1

2

1

6 66

16

39

Simplifying

3

2

2

3

13

11

2

1

113

1

2

1

3

1

2

1

ba

ba

babaabba

40

Simplifying

2

1121089 zyx

41

Simplifying

5

646542

1

2

112108 3

99y

zxzyxzyx

42

Simplified Form for Radicals of Index n

A radical expression of index n is in Simplified Radical Form if it has

1. No perfect nth powers as factors of the radicand,

2. No fractions inside the radical, and

3. No radicals in the denominator.

43

Warm-Ups 7.2

44

7.2 T or F

1. T 6. T

2. F 7. T

3. F 8. F

4. T 9. T

5. T 10. T

45

California Growing

1

1

n

P

Sr

46

Growth Rate

1

1

30

1

32.51 0.0165 1.65%

19.9

nSr

P

r

47

Operations with Radicals

7.3

48

Addition and Subtraction

Like Radicals

5555 2422223

2422223

49


Like Radicals

5322353223

2422223

24222235555

but

50

Simplifying Before Combining

188

51


25

2322

2924188

52


205

1

53


545

120

5

1

54


525

5

5

1

545

120

5

1

55


5

511

5

510

5

5

525

5

525

5

5

1

545

120

5

1

56


3 343 34 5416

5

51120

5

1

yxyx

57


3 333 333

3 333 333

3 343 34

227

28

5416

yxx

yxx

yxyx

58


yxxyxx

yxx

yxx

yxyx

3333

3 333 333

3 333 333

3 343 34

2322

227

28

5416

59


3

33

3333

3 333 333

3 333 333

3 343 34

2

2322

2322

227

28

5416

xxy

xxyxxy

yxxyxx

yxx

yxx

yxyx

60

Multiplying Radicals

Same index

7 457 37 32

10125423

1052

yxyxyx

61


Same index

4

2

4

3

7 457 37 32

42

10125423

1052

xx

yxyxyx

62


Same index

4

5

4

2

4

3

842

xxx

63


Same index

4

44 4

4

5

4

2

4

3

8842

xxxxx

64


Same index

4

4

4

44 4

4

5

4

2

4

3

8

8

842

xx

xx

xxx

65


Same index

4

4

4

4

4

44 4

4

5

4

2

4

3

2

2

8

8

842

xx

xx

xxx

66

Multiplying RadicalsSame index

4

4

4

4

4

4

4

5

4

2

4

3

16

2

2

2

8

842

xx

xx

xxx

67

Multiplying RadicalsSame index

2

2

16

2

2

2

8

842

4

4

4

4

4

4

4

4

5

4

2

4

3

xx

xx

xx

xxx

68

Multiplying Radicals - Binomials

1012610122354223

69

Multiplying Binomials

33333 10124354223

1012610122354223

70


542542

10124354223

1012610122354223

33333

71


2

33333

33

785162542542

10124354223

1012610122354223

x

72


366336933

785162542542

10124354223

1012610122354223

2

33333

xxxxx

conjugates

73

Multiplying Radicals – Different Indices

32

82222222

3

44 34

3

2

1

4

1

2

1

4

14

74

Multiplying RadicalsDifferent Indices

2

1

3

13

44 34

3

2

1

4

1

2

1

4

14

3232

82222222

75

Different Indices

6

3

6

2

2

1

3

13

44 34

3

2

1

4

1

2

1

4

14

323232

82222222

76

Different Indices

6 36 26

3

6

2

2

1

3

13

44 34

3

2

1

4

1

2

1

4

14

32323232

82222222

77

Different Indices

6

66

6 36 26

3

6

2

2

1

3

13

44 34

3

2

1

4

1

2

1

4

14

108

274

32323232

82222222

78

Conjugates

723723

1323232

22 yxyxyx

79

Conjugates

19749723723

1323232

22

yxyxyx

80

Warm-Ups 7.3

81

7.3 T or F

1. F 6. F

2. T 7. T

3. F 8. F

4. F 9. F

5. T 10. T

82

Area of a Triangle

83

Area of a Triangle

bhA2

1

84

Area of a Triangle

2536302

12

1

mA

bhA

85

Quotients, Powers, etc

7.4

86

Dividing Radicals

25

10

5

10510

87

Dividing Radicals

25

25

5

50

5

5

5

10510

25

10

5

10510

2

or

88

Dividing Radicals

53

2

532

2

526

2

206

89

Rationalizing the Denominator

62

32

90


62

62

62

32

91


62

1866222

62

62

62

32

92


4

6325

4

236322

62

1866222

62

62

62

32

93

Powers of Radical Expressions

33

444

42

7299813333

yy

94

Powers of Radical Expressions

4333333

3

444

32484242

7299813333

yyyyyyy

95

Warm-Ups 7.4

96

7.4 T or F

1. T 6. T

2. T 7. F

3. F 8. T

4. T 9. T

5. F 10. T

97

7.4 #102

3 36 1 6 1

6 1 6 1

6 1 6 13 3

6 1 6 1 6 1 6 1

3 6 1 3 6 1

5 5

3 6 3 3 6 3

5 5

3 6 3 3 6 3 2 3

5 5

LCD

98

Adding Fractions

xx

x 5

3.110

99

Adding Fractions

xxLCDxx

x3

5

3.110

100

Solving Equations

7.5

101

Solving Equations

The Odd Root Property

If n is an odd positive integer,

for any real number k.

nn kxkx

102

Solving Equations – Odd Powers




nn kxkx

28

83

3

x

x

103





nn kxkx

327

273

3

x

x

104





nn kxkx

3

33 3

3

1 54

1 54 27 2

1 3 2

x

x

x

105

Even-Root Property

7-11Page 419

106

Even-Root Property

7-11Page 419

2

2

2

4 2

0 0

4

x x

x x

x has no real solution

107

Solving Equations – Even PowersThe Even Root Property

If n is an even positive integer,

2

0

0 0

0 .

4

4 2 2,2

n n

n

n

k x k x k

k x k x

k x k has no real solution

x

x

108



4

4

44 4

0

0 0

0 .

1 80

81

81

3 3,3

n n

n

n

k x k x k

k x k x


x

x

x

x CHECK

109



2

2

0

0 0

0 .

3 4

3 4

3 2

3 2

n n

n

n

k x k x k

k x k x


x

x

x

x

110



2

0

0 0

0 .

3 4

3 2

3 2 5 3 2 1 1,5

n n

n

n

k x k x k

k x k x


x

x

x x CHECK

111

Isolating the Radical

2532

532

532

0532

22

x

sidesbothSquarex

radicaltheIsolatex

x

112

Squaring Both Sides

22

2 3 5 0

2 3 5

2 3 5

2 3 25

2 28

14 14

x

x Isolate the radical

x Square both sides

x

x

x CHECK

113

Cubing Both Sides

3 3

3 33 3

3 2 7

3 2 7

3 2 7

10 10

a a

a a

a a

a CHECK

114

Squaring Both Sides Twice

22112

112

112

xx

xx

xx

115


xxx

xx

xxx

xx

xx

xx

42

2

1212

112

112

112

22

22

116


xxx

xx

xxx

xx

xx

xx

42

2

1212

112

112

112

22

22

2

2

4

4 0

4 0

0 4

0,4

x x

x x

x x

x x

CHECK

117

Rational ExponentsEliminate the root, then the power

23

2

a

118

Eliminate the Root, Then the Power

2

3

3233

2

2

2

2

8

8

2 2 2 2,2 2

a

a

a

a

a

CHECK

119

Negative Exponents

11 3

2

r

120

Negative ExponentsEliminate the root, then the power

2

3

3233

2

2

1 1

1 1

1 1

1 1

1 1

2 0 0,2

r

r

r

r

r

r r

CHECK

121

Negative ExponentsEliminate the root, then the power

132 3

2

t

122

No SolutionEliminate the root, then the power

132

132

132

132

2

2

33

3

2

3

2

t

t

t

t

123

No SolutionEliminate the root, then the power

2

3

323

3

2

2

2 3 1

2 3 1

2 3 1

2 3 1

t

t

t

t

No real solution

124

Strategy for Solving Equations with Exponents and Radicals

7-12Page 424

125

Distance Formula

7-13Page 424

Pythagorean Theorem a2 + b2 = c2

126

Distance Formula

7-13Page 424

Find the distance between the points (-2,3) and (1, -4).

22 3421 d

127

Distance Formula

7-13Page 424

Find the distance between the points (-2,3) and (1,-4).

58499

73

3421

22

22

d

d

d

128

Diagonal of a SignWhat is the length of the diagonal of a rectangular

billboard whose sides are 5 meters and 12 meters?

222 cba

lengthdiagonalxLet

129

Diagonal of a SignWhat is the length of the diagonal of a rectangular

billboard whose sides are 5 meters and 12 meters?

169

14425

125

2

2

222

222

x

x

x

cba

lengthdiagonalxLet

130

Diagonal of a SignWhat is the length of the diagonal of a rectangular billboard whose

sides are 5 meters and 12 meters?

2 2 2

2 2 2

2

2

2

5 12

25 144

169

169

13 13

13 .

Let x diagonal length

a b c

x

x

x

x

x or CHECK

The diagonal is meters

131

Warm-Ups 7.5

132

7.5 T or F

1. F 6. F

2. T 7. F

3. F 8. T

4. F 9. T

5. T 10. T

133

Complex Numbers

7.6

134

Imaginary Numbers

bibiiDefine ,1,12

135

Imaginary Numbers

?81

10100100

,1,12

ii

bibiiDefine

136

Imaginary Numbers

ii

ii

bibiiDefine

98181

10100100

,1,12

137

Imaginary Numbers

2

2

1 , 1 ,

4 9 4 9 2 3 6 6 1 6

4 9 4 9 36 6

Beware

Define i i b i b

i i i i i

138

Imaginary Numbers

2 1 , 1 ,

2 8 ?

Define i i b i b

139

Imaginary Numbers

4

14

4

16

8282

,1,1

2

2

2

i

i

ii

bibiiDefine

140

Powers of i

.

1

111

1

1

1

145

224

23

2

1

etc

iiiii

iii

iiiii

i

ii

141

Complex Numbers

7-14Page 429

142

Figure 7.3

7-15Page 430 (Figure 7.3)

143


The sum and difference a + bi of c + di and are:

(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) - (c + di) = (a - c) + (b - d)i

144

(2 + 3i) + (4 + 5i)

The sum and difference a + bi of c + di and are:

(2 + 3i) + (4 + 5i) = (2 + 4) + (3 + 5)i

= 6 + 8i

(2 + 3i) – (4 + 5i) = (2 – 4) + (3 – 5)i

= – 2 – 2i

145

Multiplication

The complex numbers a + bi of c + di and are multiplied as follows:

(a + bi) (c + di) = ac + adi + bci + bdi2

= ac + bd(– 1) + adi + bci

= (ac – bd) + (ad + bc)i

146

(2 + 3i) (4 + 5i)

The complex numbers a + bi of c + di and are multiplied as follows:

(a + bi) (c + di) = (ac – bd) + (ad + bc)i

(2 + 3i) (4 + 5i) = 8 + 10i + 12i + 15i2

= 8 + 22i + 15(– 1)

= – 7 + 22i

147

Division

(2 + 3i) ÷ 4 = (2 + 3i) / 4

= ½ + ¾ i

148

Complex Conjugates

The complex numbers a + bi and a – bi are called complex conjugates. Their product is a2 + b2.

149

Division

We divide the complex number a + bi by the complex number c + di as follows:

dicdic

dicbia

dic

bia

150

Division

We divide the complex number a + bi by the complex number c + di as follows:

i

i

54

32

151

DivisionWe divide the complex number 2 + 3i by the

complex number 4 + 5i.

iii

i

iii

ii

ii

i

i

41

2

41

23

41

223

2516

2232516

1512108

5454

5432

54

32

2

2

152

Square Root of a Negative Number

For any positive real number b,

.bib

153

Imaginary Solutions to Equations

04325 22 xx

154

Complex Numbers1. Definition of i: i = , i2 = -1.

2. Complex number form: a + bi.

3. a + 0i is the real number a.

4. b is a positive real number

5. The numbers a + bi and a - bi are complex conjugates. Their product is a2 + b2.

6. Add, subtract, and multiply complex numbers as if they were algebraic expressions with i being the variable, and replace i2 by -1.

7. Divide complex numbers by multiplying the numerator and denominator by the conjugate of the denominator.

8. In the complex number system x2 = k for any real number k is equivalent to

.bib

.kx

155

Complex Numbers

156

Warm-Ups 7.6

157

7.6 T or F

1. T 6. T

2. F 7. T

3. F 8. F

4. T 9. T

5. T 10. F

1 roots & radicals intermediate algebra. 2 roots and radicals radicals rational exponents...

Documents