Chapter 4dgriffinresources.net/m102LN_Ch4.pdf · Chapter 4 Radical Expressions ... subtract, multiply complex numbers, rationalize denominators of complex numbers, powers of i ...

Math 102, Intermediate Algebra Name___________________________________ Author: Debra Griffin Circle one: Red Bluff Main Campus Burney Weaverville

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Chapter 4 Radical Expressions 4.1 Review of Powers and Roots 2

4.2 Simplifying Radicals 5 4.3 Operations with Radical Expressions 12 Multiplying/dividing, adding/subtracting radical expressions, rationalizing denominators

4.4 Rational Exponents 18 Convert between rational exponents and radical notation, simplify rational exponent expressions, simplify radical expressions

4.5 Equation Solving 26 Solve equations with radicals, applications of the Pythagorean Theorem

4.6 Complex Numbers 33 Simplifying complex numbers in radical form, add, subtract, multiply complex numbers, rationalize denominators of complex numbers, powers of i Lecture Note-Taking Guide Scoring Rubric Completion of these lecture notes for extra credit points is optional. Because it is an extra credit option, the expectation of work quality is very high. There is no partial extra credit for this assignment. Any of the following criteria that are not met will result in no extra credit. Extra Credit Points Condition

5 • complete • all worksteps shown • correct answers • neatly done in pencil • correctly ordered

• hole punched • fastened in folders with

fasteners • turned in on time • labeled with name and

site

0 Any of the required criteria have not been met

Math 102, Intermediate Algebra Section 4.1

Page 2 of 39

4.1 Review of Powers and Roots

Using Exponential Notation

Consider the expression 5 • 5 • 5

We can represent this product with

the following shorthand notation: The exponent, 3, indicates how many times the base, 5, is used as a factor.

We say 53 is written in exponential notation.

Demonstration Problems Practice Problems Write using exponential notation.

1. (a) 8 • 8 • 8 • 8 • 8

2. (a)

€

23

•23

•23• 23

Evaluate 3. (a) (–1.2)2

4. (a) – 12

⎛⎝⎜

⎞⎠⎟5

Write using exponential notation.

1. (b) 7 • 7 • 7 • 7

2. (b)

€

14

•14

•14

Evaluate 3. (b) (–1.1)2

4. (b) – 25

⎛⎝⎜

⎞⎠⎟3

Answers: 1. (b) 74; 2. (b) ( 14

)3; 3. (b) 1.21; 4. (b) – 8125

exponent

base

53 = 5 • 5 • 5


Page 3 of 39

Square Roots a2 = a

Note that a is not a real number for a < 0. Problem Solution Answer Example (a) Calculate 9 9 = 32 = 3 9 = 3

Example (b) Calculate 49

49= 2

3⎛⎝⎜

⎞⎠⎟2

= 23

49= 23

Example (c) Calculate 3Round to 4 decimal places

In a calculator, press ⇒ 3⇒ = /enter 3 ≈ 1.7320508075688…

3 ≈ 1.7321

Example (d) Calculate −9 There is no real number such that its square is –9.

−9 is not a real number

Demonstration Problems Practice Problems

Calculate (round to 4 decimal places if necessary) 5. (a) 25

6. (a) 149

7. (a) 24

8. (a) − 4

Calculate (round to 4 decimal places if necessary) 5. (b) 36

6. (b) 9100

7. (b) 5 8. (b) − 9

Answers: 5. (b) 6; 6. (b) 310

; 7. (b) 2.2361; 8. (b) –3


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nth Roots

Problem Solution Answer Example (a) Calculate 83 83 = 233 = 2 83 = 2

Example (b) Calculate 132

5 132

5 = 12

⎛⎝⎜

⎞⎠⎟5

5 = 12

132

5 = 12

Example (c) Calculate −273 −273 = (−3)33 = −3 −273 = −3

Example (d) Calculate 404 . Round to 4 decimal places

In a calculator, press 4 ⇒ ⇒ 40 ⇒ = /enter 404 ≈ 2.514866859…

404 ≈ 2.5149


Calculate (round to 4 decimal places if necessary) 9. (a) 814

10. (a) − 8125

3

11. (a) 203

12. (a) 16

Calculate (round to 4 decimal places if necessary) 9. (b) 646

10. (b) − 18

3

11. (b) 503 12. (b) −15

Answers: 9. (b) 2; 10. (b) − 12

; 11. (b) 3.6840; 12. (b) –1

Index Radical Sign

Radicand

(when n is even, )


Page 5 of 39

4.2 Simplifying Radicals

We say that a square root radical expression is in simplest form when

1. its radicand has no square factors 2. the radicand is not a fraction 3. no denominator contains a radical

Example (a) Which of the following radicals are in simplest form?

(i) 33 (ii) 20 (iii) 13

(iv) 15

33 = 3•11 The radicand has no square factors, so the radical

is in simplest form.

20 = 4 • 5 4 is a square factor of the radicand, so the radical

is not in simplest form.

The radicand is a fraction, so the radical


The denominator contains a radical, so the expression


To simplify square root radicals we can use the following rules of square roots:

Product Rule Quotient Rule

ab = a • b (for a ≥ 0 and b ≥ 0) ab= a

b (for a ≥ 0 and b > 0)

Example (b) Simplify 20 .

20 = 22 • 5

= 22 • 5

= 2 • 5

= 2 5

Example (c) Simplify 180 .

180 = 22 • 32 • 5

= 22 • 32 • 5

= 2 • 3• 5

= 6 5


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Demonstration Problems Practice Problems Simplify

1. (a) 28 2. (a) 50 3. (a) 300

4. (a) 15049

Simplify

1. (b) 40 2. (b) 75 3. (b) 500

4. (b) 1759

Answers: 1. (b) 2 10 ; 2. (b) 5 3 ; 3. (b) 10 5 ; 4. (b) 5 73


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In general, an nth root radical is considered to be in simplest form when

1. its radicand has no factors raised to a power greater than or equal to the index 2. the radicand is not a fraction 3. no denominator contains a radical

Example (d) Which of the following radicals are in simplest form?

(i) 183 (ii) 484 (iii) 14

5 (iv) 5 75

34

183 = 2 • 323 The radicand has no cube factors, so the radical

is in simplest form.

484 = 24 • 34 24 is 4th power factor in the radicand, so the radical


The radicand is a fraction, so the radical


The denominator contains a radical, so the radical


To simplify nth root radicals we can use the following rules of nth roots:

Product Rule Quotient Rule

abn = an • bn (for a ≥ 0 and b ≥ 0 when n is even)

ab

n = an

bn

(for a ≥ 0 and b > 0 when n is even)

Example (e) Simplify 484 .

484 = 24 • 34

= 244 • 34

= 2 • 34

= 2 34

Example (f ) Simplify −645 .

−645 = −1• 265

= −1• 25 • 25

= −15 • 255 • 25

= −1• 2 • 25

= −2 25


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Simplify 5. (a) 403 6. (a) −813

7. (a) 1124

8. (a) − 965

Simplify 5. (b) 543 6. (b) −2503 7. (b) 1624 8. (b) − 1605

Answers: 5. (b) 3 23 ; 6. (b) −5 23 ; 7. (b) 3 24 ; 8. (b) −2 55


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Variable Radical Expressions To simplify a radical that contains variables, we continue to use the rules: a2 = a (for a ≥ 0) ann = a (for a ≥ 0 when n is even)

abn = an • bn (for a ≥ 0 and b ≥ 0 when n is even)

ab

n = an

bn (for a ≥ 0 and b > 0 when n is even)

In all of the exercises in this chapter, assume all variables represent positive real numbers.

Example (g) Simplify 8y3 .

8y3 = 23y3

= 22 • 2 • y2 • y

= 22 • y2 • 2 • y

= 22 • y2 • 2 • y

= 2 • y • 2 • y

= 2y 2y

Example (h) Simplify x83 .

x83 = x3 • x3 • x23

= x33 • x33 • x23

= x • x • x23 = x2 x23

Alternative method:

x83 = x6 • x23

= x2( )3 • x23

= x2( )33 • x23

= x2 x23


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Example (i) Simplify 9a2

49b2.

9a2

49b2= 9a2

49b2

= 32a2

72b2

= 32 • a2

72 • b2

= 3a7b


Simplify 9. (a) 36a2b2

10. (a) 25x2

Simplify 9. (b) 4x4

10. (b) 8x3

3

Answers: 9. (b) 2x2; 10. (b) 2x ;


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Demonstration Problems Practice Problems Simplify 11. (a) 27x53 12. (a) 40x5y6z8

13. (a) a5

16b84

Simplify 11. (b) 8a43

12. (b) 90x4y5z6

13. (b) a6

27b123

Answer: 11. (b) 2a a3 ; 12. (b) 3x2y2z3 10y ; 13. (b) a2

3b4


Page 12 of 39

4.3 Operations with Radical Expressions

To simplify a radical expression that contains variables and any of the operations +, – , ×, or ÷, we continue to use the rules introduced in the previous section with one more:

Definition of square root a2 = a (for a ≥ 0)

Definition of nth root ann = a (for a ≥ 0 when n is even) Product Rule abn = an • bn (for a ≥ 0 and b ≥ 0 when n is even)

Quotient Rule

ab

n = an

bn (for a ≥ 0 when n is even and b ≠ 0

always)

Distributive Property a x + b x = (a + b) x In all of the exercises in this chapter, assume all variables represent positive real numbers.

Example (a) Simplify 7 • 7 .

7 • 7 = 49

= 72

= 7

Example (b) Simplify 9a4b23 • 24a5b83 .

9a4b23 • 24a5b83 = 9a4b2 • 24a5b83

= 216a9b103

= 23 • 33 • a9 •b9 •b3

= 233 • 333 • a93 • b93 • b3

= 2 • 3• a3 •b3 • b3

= 6a3b3 b3

Example (c) Simplify 4812

.

4812

= 4812

= 4

= 2


Page 13 of 39


1. (a) 50 • 2 2. (a) 2xy3 • 54xy

3. (a) 50010

4. (a) 75x2y48x4y9

Simplify

1. (b) 40 • 10 2. (b) 3x3y4 • 24xy

3. (b) 3507

4. (b) 72x6y8y

Answers: 1. (b) 20; 2. (b) 6x2y2 2y ; 3. (b) 5 2 ; 4. (b) 3x3


Page 14 of 39

Example (d) Simplify 5 3 − 3 . 5 3 − 3 = 5 3 −1 3 = (5 −1) 3 = 4 3

Example (e) Simplify 5 x − x + x2

.

5 x − x + x2

= 5 x1

− x1

+ x2

= 5 x1• 22− x1• 22+ x2

= 10 x2

− 2 x2

+ 1 x2

= 9 x2

Example (f ) Simplify 8 12 − 4 27 .

8 12 − 4 27 = 8 4 • 3 − 4 9 • 3

= 8 • 2 3 − 4 • 3 3

= 16 3 −12 3

= 4 3

Example (g) Simplify x 50x + 2x3 .

x 50x + 2x3 = x 25 • 2x + 2x2 • x

= x • 5 2x + x 2x

= 5x 2x +1x 2x

= (5 +1)x 2x

= 6x 2x


Page 15 of 39


Simplify

5. (a) 5 + 52

− 54

6. (a) 2 40 + 90

7. (a) 3 x4y3 + 2x xy3 Multiply and simplify

8. (a) 3+ 2( ) 5 − 3 2( )

9. (a) 3 + 2 5( ) 3 − 2 5( )

Simplify

5. (b) 3 − 35

+ 325

6. (b) 2 45 + 6 20 7. (b) 5y xy3 − xy43 Multiply and simplify

8. (b) 6 + 3( ) 4 − 3( )

9. (b) 5 + 11( ) 5 − 11( )

Answers: 5. (b) 21 325

; 6. (b) 18 5 ; 7. (b) 4y xy3 ; 8. (b) 21− 2 3 ; 9. (b) –6


Page 16 of 39

Rationalization Recall that a simplified radical expression cannot contain a radical in any denominator. There is a process to simplify such a radical called rationalization of the denominator. The process refers to manipulating an expression until the denominator contains only rational numbers. This process is used also to remove radical signs from variables in the denominator.

Example (h) Simplify 23

.

23= 2

3• 33

= 2 33 • 3

= 2 332

= 2 33

Recall that (a + b)(a – b) = a2 – b2 and a2 = a . These two properties allow us to simplify a radical denominator of the form a b + c d .

Notice that a b + c d( ) a b − c d( ) = a2b − ac bd + ac bd − c d • c d

= a2b − c2 d 2

= a2b − c2d

We call a b − c d the conjugate of a b + c d .

Example (i) Simplify 13+ 2

.

13+ 2

= 13+ 2

• 3− 23− 2

=1• 3− 2( )

3+ 2( ) 3− 2( )

= 3− 29 – 3• 2 + 3• 2 – 2 • 2

= 3− 29 – 2

= 3− 27


Page 17 of 39


10. (a) x2y

11. (a) 12 − 5

12. (a) x1+ 2 y

Simplify

10. (b) 53x

11. (b) 110 − 2

12. (b) a5 + 3 b

Answer: 10. (b) 5 3x3x

; 11. (b) 10 + 26 ; 12. (b)

5 a − 3 ab25 − 9b


Page 18 of 39

4.4 Rational Exponents

Here is a review of exponent rules:

Rules for Exponents Examples Product Rule am • an = am+n 1. x12 • x3 =

Quotient Rule

€

am

an= am−n (a ≠ 0)

2.

€

x16

x 7 =

Power Rules

(am )n = amn (ab)m = am bm

€

ab"

# $ %

& ' m

=am

bm (b ≠ 0)

3. (x4)5 = 4. (2x)3 = 5.

€

2x"

# $ %

& ' 4

=

Zero Exponent a0 = 1 (a ≠ 0) 8.

€

35

35 = 9. (2xy)0 =

Negative Exponent a−n = 1

an (a ≠ 0)

10. x –5 =

Let’s explore using rule am • an = am+n. Fill in the boxes below.

From the inquiry above, it seems reasonable that since

312 = 3

we can then define

a12 = a (a ≥ 0)

33 • 33 = 36

3… . • 3… . = 34

3…. • 3…. = 32

3…. • 3…. = 31

Compare with:

• = 3

⇐ This uses the rule am • an = am+n. ⇐ Think: £ + £ = 4. ⇐ Think: £ + £ = 2. ⇐ Think: £ + £ = 1.


Page 19 of 39

Let’s try another example

And now, it seems reasonable that since

513 = 53

that we can define

a13 = a3

In fact, we now have a new general rule to add to our rules of exponents list:

a1n = an (for a ≥ 0 when n is even)

Example (a) Write 11with fractional exponents.

11 = 1112

Example (b) Write x3 with fractional exponents.

x3 = x3( )12

= x3 •

12

= x32

From example (b), we can see that we can generalize our new rule as:

amn = amn (a ≥ 0 when n is even)

53 • 53 • 53 = 59

5… . • 5… • 5… = 56

5…. • 5… • 5… = 53

5…. • 5… • 5… = 51

Compare with:

• • = 5

⇐ Think: £ + £ + £ = 6. ⇐ Think: £ + £ + £ = 3. ⇐ Think: £ + £ + £ = 1.


Page 20 of 39

Demonstration Problems Practice Problems Write with rational exponents

1. (a) 6 2. (a) 53 3. (a) x5

4. (a) 1x3

Simplify

1. (b) 5 2. (b) 104 3. (b) x53

4. (b) 1a5

Answers: 1. (b) 512 ; 2. (b) 10

14 ; 3. (b) x

53 ; 4. (b) a− 1

5


Page 21 of 39

Example (c) Simplify 1634 .

1634 = 24( )

34

= 24 •

34

= 23 = 8

Example (d) Simplify 16−

34 .

16−

34 = 1

1634

= 18

Example (e) Write 534 in radical notation.

534 = 534

Demonstration Problems Practice Problems Simplify.

5. (a) 2713

6. (a) 164

⎛⎝⎜

⎞⎠⎟

13

Simplify.

5. (b) 3215

6. (b) 164

⎛⎝⎜

⎞⎠⎟

12

Answers: 5. (b) 2; 6. (b) 18


Page 22 of 39


Simplify.

7. (a) 6423

8. (a) y4( )34

Write in radical notation

9. (a) y34

Simplify.

7. (b) 1632

8. (b) a3( )53

Write in radical notation

9. (b) (ab)14

Answers: 7. (b) 64; 8. (b) a5; 9. (b) ab4

Example (f ) Use properties of exponents to simplify x8( )12 .

x8( )12 = x

8 • 12

= x4

Example (g) Use properties of exponents to simplify x14 • x

− 17 .

x14 • x

− 17 = x

14

− 17

= x7

28 −

428

= x3

28


Page 23 of 39

Demonstration Problems Practice Problems Use properties of exponents to simplify

10. (a) x5y10( )15

11. (a) x34 x

54

12. (a) x− 34 x

54

13. (a) x12

x14

Use properties of exponents to simplify

10. (b) w2z6( )12

11. (b) a35a

25

12. (b) a−25a

35

13. (b) a23

a13

Answer: 10. (b) wz3; 11. (b) a; 12. (b) a15 ; 13. (b) a

13


Page 24 of 39

Example (h) Use rational exponents to simplify 86 .

86 = 236

= 236

= 212

= 2

Example (i) Use properties of exponents to simplify 49x24 .

49x24 = 72 x24

= 72 x2( )14

= 72•14 • x

2•14

= 712 • x

12

= 7 • x

7x

Example (j) Use rational exponents to write 84 • 25 as a single radical.

84 • 25 = 234 • 25

= 234 • 2

15

= 21520

+ 420

= 21920

= 21920

Example (k) Use rational exponents to write x345

as a single radical.

x345 = x3( )14

⎛⎝⎜

⎞⎠⎟

15

= x3 •

14

• 15

= x3

20


Page 25 of 39


Use rational exponents to simplify 14. (a) 25x8 15. (a) 16x4y124 Use rational exponents to write each expression as a single radical

16. (a) x23 • x

17. (a) x33

Use rational exponents to simplify 14. (b) 49a4 15. (b) 8x6y123

Use rational exponents to write each expression as a single radical

16. (b) a23 • a5

17. (b) a45

Answer: 14. (b) 7a2; 15. (b) 2x2y4; 16. (b) a1315 ; 17. (b) a25


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4.5 Solving Equations with Radicals

Let’s explore.

Suppose x = 4 .

What number has the square root, 4?

We know 16 = 4 .

So then x = 16.

We could have solved this equation using the Power Rule for Solving Equations.

For any real numbers, a, b, and n, such that an is a real number,

if a = b, then an = bn

With this rule, we could have solved the first equation as follows:

x = 4

x( )2 = 42

x( ) x( ) = 16

x2( ) = 16

x = 16

In general, to solve an equation that contains radicals,

Step 1: Isolate the radical.

Step 2: Apply the power rule.

Step 3: Solve the resulting equation. If it still contains a radical repeat steps 1 and 2.

Step 4: Check all solutions for extraneous solutions.


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Example (a) Solve 2x −1 = 9

Example (b) Solve 2x + 8 − x = 0

2x + 8 − x = 0 +x +x 2x + 8 + 0 = x

2x + 8( )2 = x2

2x + 8 = x2 –2x –2x 0 + 8 = x2 – 2x –8 –8 0 = x2 – 2x – 8 0 = (x – 4)(x + 2) x – 4 = 0 or x + 2 = 0 +4 +4 –2 –2 x + 0 = 4 x + 0 = –2 x = 4 or x = –2

Example (c) Solve 5x − 23 = 2

2x – 1 = 81 +1 +1 2x + 0 = 82

x = 41

Check 4:

?

?

?

?

0 = 0 Yes

Check –2:

?

?

?

? 4 = 0? No

Solution set: {4}

Solution set: {41}

Check 41:

?

?

? Yes.

Solution set: {2}

5x – 2 = 8 +2 +2 5x + 0 = 10

x = 2

Check 2:

?

?

? Yes.


Page 28 of 39

Example (d) Solve 5x + 6 + 3x + 4 = 2

5x + 6 + 3x + 4 = 2

− 3x + 4 − 3x + 4 5x + 6 + 0 = 2 − 3x + 4

5x + 6( )2 = 2 − 3x + 4( )2

5x + 6 = 2 − 3x + 4( ) 2 − 3x + 4( )

5x + 6 = 4 − 2 3x + 4 − 2 3x + 4 + 3x + 4( )2

5x + 6 = 4 − 4 3x + 4 + 3x + 4 5x + 6 = −4 3x + 4 + 3x + 8 –3x –3x 2x + 6 = −4 3x + 4 + 0 + 8 –8 –8 2x − 2 = −4 3x + 4 + 0

2x − 2( )2 = −4 3x + 4( )2

2x − 2( ) 2x − 2( ) = −4( )2 3x + 4( )2

4x2 – 4x – 4x + 4 = 16(3x + 4) 4x2 – 8x + 4 = 48x + 64 –48x –48x 4x2 – 56x + 4 = 0 + 64 –64 –64 4x2 – 56x – 60 = 0 4(x2 – 14x – 15) = 0 4(x – 15)(x + 1) = 0

4(x −15)(x +1)4

= 04

(x – 15)(x + 1) = 0 x – 15 = 0 or x + 1 = 0

+15 +15 –1 –1 x + 0 = 15 or x + 0 = –1

x = 15 or x = –1

Check 15:

? ? ? 9 + 7 = 2? No

Check –1:

?

? ? 1 + 1 = 2? Yes

Solution set: {–1}


Page 29 of 39

Demonstration Problems Practice Problems Solve.

1. (a) x + 2 + 2 = 5 2. (a) x +13 = 1 3. (a) 12 − x = x

Solve.

1. (b) x − 3 +1= 4 2. (b) x − 23 = 2 3. (b) 4x + 5 = x

Answers: 1. (b) {12}; 2. (b) {10}; 3. (b) {5}


Page 30 of 39

Demonstration Problems Practice Problems Simplify. 4. (a) x − 4 = x − 2 5. (a) 3x + 4 = −5

Simplify. 4. (b) x + 3 = x +1 5. (b) 2x + 5 = −1

Answers: 4. (b) {1}; 5. (b) ∅


Page 31 of 39

Pythagorean Theorem

By the Pythagorean Theorem, in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. That is,

a2 + b2 = c2

We can use this theorem to find the lengths of unknown sides in a right triangle, when two side lengths are given.

Example (e) Find the length of the unknown side of the right triangle.

By the Pythagorean Theorem, we have that a2 + b2 = c2

a2 + 102 = 202

a2 + 100 = 400

–100 –100 a2 + 0 = 300

a2 = 300

a = 300

a = 100 • 3

a = 10 3

or a ≈ 17.3

10

20

10

20

The length of the unknown side is approximately 17.3


Page 32 of 39

Demonstration Problems Practice Problems 6. (a) Find the length of the unknown side.

6. (b) Find the length of the unknown side.

Answers: 6. (b) 2 11

10

12

7

9


Page 33 of 39

4.6 Complex Numbers

Common Number Sets

Symbol Name Description Notation or examples

Natural

Numbers The counting numbers beginning with 1. = {1, 2, 3, …}

Whole Numbers The counting numbers and zero. {0, 1, 2, 3, …}

Integers The whole numbers and their opposites.

={… –3, –2, –1, 0, 1, 2, 3, …} Z comes from the German word for number, zahlen, since I is used for Imaginary Numbers.

Rational Numbers

Numbers of the form ab

where a is

an integer and b is a nonzero integer.

Examples: 13

, 10.5, –132

, 12, 4

Q is for quotient, since R is used for Real Numbers.

Irrational Numbers

Real numbers that are not rational numbers.

Examples: π, 2 , 5 15

Real

Numbers All rational and irrational numbers. All of the above.

At the right is a Venn Diagram of the number sets from above. The diagram shows that natural numbers are contained in whole numbers which are contained in the integers, which are contained in the rational numbers. The irrational numbers are disjoint from the rational numbers. The real numbers are made up of both the rational and irrational numbers.

Until now, we have been using these number sets to solve equations. But consider the equation

x2 = − 4 What number has the square, – 4?

We know 22 = 4 and (–2)2 = 4. But no real number squared equals – 4.

Using real numbers only the solution set to this equation is empty.


Page 34 of 39

An Italian mathematician Gerolamo Cardan (1501, 1576) was curious about solutions to cubic polynomial equations of the form ax3 + bx + c = 0. Determined not to let the problem of square roots of negative numbers stop him, he developed the concept of −1 . Today, complex numbers have real world applications in many fields, two of which are electronics and electrical engineering.

We define

i = −1

Complex numbers = {a + bi | a ∈ , b ∈ , and i = −1 }. When b = 0, then the complex number becomes a real number. Thus, the complex numbers include all of the real numbers.

We can now extend

the number set Venn Diagram to

Example (a) Write −9 as a product of a real number and i.

−9 = −1• 9 = −1 • 9 = i • 3 = 3i

Example (b) Write −20 as a product of a real number and i.

−20 = −1• 22 • 5 = −1 • 22 • 5 = i • 2 • 5 = 2i 5


Page 35 of 39


Write as a product of a real number and i.

1. (a) −36 2. (a) −40 Assume x ∈ .

3. (a) −x6

Write as a product of a real number and i.

1. (b) −49 2. (b) −50 Assume x ∈ .

3. (b) −x8

Answers: 1. (b) 7i; 2. (b) 5i 2 ; 3. (b) x4i


Page 36 of 39

Example (c) Simplify (2 + 3i) + (5 – 4i). Write your answer in the form a + bi. (2 + 3i) + (5 – 4i) = 2 + 3i + 5 + –4i = 2 + 5 + 3i + –4i = 7 + –i = 7 – i Since i = −1 , then i2 = −1( )2 = −1 .

Example (d) Simplify (2 + 3i)(5 – 4i). Write your answer in the form a + bi. (2 + 3i)(5 – 4i) = 10 + –8i + 15i + –12i2 = 10 + 7i + –12(–1) = 10 + 7i + 12 = 22 + 7i

Demonstration Problems Practice Problems Simplify. 4. (a) (5 + 2i) – (3 – i) 5. (a) (1 + 4i)(1 – 4i)

Simplify. 4. (b) (4 + 6i) – (2 – 3i) 5. (b) (2 + 3i)(2 – 3i)

Answers: 4. (b) 2 + 9i; 5. (b) 13


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Example (e) Rationalize the denominator and simplify 35i

.

35i

= 35i• ii

= 3i5i2

= 3i5(−1)

= 3i−5

= − 35i

Example (f ) Rationalize the denominator and simplify 13+ i

.

Write your answer in the form a + bi.

13+ i

= 13+ i

• 3− i3− i

= 1(3− i)(3+ i)(3− i)

= 3− i9 − 3i + 3i − i2

= 3− i9 − (−1)

= 3− i10

= 310

− 110i


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Demonstration Problems Practice Problems Rationalize the denominator and simplify. Write your answer in the form a + bi.

6. (a) 13i

7. (a) 1+ 2i1− 2i

Rationalize the denominator and simplify. Write your answer in the form a + bi.

6. (b) 25i

7. (b) 1+ 2i1+ 3i

Answers: 6. (b) −25i ; 7. (b)

710

− 110i


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Powers of i Complete the following:

i = i

i2 = –1

i3 = i2 • i = –1 • i = –i

i4 = i2 • i2 = (–1)(–1) = 1

i5 = i4 • i = = i

i6 = i2 • i2 • i2 = =

i7 = i6 • i = =

i8 = i2 • i2 • i2 • i2 = =

i9 = i8 • i = =

i10 = i2 • i2 • i2 • i2 • i2 = =


Simplify 8. (a) i12 = 9. (a) i85 =

Simplify 8. (b) i15

9. (b) i90 =

Answers: 8. (b) –i; 9. (b) –1

Chapter 4dgriffinresources.net/m102LN_Ch4.pdf · Chapter 4 Radical Expressions ... subtract, multiply complex numbers, rationalize denominators of complex numbers, powers of i ...

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