WARM-UPFind the prime factorization for the following numbers. Write your answer as a product of primes.

1. 72 2. 120

5.1a Rational Exponents & Simplify Radicals

n n2 n3 n4 n5 n6 n7 n8

2 4 8 16 32 64 128 256

3 9 27 81 243 729 2187

4 16 64 256 1024

5 25 125 625

6 36 216 1269

7 49 343 2401

8 64 512

9 81 729

Objective: To simplify rational exponents

Radicalsn radicandRoot index

even10

86

42

xx

xx

xx

What should you do if the exponent is not even?

1559 xxx

1evenxx

Simplify: All variables are positive.

105

72

yx20

yx25

Cube roots: Look for perfect cubes in the coefficient.

How can you determine if the variable is a perfect cube?

4 1542

3 20127

3 106

zyx32

zyx40

yx27

Rational Exponentsxofrootnthex thn

n1

n xx root

power

na

n a xxx:General

23 = 8 28831

3

54 = 625 562562541

4

52

32

23

43

65

32

243278

94

2566427

Now – Try some fun problems!

Remember: Root first makes the number smaller.

Can you simplify rational exponents?

Assign 5.1a: 17-39 odd, 41-58 all

WARM-UP

Hmmm…do you remember??

1. x2x3 = 2. (x2)3 =

3. a-3 4. =2

5

xx

5.1a Answers42. 16

44. 27

46. 27

48. 1/3

50. 1/8

52. 7/4

54. 4/9

56. 15

58. 12/35

5.1b Simplifying Radical Expressions

Recall the exponent properties.

xaxb = b+ax

(xa)b = abx

a-n = na

1

=x

xb

abax

x0 = 1, 0x

Objective: To simplify rational expressions using exponent properties

Now try these!

21

32

1. xxEx

32

22. xEx

23

43.

Ex

xx

Ex32

4.

23

25. xxEx

23

2446. Ex

Simplify. All variables are positive.

32

96 )(37. yxEx

31

8

2

324

8.

yxxy

Ex

32

4

34

32

49.

x

xEx

32

432

21

810.

zyxEx

Can you simplify rational expressions using exponent properties?

Homework: 5.1: 59-67 odd, 68-78 all, 93, 94, 107, 108 Quiz after 5,3

5.1b Answers68.

70.

72.

74.

76.

78.

94.

108.

32

101

53

zyx

154

311

a

b

61

y

531ba

2132

1

sr

ab3

40014529 ,d,.t

1034

43

xervalinttheonxx

5.2 WARM-UP

3

1

21

129

82

yx27

yx163

Simplify:

5.2 More Rational Exponents

Multiply:

1x2x4x2.2Ex

)1x2x3(x4.1Ex

21

35

32

32

4x3x.3Ex 3

232

5x5x.4Ex 2

121

2

332

23

32

4x4x.5Ex

Objective: To continue multiplying rational exponents

3

231

31

32

31

31

yyxxyx.6Ex

Multiply:

Factor completely:

Ex 7. 4(x – 3)2 + 5x(x – 3) =

Factor with Rational Exponents

Determine the smallest exponent and factor this from all terms.

21

52

32

35

xy8yx4.9Exx2x.8Ex

6xx

6xx

6xx

6xx

74

78

52

54

31

32

21

Try these:

6x13x6

4xx3

12x5x2

51

52

32

34

35

310

Last one!

Add: Don’t forget the common denominator!

21

21

3

21

1x2

1x

x

x

x

4

xx4

Can you multiplying rational exponents?

Assign 5.2: 3-69 (x3), 77, 81, 97-100

5.2 Answers

23

x

61

k

6. 12. 18.

24. 30. t - 125 36. a + 27

42. 48. 54.

60. 66. 98.

100.

xxx 101520 23 128 21

aa 96 21

tt

yyxx 164025 21

21

21

65 x 314 31 xx

32 3

13

1xx

7x47t4 51

51

41

7

x

x

5.3 Simplified Radical Form

Properties for radicals: a, b > 0

bdacdcba.3

0bforb

a.2baba.1 n

nn

bannn

Simplify each radical means:

No perfect squares left under the

No perfect cubes left under the 3

No factors in the radicand can be written as powers of the index.

No fractions under the radical

No radicals in the denominator

Objective: To write Radicals in simplest radical form

When you simplified radicals to this point the book said that all variables were positive.

What if they do not tell us all variables are positive?

:xx 2

xxxxx 3736

The first one needs absolute value symbols to insure the answer is positive

The second does not because if x was negative, it could not be under an even root.

4 2110441710 zyx8zyx183

Simplify each: Do not assume variables are positive.

When you have an even root and an even exponent in the radicand that becomes an odd exponent when removed, you must use absolute value.

Type 1: Similar to section 5.1

3 2243

4 41127

zyx48.2Ex

zyx1622.1Ex

Type 2: No radicals in the denominator.

615

516

35 2

3y3x4

3y3x4

2

4

4

7

Try these:

310

84 23

72

32

5

4

xy8

x2

yzx9

1

yx5

xy8x2

xy9

x4

How do you know what degree to make the exponents in the denominator?

Can you write Radicals in simplest radical form?

Assign 5.3: 3-21 (x3), 23-33 odd, 48-69 (x3), 71-77 odd, 85-87 all, 89, 105

GROUP ACTIVITY

Learning Target: Find a set containing 3equivalent forms of

the same number on the face.

You will work with the 1 or 2 people sitting beside you. Begin with all of the cards face-up spread out on the desk. Take turns gathering sets of 3 cards.

5.3 Answers

28 318

3 38

6. 12.

18. 48.

54. 60.

66. 86.

66

xx

3

1354 2

y

xy

2

103 2

z

yzxy

3

65feet

22

84733

Review 5-1 to 5-3Questions? Remember NO CALCULATOR!

5.1: Simplify radicals and rational exponents Write radical expressions with rational exponentsEvaluate rational exponentsSimplify expressions with rational exponents

5.2: Multiply and factor using rational exponents Add by making common denominators with rational exponents

5.3: Simplify radicals if the variables may not be positive No fractions under the radical No radicals in the denominator Be able to do these for any root

Now let’s try some problems!

Write using rational exponents:

8 32 5 xx

Simplify:

?yxyx:Doesyx

yx

yx93aazyx272

zyx20016169

21

21

21

61

41

32

23

41

432

3

423 1492

15104

Multiply:

5x3x2x21a35a 3

152

31

31

9x24x165xx45xx2

x15x46x5x

:Factor

52

54

31

32

51

52

31

32

3

21

21

32

2x42x

3x3

x

4

x4

y3x62

124

:Simplify

42

123

Assign: Review WS

5.4 – Addition and Subtraction of Radicals

Objective: To add and subtract radicals

We all know how to simplify an equation such as: 2x +3y – 5x = 3y – 3x

The process for addition and subtraction of radicals is very similar. To do so you must have the same index and the same radicand.

225325 5323

Lets try some! **You may need to simplify first!

271251831. Ex

33 x43x4.2Ex

3 33 63 3 y6yx65yx48x3.Ex

2

1

3

44. Ex

821845. xEx

Can you add and subtract radicals?

Homework: 5.4

5.4 Answers54.

23

WARM-UP

)37)(37(

)35)(35(

Simplify.

What did you notice about the above? These are called CONJUGATES!

5.5 Multiplication and Division of Radicals

Objective: To multiply and divide radicals

Recall the radical properties we learned earlier in the chapter.

bdacdcba Then simplify if possible.

152541 .Ex 348252. Ex

636323. Ex 334. xxEx

Therefore factorable!!!

2315. xEx

Now for division. Don’t forget to rationalize the denominator!!

31

46.

Ex

Multiply the numerator & denominator by the conjugate of the denominator. Then FOIL.

24

257.

Ex2

38.

xEx

Can you multiply and divide radicals?

Homework: 5.5

5.5 Answers

6. 4200 12. 18.

24.

30. 3 36. x - 22 42.

48. 54. 60.

66.

3 514105 211050

baba 42025

6

77

yx

yx

22

bababa

2

xxx

4

2511

4914777 2 xxxxx

5.6a Equations with RadicalsObjective: To solve basic radical equations

Recall: 4x – 5 = 23 Locate the variable. Undo order of operations to isolate the variable.

+5 +54x = 28

4 4x = 7

How is similar?235x4 Procedure: Locate and isolate the radical. +5 +5

4 x = 28 4 4

x = 7 How do you undo the radical?( )2 ( )2

x = 49 Always check these answers. When you square, you may get extraneous

roots.

Squaring Property: If both sides of an equation are squared, the solutions to the original equation are also solutions to the new equation.

*You must square the entire side.

*You must check for extraneous(extra) roots.BASIC:

843x2:1Ex 123x24:2Ex

3x34:3Ex

Medium:

3x3x:4Ex * Isolate the radical on one side.* Square both sides (the entire side- FOIL)* Solve the quadratic. (How?)

( )2 ( )2

x2 – 6x + 9 = x – 3 x2 – 7x + 12 = 0 (x – 4) (x – 3) = 0 x = 4, 3 Check both answers – one generally does not

work.You try these:

010a2a:5Ex 43x:6Ex 3

33 7x25x:7Ex

Can you solve basic radical equations?

Assign: 5-6 to # 35

5.6b Warm - up

Solve: 57 tt

5.6b More Solving Radical Equations

Objective: To solve radical equations with radicals on both sides and identiry extaneous rootsWhat happens when you have two radicals that

you cannot combine?

25 xx * Two different roots & something else

*Isolate the more complicated radical on one side and square both sides. (The entire side.)

* Isolate the radical that is left and square both sides again.

You try this one:

835 xx

Graphing

xy x y 0 0 1 1 4 2

What is the domain:What is the range:

How would each change affect the graph? Give the domain and range for each.

xy:Ex 31

Domain:

Range:

22 xy:Ex 23 xy:Ex 134 xy:Ex 3 xy:OneLast

Summary:

Can you solve radical equations with radicals on both sides and identity extaneous roots?

Assign: Rest of 5.6

How can you make the root open left? Upside down?

5.6b Solutions

42. 4

48. 5, 13

54.

56. 10

58. 10,000

60. The plume would be smaller if there was a current.

2512 1x

5.6c Solving Equations with Rational Exponents

Objective: To solve equations with rational exponents and understand extraneous roots

4x4x31

3

* To solve an equation with a rational exponent, you must first solve for the variable or parenthesis with the rational exponent.* You must undo the exponent, by taking it to a power that will cancel the exponent to a 1.

( )3 ( )3

8x:1Ex 4 3

x1 = 64

45x4:2Ex32

How do you know when you should use for your solution?

When solving an equation and you must take an even root, you must use x = answer.

You try these:

2611x3:3Ex43

36x4:4Ex52

08x6x:5Ex 40x3x:6Ex

036x13x:7Ex32

34

Miscellaneous

Completely factor: x2n – 5xn + 6

Now try:

Ex: 2x4n + x2n - 6 Ex: x2n+1 - 5xn+1 + 6x

Cancel:

2

n3

x

xn

5n3

x

x

1

52

n

n

xx

Assignment: Worksheet and begin test review.Can you solve equations with rational exponents and understand extraneous roots?

5.6c Worksheet Solutions

1. 27 10. 19. 252. 16 20. 27 & -64 3. 32 11. 81 21. 49 & 254. -32 12. 1024 22. 255. 64 13. 63, -62 23. 32,768 & -326. 14. 341 24. 9 & .25

25.7. 15. 26.

8. 14 16. 81 27.17. 32 28.

9. 18. 35 & -29 29.

30.

31.

64

1

16

1

3

7

3

4

2

15

)2)(3( nn xxx)3)(3(2 3

2

xxx

)5)(12( 221 nn xnx12 nx12 nx

nx2

6nx

5.7a Introduction to Complex Numbers

Objective: To define imaginary and complex numbers and perform simple operations on each

Real Numbers (R): the set of rational and irrational numbers.

Rational (Q) : any number that can be written as a fraction

Irrational (Ir) : non-repeating, non-terminating decimals

Integers (I or Z): positive and negative Whole numbers: no fractions or decimalsWhole (W) : {0, 1, 2, 3, …} no fractions, decimals or negatives.

Natural (N) : counting numbers no decimals, fractions, negatives, 0

Complex Numbers (C) : a + bi a = real part b = imaginary part

Imaginary (Im):

negative i 1

What are imaginary numbers?

Symbol?

Value?

Square roots of negative numbers.

i

1

A complex number is in the form of a + bi

where a = real part and bi = imaginary part.

A pure imaginary number only has the imaginary part, bi.

**Always remove the negative from the radical first!**

Ex1: Ex2: Ex3:

Ex4: Ex5: Ex6:

4 94 27

82 25

9

55

What is the value of i 2 ?

i 1( )2 ( )2

-1 = i2

When you get an i2 , always replace it with a -1.

i = i i2 = -1 i3 = i4 = i5 = i6 = i7

= i8 =

i9 = i10 = i11 = i12 = Ex1: i20 = Ex2: i30 =

Ex3: i57 = Ex4: i101 =

Ex5: i12 i25 i-3 =

For 2 complex numbers to be equal, the real parts must be equal and the imaginary parts must be equal.

Ex1: 3x + 2i = 6 + 8yi

Ex2: 4x – 3 + 2i = 9 – 6yi

Ex3: 5 – (4 + y)i = 2x + 3 – 6i

Assign: 1-24 all

Can you define imaginary and complex numbers and perform simple operations on each?

5.7a Solutions

2. 7i 20. x = -.5 y = -5/3

4. -9i 22. x = 11/4 y = -2

6. 24. x = 2/5 y = -4

8.

10. -i

12. i

14. -1

16. x = 1 y = -4

18. x = 2/3 y = -.5

34i

35i

5.7b Operations on Complex Numbers

Objective: To perform operations on complex numbersAdd/subtract:

Compare to: (4 – 3x) + (2x – 8) =

Add: (2 + 4i) + (6 – 9i) = Subtract: (5 – 3i) – (7 – 5i) =

-x - 4

Add real part to real part and imaginary part to imaginary part

Multiply: Distribute or FOIL – all answers should be in standard complex form: a + bi

Don’t forget i2 = -1Ex1: 2i(3 – 4i) = Ex2: -4i(5 + 6i) =

Ex3: (2 – 3i)(4 + 5i) = Ex4: (4 – 2i)2 =

Division:

This is similar to rationalizing the denominator with radicals.

Type 1: -or- i4

i2

5

Type 2: -or-i23

4

ii

49

812

Recall: How did we rationalize the denominator? 32

4

Use the complex conjugate to divide complex numbers. a + bi a - bi

Assign: 5.7b: 25-77 odd 87-90

Can you perform operations on complex numbers?

5.7 b Solutions88. i if n is even

90. x = 1 – i is a solution to the equation.

5.7c and Review

FOIL: (x - 3)(x + 3) -compare to- (x – 3i)(x + 3i)

This means the following can be factored. How?

1. 4x2 - 25 2. 4x2 + 25

3. x2 + 4 4. 2x2 + 98

Objective: To factor and simplify using complex numbers

Just for fun. (And they make great essay questions.)*What are imaginary numbers?*What symbol is used to designate imaginary units?*What is the value of the imaginary unit?*Give the definition of a complex number?*What is the complex conjugate and when should it be used? Give an example.*What is a pure imaginary number?

53021 iii:Ex 50821132 iiii:Ex 30221253 iiii:Ex

Ex4: 4x – 3 + 2yi = x + 2y – 8i Ex5: 68 ii

:Ex3

46

ii

:Ex34

27

iii:Ex 658548 2239 i:Ex

iii:Ex 23510 2 0411 2 x:Solve:Ex

Can you factor and simplify using complex numbers?

Assign Worksheet:

Mini-Quiz Tomorrow!!

Worksheet Answers1. 10. 17i 19. 28. 0

2. 10i 11. 20. 29. 1

3. 12. 6 21. 17-6i 30. 1

4. -20 13. -5 22. 50 31. x2 - 36

5. -8 14. 23. 32. x2 + 36

6. i 15. -8 + 6i 24. -3 + 4i 33. x2 - 4

7. 15i 16. 2/3 25. 1 + 21i 34. x2 + 4

8. -35 17. -4i 26. -45 + 30i 35. (x+3)(x-3)

9. -36 18. 27. 1 36. (x+3i)(x-3i)

37. (x+7)(x-7)

38. (x+7i)(x-7i)

11i

22i

28i

25i

5

48 i

41

232 i

2

43 i

341 i

Ch 5 Review Answers

.274x.26

13x.2526,28x.24517x.236x.22

9x.211y.208

27,1x.19100x.186x.17

35

x.162.15424.1420x3x2.13

1x145x3.122362336.11548.10

775

.9y2

y12x.877x2x.7

1526.6380.5yx7.4

326.3z2

xyz3x.2b6cba2.1

331

32

2

4 322

32

3653

Ø

2xy.c

2xy.b

xy.a.28

3xy.b

2xy.a.29

xy.b

x2y.a.30

21

ab

ca b

a

b