LAST REVISED November, 2008

Copyright This publication © The Northern Alberta Institute of Technology 2002. All Rights Reserved.

Algebra Module A40

Exponents

Module A40 − Exponents

1

Exponents Statement of Prerequisite Skills Complete all previous TLM modules before completing this module.

Required Supporting Materials Access to the World Wide Web. Internet Explorer 5.5 or greater. Macromedia Flash Player.

Rationale

Why is it important for you to learn this material? Exponents are seen in many different practical applications. Scientific notation makes use of exponents to express numbers and exponents are used in applications as diverse as business, finance, pH levels, the Richter scale, decibel levels, and astronomy. Learning how to manipulate exponents will assist the student in many different technologies.

Learning Outcome

When you complete this module you will be able to… Simplify expressions containing exponents.

Learning Objectives 1. Use laws of exponents to simplify expressions with integral exponents. 2. Use laws of exponents to simplify expressions with zero or negative exponents. 3. Use laws of exponents to simplify expressions with rational exponents.

Connection Activity You don’t actually need exponents. You could just write 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5. Exponents do make it easier to write such expressions but they are not essential. Many would rather write 523 than the expression you see above. Can you think of any other examples of where exponents may make expressing a number more convenient?

Module A40 − Exponents 2

OBJECTIVE ONE When you complete this objective you will be able to… Use laws of exponents to simplify expressions with integral exponents.

Exploration Activity An expression such as 25 is called a power. The 2 is called the base of the power, while 5 is called the exponent of the power. The expression is read as 2 to the exponent of 5, or 2 to the fifth, or the fifth power of 2. The expression 25 means 2 · 2 · 2 · 2 · 2 and is equal to 32. GENERAL LAWS of EXPONENTS The general laws of exponents enable you to evaluate or simplify expressions that include power terms. First Law of Exponents – Product Rule When you multiply power terms that have the same base you add the exponents.

xm · xn = xm+n

Note that the base in the answer is identical to the base of each factor of the product.

EXAMPLE 1 34 · 37 = __________ To simplify: Add the exponents. = 34+7 = 311 Evaluate: = 177147

EXAMPLE 2 (−2)6 · (−2)3 = ___________ To simplify: Add the exponents. = (−2)6+3 = (−2)9 Evaluate: = −512

Module A40 − Exponents

EXAMPLE 3 a3 · al0 = To simplify: Add the exponents. = a3+10 = a13 Second Law of Exponents – Quotient Rule When you divide one power term by another power term with the same base, you subtract the exponents.

xm ÷ xn = xm−n

EXAMPLE 1 214 ÷ 28 = __________ To simplify: Subtract the exponents. = 214−8 = 26 Evaluate: = 64

EXAMPLE 2

_________77

3

5=

To simplify: Subtract the exponents. = 75−3 = 72 Evaluate: = 49

EXAMPLE 3 y12 ÷ y7 = To simplify: Subtract the exponents. = y12−7 = y5

3

Module A40 − Exponents 4

Third Law of Exponents – Power Rule

(xm)n = xmn

EXAMPLE 1 (23)2 = _________ To simplify: Multiply the exponents. = 23·2 = 26 Evaluate: = 64

EXAMPLE 2 (32)4 = _________ To simplify: Multiply the exponents. = 32·4 = 38 Evaluate = 6561

EXAMPLE 3 (b4)5 = ________ To simplify: Multiply the exponents. = b4·5 = b20

Module A40 − Exponents

5

Fourth Law of Exponents When you have a power of a product of factors, you can write the expression as a product of power factors.

(xy)n = xnyn

EXAMPLE 1 (4a)3 = ________ To simplify: Raise each factor to the exponent 3. = 43a3 Evaluate: = 64a3

EXAMPLE 2 (−3b)5 = ________ To simplify: Raise each factor to the exponent 5. = (−3)5b5 Evaluate: = −243b5

EXAMPLE 3 (2a2bc3)4 = ________ To simplify: Raise each factor to the exponent 4. = 24a2·4b4c3·4 = 16a8b4c12

Module A40 − Exponents

Fifth Law of Exponents When you have a power of a quotient you can write the expression as the power of the numerator divided by the power of the denominator.

n

nn

yx

yx

=⎟⎟⎠

⎞⎜⎜⎝

⎛

EXAMPLE 1

________32 5

=⎟⎠⎞

⎜⎝⎛

To simplify: Raise both the numerator and the denominator to the exponent 2.

5

5

32

=

Evaluate:

24332

=

EXAMPLE 2

_______4 2

=⎟⎠⎞

⎜⎝⎛

a

To simplify: Raise both the numerator and the denominator to the exponent 2.

2

24a

=

Evaluate:

216a

=

EXAMPLE 3

________3

3

2=⎟

⎟⎠

⎞⎜⎜⎝

⎛

cba

To simplify: Raise both the numerator and the denominator to the exponent 3.

33

332

⋅

⋅

=c

ba

9

36

cba

=

6

Module A40 − Exponents

Experiential Activity One Simplify each of the following expressions: (Whenever numbers are involved evaluate the numbers completely) 1. x4 · x7 2. 59 ÷ 56 3. (a6)7 4. (4x)5

5. 4

43⎟⎠⎞

⎜⎝⎛ 6.

6

3

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

yx

7. 43 · 45 8. a6 ÷ a4 9. (23)5 10. (6a2b3)3 11. (−7)12 ÷ (−7)9 12. (−6y)3 13. (53)2 14. (a + b)5 · (a + b)8 Show Me. 15. (x6)6 16. (−3)6 · (−3)9

17. (10ab2)4 18. 5

38

⎟⎟⎠

⎞⎜⎜⎝

⎛yx Show Me.

19. (x − y)8 ÷ (x − y)5 20. 5

4

32

3 ⎟⎟⎠

⎞⎜⎜⎝

⎛

cba

Experiential Activity One Answers 1. x11 2. 125 3. a42 4. 1024x5

5. 25681 6. 18

12

yx

7. 65536 8. a2 9. 32768 10. 216a6b9 11. −343 12. −216y3 13. 15625 14. (a + b)13 15. x36 16. −14348907

17. 10000a4b8 18. 5

5

24332768

yx

19. (x −y)3 20. 20

1510

243cba

7

Module A40 − Exponents 8

OBJECTIVE TWO When you complete this objective you will be able to… Use laws of exponents to simplify expressions with zero or negative exponents.

Exploration Activity General Zero Exponent Law According to the second law of exponents 43 ÷ 43 = 40. However, we also know that 43 ÷ 43 = 1

Therefore: 40 = 1 In general, for all x ≠ 0, (x)0 = 1 . However, an expression such as −60 is evaluated as follows:

−60 = (−1)(6)0 [Regarded as the negative value 60.] = (−1)(1) = −1

Observe −60 is not the same as (−6)0. Also, 00 is undefined.

EXAMPLE 1 80 = __________ To simplify: Apply the zero exponent law. 80 = 1

EXAMPLE 2 (a2b2)0 = __________ To simplify: Multiply exponents and apply the zero exponent law. = a0b0 = (1)(1) = 1

Module A40 − Exponents

EXAMPLE 3 (−6x)0 = _________ To simplify: Raise each factor to the exponent 0 and then apply the zero exponent law. = (−6x)0 = (1)(1) = 1

EXAMPLE 4 4x0 = __________ To simplify: Substitute 1 for x0. = 4(1) = 4

EXAMPLE 5 −9x0 = ___________ To simplify: Substitute 1 for x0. = −9(1) = −9 Negative Exponents According to the second law of exponents, 64 ÷ 67 = 6−3. However,

37

4

61

6661

66666666666

66

=⋅⋅

=⋅⋅⋅⋅⋅⋅

⋅⋅⋅=

Therefore: 33

616 =−

In general, we can say nn

xx 1

=− so that any power term with a negative

exponent can be rewritten as a power term with a positive exponent.

9

Module A40 − Exponents

EXAMPLE 1 x−4 = _________ To simplify: Rewrite with a positive exponent.

41x

=

EXAMPLE 2 (2x)−3 = _________ To simplify: Rewrite with a positive exponent.

( )321x

=

Expand and evaluate.

333 81

21

xx==

EXAMPLE 3 5x−2 = _________ To simplify: Rewrite with a positive exponent.

25x

=

10

Module A40 − Exponents

EXAMPLE 4

_________3

=⎟⎠⎞

⎜⎝⎛

−

ba

To simplify: Raise both the numerator and denominator to the exponent −3.

3

3

−

−

=ba

Rewrite the numerator and denominator with positive exponents.

33

3

3 11or 1

1

bab

a ÷=

Now applying the rules for dividing fractions we invert and multiply.

3

33

3 11

abb

a=×=

This last example leads to the generalization:

nn

xy

yx

⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

EXAMPLE 5

_________32 5

=⎟⎠⎞

⎜⎝⎛

−

To simplify: Rewrite with a positive exponent.

5

23⎟⎠⎞

⎜⎝⎛=

Expand and evaluate.

32243

23

5

5==

Here are some further examples of questions involving power terms with negative exponents.

11

Module A40 − Exponents

EXAMPLE 6 (b4)(b−8) = _________ To simplify: Add exponents. = b−4 Rewrite with a positive exponent.

41

b=

EXAMPLE 7 a3 ÷ a−7 = _________ To simplify: Subtract exponents and evaluate. = a3−(−7) = al0

EXAMPLE 8 (32)−2 = _________ To simplify: Multiply exponents. = 3−4 Rewrite with a positive exponent and evaluate.

431

=

811

=

12

Module A40 − Exponents

EXAMPLE 9

__________4

2

53=⎟

⎟⎠

⎞⎜⎜⎝

⎛−

−

−

zyx

To simplify: Multiply exponents.

8

2012

zyx −

=

Rewrite with positive exponents.

820

12

zyx

=

EXAMPLE 10 (3x−4)2 = __________ To simplify: Raise each factor to the exponent 2. = 32x−8 Rewrite with positive exponents and evaluate.

8

23x

=

89x

=

13

Module A40 − Exponents

Experiential Activity Two Simplify and/or evaluate the following expressions. Answers that are power terms should be expressed with positive exponents. 1. a−5 · a−8 2. b−6 ÷ b10 3. (c3)−4 4. (8a)−3

5. 5

3

2 −

− ⎟⎟⎠

⎞⎜⎜⎝

⎛

yx 6. 2−5 · 23

7. 10−5 ÷ 10−6 8. (2−1)−4 9. (a−3b2)0 10. (−133x2y5)0

11. 3

54 −

⎟⎠⎞

⎜⎝⎛ 12. (9a−2)−2

13. (x + y)5 · (x + y)−2 14. (a − b)−4 ÷ (a − b)−7 15. [(a + b)2]−1 16. (3x3y6z−2)−4

17. 2

37

−

⎟⎟⎠

⎞⎜⎜⎝

⎛yx Show Me. 18. a−9 · a3 ÷ a−6

19. 8x−2y3z0 20. 4

32 −

⎟⎠⎞

⎜⎝⎛ −

Experiential Activity Two Answers

1. 131

a 2. 16

1b

4. 35121a

3. 121

c

5. 15101yx

6. 41

7. 10 8. 16 9. 1 10. 1

11. 64

125 12. 81

4a

13. (x + y)3 14. (a − b)3

15. ( )2

1ba +

16. 2412

8

81 yxz

17. 2

2

499

xy 18. 1

19. 2

38xy 20.

1681

14

Module A40 − Exponents

OBJECTIVE THREE When you complete this objective you will be able to… Use laws of exponents to simplify expressions with rational exponents.

Exploration Activity The General Laws of Exponents hold true for rational exponents just as they do for integral exponents. One simply must remember the rules for the arithmetic operations involving rationals in order to simplify or evaluate these expressions.

EXAMPLE 1

_________32

43

=⋅ xx To simplify: Rewrite exponents with a common denominator.

128

129

xx ⋅= Add exponents.

1217

x=

EXAMPLE 2

_________65

52

=÷ xx To simplify: Rewrite exponents with a common denominator.

3025

3012

xx ÷= Subtract exponents.

1330x

−= Rewrite with a positive exponent.

30131

x=

15

Module A40 − Exponents

EXAMPLE 3

_________4

33

2=⎟

⎠⎞

⎜⎝⎛a

To simplify: Multiply exponents and reduce the fraction.

( )323 4a=i

21

a=

EXAMPLE 4

( ) __________3263 =

−− yx To simplify: Multiply exponents and reduce the fraction.

( ) ( )2 23 63 3

2 4

x yx y

− − −

−

=

=

i i

Rewrite with positive exponents.

4

2

yx

EXAMPLE 5

_________5

4

15

10=⎟

⎟⎠

⎞⎜⎜⎝

⎛−

yx

To simplify: Multiply exponents.

( )

( )

410 5

415 5

8

12

x

yxy

−

−

−

−

=

=

i

i

Rewrite with positive exponents.

8

12

xy

=

16

Module A40 − Exponents

Summary of Exponent Rules: Product Rule: xm · xn = xm+n

Quotient Rule: xm ÷ xn = xm−n

Power Rules: (xm)n = xmn

(xy)n = xnyn

n

nn

yx

yx

=⎟⎟⎠

⎞⎜⎜⎝

⎛

Zero Power Rule: (x)0 = 1 where x ≠ 0,

nn

xx 1

=−Negative exponents:

17

Module A40 − Exponents

Experiential Activity Three Simplify the following expressions. Power term answers should be expressed with positive exponents.

107

32 −−

÷ bb41

53

xx ⋅−1. 2.

34

65 −

⎟⎠⎞

⎜⎝⎛a ( ) 7

22xy3. 4.

36

43

91

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

b

a61

18

12 −

− ⎟⎟⎠

⎞⎜⎜⎝

⎛

ba 6. 5.

48

3 −−

⎟⎠⎞

⎜⎝⎛b8

57

4 −−⋅ aa 8. 7.

52

41

31 −

−⎟⎠⎞

⎜⎝⎛ yx 3

27

12 −÷ xxShow Me.

18

9. 10.

Experiential Activity Three Answers

2071

x1. 30

1b2.

9101

a 3. 7

47

2yx 4.

321ba 274

1ba

5. 6.

56671

a 7. 2

3b8.

152

101

x

y21

50x 9. 10.

Practical Application Activity Complete the Exponents assignment in TLM.

Summary This module dealt with theory on exponents.