Download Module - Exponents

Algebra
Module A40
Exponents
Copyright
This publication © The Northern
Alberta Institute of Technology
2002. All Rights Reserved.
LAST REVISED November, 2008
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
1
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
2
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
75
73
= _________
To simplify: Subtract the exponents.
= 75−3 = 72
Evaluate:
= 49
EXAMPLE 3
y12 ÷ y7 =
To simplify: Subtract the exponents.
= y12−7 = y5
Module A40 − Exponents
3
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
4
Module A40 − Exponents
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
5
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
⎛ x⎞
xn
⎜⎜ ⎟⎟ = n
y
⎝ y⎠
EXAMPLE 1
5
⎛ 2⎞
⎜ ⎟ = ________
⎝ 3⎠
To simplify: Raise both the numerator and the denominator to the exponent 2.
=
25
35
Evaluate:
=
32
243
EXAMPLE 2
2
⎛4⎞
⎜ ⎟ = _______
⎝a⎠
To simplify: Raise both the numerator and the denominator to the exponent 2.
=
42
a2
Evaluate:
=
16
a2
EXAMPLE 3
3
⎛ a 2b ⎞
⎜
⎟
⎜ c 3 ⎟ = ________
⎝
⎠
To simplify: Raise both the numerator and the denominator to the exponent 3.
=
=
a 2⋅3b 3
c 3⋅3
a 6b 3
c9
6
Module A40 − Exponents
Experiential Activity One
Simplify each of the following expressions: (Whenever numbers are involved evaluate
the numbers completely)
1. x4 · x7
3. (a6)7
2. 59 ÷ 56
4. (4x)5
5.
7.
9.
11.
13.
15.
43 · 45
(23)5
(−7)12 ÷ (−7)9
(53)2
(x6)6
6.
8.
10.
12.
14.
16.
a6 ÷ a4
(6a2b3)3
(−6y)3
(a + b)5 · (a + b)8
(−3)6 · (−3)9
⎛ 8x ⎞
18. ⎜⎜ ⎟⎟
⎝ 3y ⎠
2 4
17. (10ab )
8
19. (x − y) ÷ (x − y)
5
6
⎛ x2 ⎞
⎜ ⎟
⎜ y3 ⎟
⎝ ⎠
4
⎛ 3⎞
⎜ ⎟
⎝ 4⎠
5
⎛ a 2b 3 ⎞
20. ⎜⎜ 4 ⎟⎟
⎝ 3c ⎠
Show Me.
Show Me.
5
Experiential Activity One Answers
1. x11
3. a42
2. 125
4. 1024x5
5.
81
256
6.
7.
9.
11.
13.
15.
65536
32768
−343
15625
x36
8.
10.
12.
14.
16.
17. 10000a4b8
18.
19. (x −y)3
20.
x12
y 18
a2
216a6b9
−216y3
(a + b)13
−14348907
32768 x 5
243 y 5
a10 b15
243c 20
Module A40 − Exponents
7
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
8
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,
64
6
7
=
6⋅6⋅6⋅6
1
1
=
=
6 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 6 ⋅ 6 ⋅ 6 63
Therefore: 6 −3 =
1
63
In general, we can say x −n =
1
xn
so that any power term with a negative
exponent can be rewritten as a power term with a positive exponent.
Module A40 − Exponents
9
EXAMPLE 1
x−4 = _________
To simplify: Rewrite with a positive exponent.
=
1
x4
EXAMPLE 2
(2x)−3 = _________
To simplify: Rewrite with a positive exponent.
=
1
(2 x )3
Expand and evaluate.
=
1
23 x 3
=
1
8x 3
EXAMPLE 3
5x−2 = _________
To simplify: Rewrite with a positive exponent.
=
5
x2
10
Module A40 − Exponents
EXAMPLE 4
⎛a⎞
⎜ ⎟
⎝b⎠
−3
= _________
To simplify: Raise both the numerator and denominator to the exponent −3.
=
a −3
b −3
Rewrite the numerator and denominator with positive exponents.
1
3
1
1
= a or 3 ÷ 3
1
a
b
b3
Now applying the rules for dividing fractions we invert and multiply.
=
1
a3
b3 b3
= 3
1
a
×
This last example leads to the generalization:
⎛ x⎞
⎜⎜ ⎟⎟
⎝ y⎠
−n
⎛ y⎞
=⎜ ⎟
⎝ x⎠
n
EXAMPLE 5
⎛ 2⎞
⎜ ⎟
⎝ 3⎠
−5
= _________
To simplify: Rewrite with a positive exponent.
⎛ 3⎞
=⎜ ⎟
⎝ 2⎠
5
Expand and evaluate.
=
35
2
5
=
243
32
Here are some further examples of questions involving power terms with negative
exponents.
Module A40 − Exponents
11
EXAMPLE 6
(b4)(b−8) = _________
To simplify: Add exponents.
= b−4
Rewrite with a positive exponent.
=
1
b4
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.
=
=
1
34
1
81
12
Module A40 − Exponents
EXAMPLE 9
⎛ x −3 y 5 ⎞
⎜
⎟
⎜ z −2 ⎟
⎝
⎠
−4
= __________
To simplify: Multiply exponents.
=
x12 y −20
z8
Rewrite with positive exponents.
=
x12
y 20 z 8
EXAMPLE 10
(3x−4)2 = __________
To simplify: Raise each factor to the exponent 2.
= 32x−8
Rewrite with positive exponents and evaluate.
=
=
32
x8
9
x8
Module A40 − Exponents
13
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
3. (c3)−4
5.
⎛ x2 ⎞
⎜
⎟
⎜ y −3 ⎟
⎝
⎠
2. b−6 ÷ b10
4. (8a)−3
−5
6. 2−5 · 23
7. 10−5 ÷ 10−6
9. (a−3b2)0
⎛ 4⎞
⎝5⎠
8. (2−1)−4
10. (−133x2y5)0
−3
11. ⎜ ⎟
12. (9a−2)−2
13. (x + y)5 · (x + y)−2
15. [(a + b)2]−1
14. (a − b)−4 ÷ (a − b)−7
16. (3x3y6z−2)−4
⎛ 7x ⎞
⎟⎟
⎝ 3y ⎠
−2
17. ⎜⎜
19. 8x−2y3z0
Show Me.
18. a−9 · a3 ÷ a−6
⎛ − 2⎞
⎟
⎝ 3 ⎠
−4
20. ⎜
Experiential Activity Two Answers
1.
3.
5.
1
2.
13
a
1
4.
c12
1
6.
x10 y 15
7. 10
9. 1
11.
17.
19.
125
64
12.
1
(a + b )
9y2
49 x 2
8y3
x
b16
1
512a 3
1
4
8. 16
10. 1
13. (x + y)3
15.
1
2
2
a4
81
14. (a − b)3
16.
z8
81x12 y 24
18. 1
20.
81
16
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
x
3
4
⋅x
2
= _________
3
To simplify: Rewrite exponents with a common denominator.
9
=x
12
⋅x
8
12
Add exponents.
=x
17
12
EXAMPLE 2
x
2
5
÷x
5
6
= _________
To simplify: Rewrite exponents with a common denominator.
=x
12
30
÷x
25
30
Subtract exponents.
=x
−13
30
Rewrite with a positive exponent.
1
=
x
13
30
Module A40 − Exponents
15
EXAMPLE 3
⎛ 23 ⎞
⎜a ⎟
⎝
⎠
3
4
= _________
To simplify: Multiply exponents and reduce the fraction.
( 23 i 34)
=a
1
=a
2
EXAMPLE 4
(x
)
2
−3 6 − 3
y
= __________
To simplify: Multiply exponents and reduce the fraction.
=x
( −3i − 2 3 ) ( 6 i − 2 3 )
y
=x y
2
−4
Rewrite with positive exponents.
x2
y4
EXAMPLE 5
⎛ x10 ⎞
⎜
⎟
⎜ y15 ⎟
⎝
⎠
−4
5
= _________
To simplify: Multiply exponents.
=
x
y
=
(10 i − 4 5 )
(15i − 4 5 )
x −8
y −12
Rewrite with positive exponents.
=
y 12
x8
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
⎛ x⎞
xn
⎜⎜ ⎟⎟ = n
y
⎝ y⎠
Zero Power Rule: (x)0 = 1 where x ≠ 0,
Negative exponents: x −n =
1
xn
Module A40 − Exponents
17
Experiential Activity Three
Simplify the following expressions. Power term answers should be expressed with
positive exponents.
1.
x
−3
5
⋅x
1
−4
4
3.
⎛ 56 ⎞
⎜a ⎟
⎝
⎠
5.
⎛ a12 ⎞
⎜
⎟
⎜ b −18 ⎟
⎝
⎠
7.
a
9.
⎛ 13 − 1 4 ⎞
⎜x y
⎟
⎝
⎠
−4
7
3
−1
⋅a
6
−5
8
−2
5
Show Me.
−2
÷b
2.
b
4.
(xy )
7
6.
⎛ − 19
⎜a
⎜⎜ 3
4
⎝ b
⎞
⎟
⎟⎟
⎠
8.
⎛ − 38 ⎞
⎜b
⎟
⎝
⎠
3
2
2
10. x
12
7
−7
36
−4
÷x
−2
Experiential Activity Three Answers
1
1.
7
x
1
3.
a
5.
7.
10
9
1
a b
1
y
x
67
2.
b
4.
x
6.
2 3
a
9.
20
8.
56
4
7y 7
1
a b
b
10. x
15
2
30
4 27
1
10
2
1
3
2
50
10
21
Practical Application Activity
Complete the Exponents assignment in TLM.
Summary
This module dealt with theory on exponents.
18
Module A40 − Exponents
3