Introduction
Prepared by Sa’diyya Hendrickson
Name:
Date:
c Cengage Learning
Figure 1: Package Summary
• Definition and Properties of Exponents
• Understanding Properties (Frayer Models)
• Discovering Zero and Negative Exponents
• Working with Fractions in Disguise
• Let’s Play! (Exercises)
• Notes and Resources
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Some Properties
Recall the following definition of a positive exponent:
1. The Exponential Form bn :
For some number b and some positive integer n, bn simply means to multiply b exactly
n times (i.e. repeated multiplication). Using algebra, we have:
bn = b| · b · · {z
· · · · b · }b
n times
The left side is the simplified form and the right side is the expanded form.
Here, the base b has been “raised to the exponent/power of n.”
e.g. (−2)3 = (−2)(−2)(−2) = −(2 · 2 · 2) = −8
Let a and b be any numbers and let m and n be positive integers. Then, our Properties
of Exponents are as follows:
1. Product Property: (expanded form to simplified form)
bm bn = bm+n
e.g.34 · 36 = 310
2. Quotient Property: (expanded form to simplified form)
59
bm
m−n
=
b
e.g.
= 56
n
3
b
5
3. A Power to a Power Property: (expanded form to simplified form)
(bm )n = bmn
e.g.(62 )5 = 610
4. Product to a Power Property: (simplified form to expanded form)
(ab)n = an bn
e.g.(2 · 7)3 = 23 · 73
5. Quotient to a Power Property: (simplified form to expanded form)
4
a n an
7
74
= n
e.g.
= 4
b
b
8
8
**Let’s use Frayer Models to better understand the properties above!**
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Understanding Properties
I. Product Property (PP)
II. Quotient Property (QP)
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Understanding Properties
III. Power to a Power Property (Pp P)
IV. Product to a Power Property ((Pr)p P)
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Understanding Properties
V. Quotient to a Power Property (Qp P)
1. Raising a number or variable to a positive exponent n means the repeated multiplication of the number/variable exactly n times.
2. If you’re ever in doubt of a property, use the definition of repeated multiplication (i.e.
write out the longer, expanded form of the expression as we did when we explained
the properties with numbers) to verify that the one you are using is correct.
3. Remember that depending on the question, you may need to go from the expanded
form of a property to the simplified form, or vice versa. Because the properties say
“equal to” you can move in either direction!
4. These properties increase your list of rules to the game of math. The better you
know these rules, the better you will be at playing this game!
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Zero and Negative Exponents
Consider the following examples:
1. We know that:
25
= 1,
25
while on the other hand:
25
52 QP
= 2 = 5(
25
5
)
= 50 .
Therefore:
50 = 1
2. We also know that:
25
5·5
=
125
5·5·5
reducing
=
1
5
while on the other hand:
25
52 QP
= 3 = 5(
125
5
)
= 5−1 .
Therefore:
5−1 =
1
5
(the reciprocal of 5!)
The examples above show that the definition of repeated multiplication no
longer applies to the zero and negative exponents. To get a better understanding of these exponents, let’s complete the pattern below:
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Zero and Negative Exponents
1. zero exponent:
b0 = 1 for any number b as long as b 6= 0.
General explanation: 1 =
bn QP
=
bn
b(n−n) = b0
2. negative exponent:
b−n =
General explanation:
1
bn
for any number b as long as b 6= 0.
b0
1
=
bn
bn
QP (0−n)
= b
by the definition of the zero exponent
= b−n
= b(−1)(n)
by definition of multiplying by −1
= b(n)(−1)
by Commutative Property of Multiplication
Pp P
= (bn )−1
n −1
b
=
1
So, the “negative part” of the exponent just tells us to flip the fraction.
i.e. it creates the reciprocal of the original number!
• Positive exponents cause repeated multiplication!
• The zero exponent creates the number 1!
Note: Exponents do not have the power to make a number equal 0.
• The negative part of an exponent creates the reciprocal of the given number (i.e.
makes numbers flip)! Note: Exponents do not have the power to make a number
negative.
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Fractions in Disguise
Because negative exponents create reciprocals, a number raised to a negative power may
be a fraction in disguise! Let’s explore a few examples.
1. 2(5
−1
−1
5
) = 2
1
1
2
=
1
5
by definition of the negative exponent
2·1
1·5
2
=
5
=
−2
2. 2
+ 3(2
−3
1
) = 2+
2
=
by the definition of fraction multiplication
3
1
by definition of the negative exponent
1
23
3
1
+
22 23
by the definition of fraction multiplication
1 · 21
3
+ 3
2
1
2 ·2
2
2
3
= 3+ 3
2
2
2+3
=
23
5
=
8
by equivalent fractions and LCD = 23
=
3.
5
5
÷ 10(7−2 ) =
÷
42
42
10
1
5
10
÷
42 49
5 49
·
=
42 10
1 7
= ·
6 2
7
=
12
=
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by Product Property (i.e. 22 · 21 = 2(2+1) = 23 )
by definition of adding fractions
already in reduced form!
1
72
by definition of the negative exponent
by the definition of fraction multiplication and squaring
by theorem of fraction division
by reducible pairs: (5, 10) and (42, 49)
by definition of fraction multiplication
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Let’s Play!
1. Use Properties of Exponents to simplify the following:
(a) 32 · 37
(b) 24 · 54
(d) (75 )8
(e)
138
135
(g) m2 n3 × m5 n7
(h)
15q 5 r3 ×q 2
3q 6 r2
(c)
36
46
(f) 75 ÷ 73 × 72
2. Use Properties of Exponents to expand the following:
(a) (11 · 12)2
12 8
(d) 13
(b) 5(7·6)
(c) 9(9−7)
(e) 25(3+6)
3. Use your rules to prove the following results. State each rule explicitly.
(a)
5 −3
4
=
43
53
(b)
34
810
7
=
328
870
4. Evaluate and express in simplified/reduced form:
(a) 4(15−1 ) ÷ 16(5−2 )
(b) 3(4−1 ) + 4(3−1 )
−1 12 (d) 74
72
(c) 4(3−2 ) − 3−3
−2 25 −1
(e) 34
33
Solutions:
1. (a) 39
(b) (2 · 5)4
2. (a) 112 · 122
(c)
(b) (57 )6
3 6
4
(c)
99
97
(d) 740
(d)
128
138
(e) 133
(f) 74
(g) m7 n10
(h) 5qr
(e) 253 · 256
3. Start with the left side and with every step taken to get to the right side, state the
rules that you are using.
4. (a)
5
12
(b)
25
12
(c)
11
27
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(d) 37
(e)
3
2
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Let’s Play!
Powers Table Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Squared
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
729
784
841
900
Cubed
1
8
27
64
125
216
343
512
729
1,000
1,331
1,728
2,197
2,744
3,375
4,096
4,913
5,832
6,859
8,000
9,261
10,648
12,167
13,824
15,625
17,576
19,683
21,952
24,389
27,000
Fourth
1
16
81
256
625
1,296
2,401
4,096
6,561
10,000
14,641
20,736
28,561
38,416
50,625
65,536
83,521
104,976
130,321
160,000
194,481
234,256
279,841
331,776
390,625
456,976
531,441
614,656
707,281
810,000
Fifth
1
32
243
1,024
3,125
7,776
16,807
32,768
59,049
100,000
161,051
248,832
371,293
537,824
759,375
1,048,576
1,419,857
1,889,568
2,476,099
3,200,000
4,084,101
5,153,632
6,436,343
7,962,624
9,765,625
11,881,376
14,348,907
17,210,368
20,511,149
24,300,000
Sixth
1
64
729
4,096
15,625
46,656
117,649
262,144
531,441
1,000,000
1,771,561
2,985,984
4,826,809
7,529,536
11,390,625
16,777,216
24,137,569
34,012,224
47,045,881
64,000,000
85,766,121
113,379,904
148,035,889
191,102,976
244,140,625
308,915,776
387,420,489
481,890,304
594,823,321
729,000,000
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