year 8 laws of indices dr j frost ([email protected]) last modified: 29 th august 2015
TRANSCRIPT
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“Power” or “index” (plural: “indices”) or “exponent”
34
?“Base”?
The whole expression is sometimes (confusingly) referred to as a ‘power’ or ‘power expression’.
Terminology Recap
!
! We say this as “3 to the power of 4” or “3 raised to the power of 4” or “3 to the 4”.
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35 × 34
How would I write this multiplication out in full?Therefore, how could I write the result of this multiplication in the form 3k?
39?
Multiplying Power Expressions with the Same Base
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ab × ac = ab+c?
1st Law of Indices
!
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47
42How would I write this multiplication out in full?Therefore, how could I write the result of this multiplication in the form 4k?
45?
Dividing power expressions with the same base
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ab
ac= ab-c?
2nd Law of Indices
!
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(42)3
How would I write this multiplication out in full?Therefore, how could I write the result of this multiplication in the form 4k?
46?
Raising a power to a power
!
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(ab)c = abc?
3rd Law of Indices
!
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3-1
At this point, it doesn’t make sense to say “Multiply 3 by itself negative 1 times”. We’ll have to use a different approach!
33 = 2732 = 931 = 330 = 1
3-1 =
3-2 =
30
13
19
?
?
Is there a pattern we can see that will help us out?
Zero and negative indices
?
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a0 = 1
a-b =
a1 = a
1ab
Final Laws of Indices
!
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Mastermind
Occupation: StudentFavourite Teacher: Dr FrostSpecialist Subject: Laws of Indices
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Instructions: Everyone starts by standing up. You’ll get a question with a time limit to answer. If you run out of time or get the question wrong, you sit down. The winner is the last man standing.Warmup:
23 × 24 = 27
?Start
Question >
(23)4 = 212
?Start
Question >
26
23 = 23?Start
Question >
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_1_2
47 × 43 = 410
?Start
Question >
(35)2 = 310
?Start
Question >
91
1
92
= 99?Start
Question >
74 × 76 = 710
?Start
Question >
57
53 = 54?Start
Question >
(46)3 = 418
?Start
Question >
(22)2 = 24
?Start
Question >
2-1 =
?Start
Question >
a b c
d e f
g h
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_1_8
_1_16
77 × 7-2 = 75
?Start
Question >
(53)-2 = 5-
6
?Start
Question >
105
102
= 103
?Start
Question >
8-2 × 84 = 82
?Start
Question >
87
8-
2
= 89?Start
Question >
2-3 =?Start
Question >
50 = 1?Start
Question >
4-2 =
?Start
Question >
a b c
d e f
g h
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_1_56
_1_27
4-2 × 4-2 = 4-4
?Start
Question >
(3-2)-2 = 34
?
Start Question
>
9-
2
9-
2
= 90 = 1
?Start
Question >
14 × 16 = 110 = 1
?Start
Question >
101
10-3
= 104
?Start
Question >
(5-3)2 =5-6 =
?Start
Question > 3-3 =
?
Start Question
>
a b c
d e
g h
INSTANT DEATH
Start Question
>
f
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_1_81
50 = 1?Start
Question >
(30)2 = 1?Start
Question >
50
5-
2
= 52?Start
Question >
(24 × 26)2 = 220
?Start
Question >
51 x 52 x 53 = 56
?
Start Question
>((41)2)3 = 46
?Start
Question >
(23 × 23)3 = 218
?Start
Question >
3-4 =
?
Start Question
>
a b c
d e f
g h
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47 × 43
42
(35
)4
33
a
= 48
?Start
Question >
= 317
?Start
Question >
b(73
)3
(72
)3
= 73
?Start
Question >
c
58 × 58
51 × 5-1
((32)2
)2
32
= 516
?Start
Question >
= 36
?Start
Question >
e(71
)3
(72
)1
×74 = 75
?Start
Question >
fd
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What is half of ? What is a ninth of ?
What is a quarter of ?
What is the square root of ?
𝟐𝟕
𝟐=𝟐𝟕
𝟐𝟏=𝟐𝟔 𝟑𝟗𝟗
𝟗=𝟑𝟗𝟗
𝟑𝟐 =𝟑𝟗𝟕
𝟒𝒙
𝟒=𝟒𝒙
𝟒𝟏=𝟒𝒙 −𝟏 𝟑𝟒𝒃𝒆𝒄𝒂𝒖𝒔𝒆 (𝟑𝟒 )𝟐=𝟑𝟖
? ?
? ?
Challenges
What is x?The LHS is .
So ?
1 2
3 4
5
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Exercise 1Simplify the following.
a) b) c) d) e) f) g)
i)
1
Please ensure you write out the question.
2a) b) c) d)
Simplify the following.
3 Evaluate the following (i.e. give as a fraction or integer with no power)
a) b) c) d) e) f) g)
4 Simplify the following.
Solve
Given that , express as a single power of 4.
N1
a) b) c)
N2
??
???
???
?
??
??
???
???
?
???
?
?
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910=3𝑥(Hint: can we express 9 as a power of 3 perhaps?)
(32 )10=3𝑥?
Changes of Base
Solve the following equation. Express as a single power.
4𝑥×8𝑥+1
(𝟐𝟐)𝒙× (𝟐𝟑 )𝒙+𝟏?
The strategy therefore is to find what both bases are a power of (e.g. 4 and 8 are both powers of 2), and replace them as such.
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A few more examples…
4𝑥+1=83
(𝟐𝟐)𝒙+𝟏=(𝟐𝟑 )𝟑
125𝑥+1=25𝑥
(𝟓𝟑)𝒙+𝟏=(𝟓𝟐 )𝒙? ?
Express as a single power of 3:
?
Solve Solve
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28 = 4x
x = 8
274 = 9x
x = 6?x = 4?
8x = 412
? x = 300
3x = 27100
?
Test Your Understanding
Express as a single power:
𝟑𝒙×𝟐𝟕𝒙 −𝟏=𝟑𝟒𝒙 −𝟑
Express as a single power:
𝟒𝟏𝟎×𝟖𝒙=𝟐𝟐𝟎+𝟑 𝒙? ?
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Exercise 2 Please ensure you write out the question.
Solve the following.
2𝑥=410 𝒙=𝟐𝟎9𝑥+1=27𝑥−1𝒙=𝟓
Solve the following.
Express as a single power.
4𝑥×23=𝟐𝟐𝒙+𝟑1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
N
?????
?
??
???
??
??
?