Year 8 Laws of Indices

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 29th August 2015

“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”.

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

ab × ac = ab+c?

1st Law of Indices

!

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

ab

ac= ab-c?

2nd Law of Indices

!

(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

!

(ab)c = abc?

3rd Law of Indices

!

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

?

a0 = 1

a-b =

a1 = a

1ab

Final Laws of Indices

!

Mastermind

Occupation: StudentFavourite Teacher: Dr FrostSpecialist Subject: Laws of Indices

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 >

_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

_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

_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

_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

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

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

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

??

???

???

?

??

??

???

???

?

???

?

?

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.

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

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? ?

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

?????

?

??

???

??

??

?