Indices and standard form PDF

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3 Indices and Standard Form 3.1 Index Notation Here we revise the use of index notation. You will already be familiar with the notation for squares and cubes 2

, and

3

this is generalised by defining:

... 3 144244 of these

Example 1 Calculate the value of: (a)

52

25

(b)

(c)

33

(d)

104

Solution (a)

5

52

5

25 (b)

2

25

2

2

3

3

2

2

32 (c)

3

33

27 (d)

10

104

10

10

10

10 000

Example 2 Copy each of the following statements and fill in the missing number or numbers: (a)

2

(b)

9

(c)

1000

(d)

53

2

2

2

2

2

2

2

3 10

37

MEP Y9 Practice Book A

3.1 Solution (a)

27

(b)

9

(c)

1000

(d)

53

2

2

3

2

2

2

2

32

3 10

10

5

5

5

2

103

10

Example 3 (a)

Determine 2 5 .

(b)

Determine 2 3 .

(c)

Determine 2 5 2 3 .

(d)

Express your answer to (c) in index notation.

Solution (a)

25

32

(b)

23

8

(c)

25

23

(d)

4

22

32 4

8

Exercises 1.

2.

Calculate: (a)

23

(b)

10 2

(c)

32

(d)

103

(e)

92

(f)

33

(g)

24

(h)

34

(i)

72

Copy each of the following statements and fill in the missing numbers: (a)

10

(b)

3

10 3

10 3

3

10

10

10

3 38

MEP Y9 Practice Book A

3.

(c)

7

7

7

7

7

7

(d)

8

8

8

8

8

8

(e)

5

5

(f)

19

(g)

6

(h)

11 11 11 11 11 11

5

19

6

19

19

6

6

6

19

6

6

6 11

Copy each of the following statements and fill in the missing numbers: (a)

8

(c)

100

(e) (g)

2

(b)

81

3

10

(d)

81

125

5

(f)

1 000 000

216

6

(h)

625

9 10

5

4.

Is 102 bigger than 210 ?

5.

Is 3 4 bigger than 4 3 ?

6.

Is 5 2 bigger than 25 ?

7.

Copy each of the following statements and fill in the missing numbers:

8.

(a)

49

2

(b)

64

3

(c)

64

6

(d)

64

2

(e)

100 000

(f)

243

5

5

Calculate: (a)

22

23

(b)

22

23

(c)

32

22

(d)

32

22

(e)

2 3 10 3

(f)

103

39

25

MEP Y9 Practice Book A

3.1 9.

10.

11.

Calculate: (a)

3

2

4

(b)

3

2

4

(c)

7

4

3

(d)

7

4

3

Writing your answers in index form, calculate: (a)

102

103

(c)

34

(e)

106

(a)

Without using a calculator, write down the values of

32

102

82

64 (b)

4

(b)

23 27

(d)

25

(f)

54 52

22

and

2

Complete the following:

2 15

32 768

2 14 (KS3/99Ma/Tier 5-7/P1)

3.2 Laws of Indices There are three rules that should be used when working with indices: When

and

are positive integers,

1. 2.

or

3.

These three results are logical consequences of the definition of , but really need a formal proof. You can 'verify' them with particular examples as below, but this is not a proof: 27

23

2 2

2 10

2 2

2 2

2 2

2 2

2

2

2

2

(here 40

2 2

2 2

2 2

7,

3 and

10 )

MEP Y9 Practice Book A

or, 27

23

2

2

2 2

2 2

2

2

2

2

2 2

2

24 Also,

27

3

27

27

2

(again

7,

3 and

(again

7,

3 and

)4

27

2 21 (using rule 1)

)21

The proof of the first rule is given below:

Proof ...3 144244 of these

...3 144244 of these

... ...3 14444 4244444 of these

The second and third rules can be shown to be true for all positive integers in a similar way.

and

We can see an important result using rule 2: 0

1, so

but

1

0

0 so, for example, 3

This is true for any non-zero value of

1001

0

1.

41

1, 270

1 and

MEP Y9 Practice Book A

3.2 Example 1

Fill in the missing numbers in each of the following expressions: (a)

24

26

2

(b)

(c)

36

32

3

(d)

37 39

10 4

3

3

10

Solution (a)

24

26

2

4

6

(b)

37 39

2 10 (c)

36

32

3

3

7

9

316 6

2

(d)

10 4

3

34

10

4

3

10 12

Example 2 Simplify each of the following expressions so that it is in the form is a number: 4

(a)

6

7

(b)

(a)

6

6

7

7

13

4

2

(b)

4

3

2 3

6 3

6

3

3

(c)

4

3

4

2 3

Solution

3

12

42

, where

(c)

4

3

MEP Y9 Practice Book A

Exercises 1.

2.

Copy each of the following statements and fill in the missing numbers: (a)

23

27

2

(b)

36

35

3

(c)

37

34

3

(d)

83

84

8

(f)

23

(h)

47 42

(e)

32

(g)

36 32

5

3

3

2

4

Copy each of the following statements and fill in the missing numbers: (a)

3

2

2 5

(c)

(b)

7

2

(d)

6

4

16

9

3

(e)

3.

Explain why 9 4

4.

Calculate:

5.

6

(f)

7

3 8.

(a)

30

40

(b)

60

70

(c)

80

30

(d)

60

20

40

Copy each of the following statements and fill in the missing numbers: (a)

36

3 17

3

(b)

46

6

4

(c) (e) (g)

19

95

8

6

(d) (f)

40

(h)

43

16

13

4 11

4 18

7

MEP Y9 Practice Book A

3.2 6.

7.

8.

Calculate: (a)

23 22

30

(b)

34 33

30

(c)

54 52

62 6

(d)

77 75

59 57

(e)

10 8 10 5

(f)

4 17 4 14

56 53

4 13 4 11

Fill in the missing numbers in each of the following expressions: (a)

82

(c)

25 6

(e)

125 4 5

(g)

81

2 5

4

(b)

81 3

(d)

47

(f)

1000 6 10

(h)

256

9

3

2

4

Fill in the missing numbers in each of the following expressions: (a)

8

2

4

2

(b)

25

625

2

(c)

3

243 9

5

5

(d)

3

2

128 16

2

3 9.

8

2

Is each of the following statements true or false? (a)

32

(c)

68 28

22

3

64

8

44

(b)

54

(d)

10 8 56

2 3 10 7

2

2

5

MEP Y9 Practice Book A

10.

Copy and complete each expression: (a)

26

23

23

24

(c)

(e)

4

62

68

4

2

(b)

36 32

2

4

2

(d)

32 9 33

4

27

5

2

4 4

6

63

6

78

(f)

72

5

3

3

4

4

3

3

5

7

73

5

3.3 Negative Indices Using negative indices produces fractions. In this section we practice working with negative indices. From our work in the last section, we see that 2

3

2

3

2

3

1

but we know that

1

So clearly,

1

1

In same way, 1

2

2

1

1

3

3

1

45

, a fraction.

7

3.3

MEP Y9 Practice Book A

and, in general, 1

for positive integer values of The three rules at the start of section 3.2 can now be used for any integers and not just for positive values.

Example 1 Calculate, leaving your answers as fractions: (a)

3

2

1

(b)

2

(b)

64

1

4

(c)

5

Solution 1 32 1 9

2

(a)

3

(b)

2

1

(c)

5

3

1 2 1 4

1

4

1 4

1 53

1 125

Example 2 Simplify: (a)

67 69

Solution (a)

67 69

6

6

(b)

64

6

3

7

9

1 62

2

6 6

4

4

1 36 3

3

61

6

46

6

3

(c)

10 2

3

3

MEP Y9 Practice Book A

(c)

3

10 2

10 1 10 6

6

1 1000 000

Exercises 1.

2.

Write the following numbers as fractions (a)

4

1

(d)

7

2

(c)

(e)

4.

(e)

2

(c)

10

4

3

(f)

6

3

2

Copy the following expressions and fill in the missing numbers: (a)

3.

(b)

:

3

1 49

1

7

(b)

7

1 81

1

9

(d)

9

1 10 000 000

1

10

(f)

10

1 100

1

10

10

1 16

1

2

2

1 1024

1

2

2

Calculate: (a)

4

(c)

5

(e)

4

1

1

3

1

10

1

10

1

1

(b)

6

(d)

10

(f)

6

1

2

1

1

2

10

3

1

7

Simplify the following expressions giving your answers in the form of a number to a power: (a)

47

(c)

74 6 7

(e)

6

(g)

72 2 7

6

4

(b) (d)

2

3

47

57

32

(f)

84

(h)

89 9 8

5

3

4

8

9

3.3

MEP Y9 Practice Book A

5.

6.

Copy each of the following expressions and fill in the missing numbers; (a)

1 9

(c)

1 125

(e)

62 63

(b)

1 100

10

5

(d)

5 54

5

6

(f)

22 210

2

3

Simplify the following expressions: 8

(a)

7

(b)

3

9

4

(c)

(e)

7.

8.

6

(d)

8

1

4

8

(f)

2

4

3

Copy and complete the following statements: (a)

0.1 10

(b)

0.25

(c)

0.0001 10

(d)

0.2

(e)

0.001 10

(f)

0.02

2 5 50

Copy the following expressions and fill in the missing numbers: 4 2

(a)

(b)

6

2

7

(c)

9

2

2

(d)

3

(e)

4

(f)

48

3

6

MEP Y9 Practice Book A

9.

10.

Copy the following expressions and fill in the missing numbers: (a)

1 8

(c)

1 81

2 9

3

If

1

and

2

, express

(b)

1 25

(d)

1 10 000

as a power of

5 10

without having any

fractions in your final answer.

3.4 Standard Form Standard form is a convenient way of writing very large or very small numbers. It is used on a scientific calculator when a number is too large or too small to be displayed on the screen. Before using standard form, we revise multiplying and dividing by powers of 10.

Example 1 Calculate: (a)

3

10 4

(b)

3 .27

10

(c)

3

10 2

(d)

4.32

10 4

Solution (a)

3

10 4

3

10 000

30 000 (b)

3.27

10 3

3.27

1000

3270 (c)

3

10 2

3 100

0.03

49

3

MEP Y9 Practice Book A

3.4 (d)

4.32

4 .32 10 000

10 4

432 1000 000 0.000432 These examples lead to the approach used for standard form, which is a reversal of the approach used in Example 1. In

, numbers are written as

10 where 1

10 and

is an integer.

Example 2 Write the following numbers in standard form: (a)

5720

(b)

7.4

(c)

473 000

(d)

6 000 000

(e)

0.09

(f)

0.000621

Solution (a)

(b)

(c)

5720

7.4

5.72
...