25 INDICES, STANDARD FORM AND SURDS

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In this chapter you will: work out the value of an expression with zero,

negative or fractional indices convert between standard form and ordinary numbers calculate with numbers in standard form manipulate surds make estimates to calculations using numbers in

standard form.

The photo shows a male Escheria coli bacteria. You may have heard of e-coli. These bacteria are commonly known in relation to food poisoning as they can cause serious illness. Each bacterium is about a millionth of a metre long. That can be written as 0.000001m, or in standard form as 1 × 106m long. Standard form allows us to write both very large and very small numbers in a more useful form.

Objectives Before you start

You need to be able to: use the index laws round numbers to one signifi cant fi gure.

HI- RES

ST ILL

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BE SU

PPL IED

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25.1 Using zero and negative powers

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25.1 Using zero and negative powers

You know that n0  1 when n ≠ 0. You know the meaning of negative indices.

Objectives

If you are x metres from a live band, the volume of sound they are producing is directly proportional to x2. This means that if you halve your distance to the band, the music will get four times as loud.

Why do this?

Work out 1. 32 2. 25 3. 43

Get Ready

For non-zero values of a a0  1

For any number n an  1 __

an

Key Points

Work out the value of a 30 b 5–1 c 6–2 d (� 2 __ 5 )

–2

a 30  1

b 51  1 __ 5

c 62  1 __ 6 2

 1 __ 36

d (� 2 __ 5 ) 2

 1

(� 2 __ 5 ) 2

 (� 5 __ 2 ) 2

 25 ___ 4

Example 1

Any number to the power of zero is 1.

Use the rule an  1 __ an

62  6  6  36.

To work out the reciprocal of a fraction, turn the fraction upside down. Square the number on the top and the number on the bo� om of the fraction.

Examiner’s Tip

Do not change the fraction to a decimal. It is much easier to square the numbers in a fraction than it is to square a decimal.

Chapter 25 Indices, standard form and surds

612 standard form integer

1 Write down the value of these expressions. a 70 b 81 c 51 d 40

e (2)3 f 92 g 104 h 1450

i (3)2 j (8)0 k 160 l 106

2 Work out the value of these expressions. a (� 1 _ 3 )

1 b (� 2 _ 7 ) 1 c (� 1 _ 7 )

2 d (� 1 _ 4 ) 3

e (0.25)2 f (� 2 _ 5 ) 3 g (� 5 _ 3 )

0 h (� 9 _ 5 ) 1

i (�1 2 _ 5 ) 2 j (�1 1 _ 3 )

3 k (0.1)4 l (0.2)3

Exercise 25A

25.2 Using standard form

You can convert ordinary numbers into standard form. You can convert numbers in standard form into

ordinary numbers. You can calculate with numbers in standard form You can convert to standard form to make sensible

estimates for calculations.

Objectives

Astronomers use standard form to record large measurements. The Sun’s diameter is about 1.392  106 km. Biologists working with micro- organisms sometimes use standard form to record their very small sizes, like 2.1  104 cm.

Why do this?

1. Work out a 103 b 10–2

2. Write 10 000 as a power of 10. 3. Work out 2.35  10 000.

Get Ready

Standard form is used to represent very large (or very small) numbers. A number is in standard form when it is in the form a  10n where 1  a  10 and n is an integer.

To use standard form you need to know how to write powers of ten in index form. 10  101 100  10  10  102 1000  10  10  10  103

A number in standard form looks like this. 6.7  104

This part is written as a This part is written as a number between 1 and 10. power of 10.

These numbers are all in standard form – 4.5  102, 9  108, 1.2657  106. These numbers are not in standard form – 67  109, 0.087  103 – because the fi rst number is not between

1 and 10. It is often easier to multiply and divide very large or very small numbers, or estimate a calculation if the numbers

are written in standard form. To input numbers in standard form into your calculator, use the 10 or EXP key.

To enter 4.5  107 press the keys 4 · 5 � 10 7 .

Key Points

Questions in this chapter are targeted at the grades indicated.

B

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25.2 Using standard form

Write these numbers in standard form. a 50 000 b 34 600 000 c 682.5

a 50 000  5  10 000  5  104

b 34 600 000  3.46  10 000 000  3.46  107

c 682.5  6.825  100  6.825  102

Write as an ordinary number a 8.1  105 b 6  108

a 8.1  105  8.1  100 000  810 000 b 6  108  6  100 000 000  600 000 000

Example 2

Example 3

Use 3.46 not 34.6 or 346 as 3.46 is between 1 and 10.

1 Write these numbers in standard form. a 700 000 b 600 c 2000 d 900 000 000 e 80 000

2 Write these as ordinary numbers. a 6  105 b 1  104 c 8  105 d 3  108 e 7  101

3 Write these numbers in standard form. a 43 000 b 561 000 c 56 d 34.7 e 60

4 Write these as ordinary numbers. a 3.96  104 b 6.8  107 c 8.02  103 d 5.7  101 e 9.23  100

5 In 2008 there were approximately 7 000 000 000 people in the world. Write this number in standard form.

6 The circumference of Earth is approximately 40 000 km. Write this number in standard form.

Exercise 25B

Write in standard form a 0.000 000 006 b 0.000 56

a 0.000 000 006  6  0.000 000 001

 6  1 __________ 1 000 000 000

 6  1 __ 10 9

 6  109

b 0.000 56  5.6  0.0001

 5.6  1 _____ 10 000

 5.6  1 __ 10 4

 5.6  104

Example 4

0.000 000 001 is equivalent to 1 ___________ 1 000 000 000 .

Using an  1 __ an

Use 5.6 rather than 56 as 5.6 is between 1 and 10.

B

Chapter 25 Indices, standard form and surds

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1 Write these numbers in standard form. a 0.005 b 0.04 c 0.000 007 d 0.9 e 0.0008

2 Write these as ordinary numbers. a 6  105 b 8  102 c 5  107 d 3  101 e 1  108

3 Write these numbers in standard form. a 0.0047 b 0.987 c 0.000 803 4 d 0.000 15 e 0.601

4 Write these as ordinary numbers. a 8.43  105 b 2.01  102 c 4.2  107 d 7.854  101 e 9.4  104

5 Write these numbers in standard form. a 457 000 b 0.0023 c 0.0003 d 2 356 000 e 0.782 f 89 000 g 200 h 0.005 26 i 6034 j 0.000 008 73

6 Write these as ordinary numbers. a 4.12  104 b 3  103 c 2.065  107 d 4  106 e 3.27  108 f 7.5  101 g 1.5623  102 h 5.12  107 i 2.7  105 j 6.12  101

7 1 micron is 0.000 001 of a metre. Write down the size of a micron, in metres, in standard form.

8 A particle of sand has a diameter of 0.0625 mm. Write this number in standard form.

Exercise 25C

Write in standard form a 40  102 b 0.008  102

Method 1 a 40  102  4  101  102

 4  1012

 4  103

b 0.008  10–2  8  103  102

 8  103  2

 8  105

Example 6

Write 40 in standard form. Use the rule am  an  amn.

Write 0.008 in standard form. Use am  an  amn.

Write as an ordinary number a 3  106 b 1.5  103

a 3  106  3 ___ 106

b 1.5  103  1.5 ___ 103

 3 _______ 1 000 000  15 ____ 1000

 0.000 003  0.0015

Example 5

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Examiner’s Tip

The power of 10 tells you how many 0s there are. 102  100 2 zeros 102  0.01 2 zeros

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25.2 Using standard form

1 Write these in standard form. a 45  103 b 980  103 c 3400  102 d 186  1010

2 Write these in standard form. a 0.009  105 b 0.045  106 c 0.3708  1012 d 0.006  107

3 Some of these numbers are not in standard form. If a number is in standard form then say so. If a number is not in standard form then rewrite it so that it is in standard form. a 7.8  104 b 890  106 c 13.2  105 d 0.56  109

e 60 000  108 f 8.901  107 g 0.040 05  1010 h 9080  1015

i 6.002  105 j 0.0046  108 k 67 000  103 l 0.004  103

4 Write these numbers in order of size. Start with the smallest number. 6.3  106, 0.637  107, 6290000, 63.4  105

5 Write these numbers in order of size. Start with the smallest number. 0.034  102, 3.35  105, 0.000033, 37  104

Exercise 25D

Work out (3  106)  (4  103) giving your answer in standard form.

(3  106)  (4  103)  3  4  106  103

 12  109

 1.2  101  109

 1.2  1010

By writing 760 000 000 and 0.000 19 in standard form correct to one signifi cant fi gur