25 INDICES, STANDARD FORM AND SURDS...Chapter 25 Indices, standard form and surds 614 1 Write these...

17
25 INDICES, STANDARD FORM AND SURDS 610 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 6 m 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 significant figure. HI-RES STILL TO BE SUPPLIED

Transcript of 25 INDICES, STANDARD FORM AND SURDS...Chapter 25 Indices, standard form and surds 614 1 Write these...

Page 1: 25 INDICES, STANDARD FORM AND SURDS...Chapter 25 Indices, standard form and surds 614 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

25 INDICES, STANDARD FORM AND SURDS

610

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 × 106m 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 STILL TO

BE SUPPLIED

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611

25.1 Using zero and negative powers

611

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 x2. This means that if you halve your distance to the band, the music will get four times as loud.

Why do this?

Work out1. 32 2. 25 3. 43

Get Ready

For non-zero values of aa0 1

For any number nan 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 51 1 __ 5

c 62 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 an 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.

Page 3: 25 INDICES, STANDARD FORM AND SURDS...Chapter 25 Indices, standard form and surds 614 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

Chapter 25 Indices, standard form and surds

612 standard form integer

1 Write down the value of these expressions.a 70 b 81 c 51 d 40

e (2)3 f 92 g 104 h 1450

i (3)2 j (8)0 k 160 l 106

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 2.1 104 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 anumber between 1 and 10. power of 10.

These numbers are all in standard form – 4.5 102, 9 108, 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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613

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 000b 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 forma 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

6 109

b 0.000 56 5.6 0.0001

5.6 1 _____ 10 000

5.6 1 __ 10 4

5.6 104

Example 4

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

Using an 1 __ an

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

B

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Chapter 25 Indices, standard form and surds

614

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 6 105 b 8 102 c 5 107 d 3 101 e 1 108

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 8.43 105 b 2.01 102 c 4.2 107 d 7.854 101 e 9.4 104

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

6 Write these as ordinary numbers.a 4.12 104 b 3 103 c 2.065 107 d 4 106 e 3.27 108

f 7.5 101 g 1.5623 102 h 5.12 107 i 2.7 105 j 6.12 101

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 forma 40 102 b 0.008 102

Method 1a 40 102 4 101 102

4 1012

4 103

b 0.008 10–2 8 103 102

8 103 2

8 105

Example 6

Write 40 in standard form.Use the rule am an amn.

Write 0.008 in standard form.Use am an amn.

Write as an ordinary numbera 3 106 b 1.5 103

a 3 106 3 ___

106 b 1.5 103 1.5 ___

103

3 _______ 1 000 000 15 ____ 1000

0.000 003 0.0015

Example 5

B

Examiner’s Tip

The power of 10 tells you how many 0s there are.102 100 2 zeros102 0.01 2 zeros

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615

25.2 Using standard form

1 Write these in standard form.a 45 103 b 980 103 c 3400 102 d 186 1010

2 Write these in standard form.a 0.009 105 b 0.045 106 c 0.3708 1012 d 0.006 107

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 13.2 105 d 0.56 109

e 60 000 108 f 8.901 107 g 0.040 05 1010 h 9080 1015

i 6.002 105 j 0.0046 108 k 67 000 103 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. 0.034 102, 3.35 105, 0.000033, 37 104

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 gure, work out an approximation for 760 000 000 0.000 19.760 000 000 8 108 correct to one signifi cant fi gure.0.000 19 2 104 correct to one signifi cant fi gure.

760 000 000 _____________ 0.000 19

8 108 _________

2 104

8 __ 2

108 _____

104

4 1084

4 1012

Example 7

Example 8

Rearrange the expression so the powers of 10 are together.Multiply the numbers.Use am an amn to multiply the powers of 10.12 109 is not in standard form.Write your fi nal answer in standard form.

Rearrange the expression so the powers of 10 are together.Divide the numbers.Use am an amn to divide the powers of 10.

B

Method 2a 40 102 40 100 4000 4 103

b 0.008 102 0.008 1 ___ 100

0.008 100 0.000 08 8 105

Work out the calculation.Change the answer into standard form.

Use the rule an 1 __ an .Multiplying by 1 ___ 100 is the same as dividing by 100.

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Chapter 25 Indices, standard form and surds

616

1 Work out and give your answer in standard form.a (4 108) (2 103) b (6 105) (1.5 103) c (4 107) (3 105)d (6 107) (3 106) e (6 109) (5 103) f (5 108) (2 103)

2 Work out and give your answer in standard form.a (4 108) (2 103) b (9 105) (2 104) c (3 109) (6 103)d (8.6 108) (2 1013) e (1 1012) (4 103) f (7 109) (7 105)

3 Express in standard form.a (2 105)2 b (5 105)2 c (4 106)2 d (7 108)2

4 By writing these numbers in standard form correct to one signifi cant fi gure, work out an estimate of the value of these expressions. Give your answer in standard form.a 600 008 598 b 78 018 4180 c 699 008 198 d 8 104 660 000 0.000 078

5 Light travels at 3 108 metres per second. Work out the time it takes light to travel: a 200 metres b 1.5 centimetres.

6 The base of a microchip is in the shape of a rectangle. Its length is 2 103 mm and its width is 1.6 103 mm. Find the area of the base. Give your answer in mm2 in standard form.

7 The distance of the Earth from the Sun is approximately 93 000 000 miles. Light travels at a speed of approximately 300 000 kilometres per second.Work out an estimate of the time it takes light to travel from the Sun to the Earth.

8 An atomic particle has a lifetime of 3.86 105 seconds. It travels at a speed of 4.2 106 metres per

second. Calculate an approximation for the distance it travels in its lifetime.

A

AO3

AO2AO3

Exercise 25E

Use a calculator to work out a (3.4 106) (7.1 104 ) b (4.56 108) (3.2 103)

a (3.4 106) (7.1 104) 2.414 1011

b (4.56 108) (3.2 103) 1.425 1011

Example 9

Use the EXP of 10x on your calculator.

Hint

In both of these cases the brackets need not be used, but in more complex expressions the brackets must be used.

Hint

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617

25.2 Using standard form

x 3.1 1012, y 4.7 1011

Use a calculator to work out the value of x y

_____ xy

.

Give your answer in standard form correct to 3 signifi cant fi gures.

(3.1 1012 4.7 1011)

_______________________ (3.1 1012 4.7 1011)

3.57 1012 _____________

1.457 1024

2.4502… 1012

2.45 1012

Example 10

Substitute the values into the expression.

Hint

Include brackets here to ensure that the answer from the calculation on the top of the fraction is divided by the answer to the calculation on the bottom of the fraction.

Hint

Write the number from your calculator correctly in standard form showing more than 3 signifi cant fi gures.

Give your answer correct to 3 signifi cant fi gures.

1 Evaluate these expressions, giving your answers in standard form.

a 500 600 700 b 0.006 0.004 c 0.08 480 _________ 180 d 89000 0.0086 _____________ 48 0.25

e 65 120 ________ 1500 f 8.82 5.007 __________ 10000 g (12.8)4 h (36.4 24.2)3

2 Evaluate these expressions. Give your answers in standard form.a (3.2 1010) (6.5 106) b (1.3 107) (4.5 106)c (2.46 1010) (2.5 106) d (3.6 1020) (3.75 106)

3 Express as a number in standard form.a (3.2 108) (6.5 106) b (1.3 107) (4.5 106)c (2.46 1010) (2.5 106) d (3.6 1020) (3.75 106)

4 Evaluate these expressions. Give your answers in standard form correct to 3 signifi cant fi gures.a (3.5 1011) (6.5 106) b (1.33 1010) (4.66 104)c (5.3 108) (6.45 106) d (3.24 108) (6.4 104)

5 Express as a number in standard form correct to 3 signifi cant fi gures.a (3.5 1011) (6.5 106) b (1.33 1010) (4.66 104)c (5.3 108) (6.45 106) d (3.24 108) (6.4 104)

Exercise 25F

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Chapter 25 Indices, standard form and surds

618

6 x 3.5 109, y 4.7 105

Work out the following. Give your answer in standard form correct to 3 signifi cant fi gures.

a x __ y b x(x 800y) c

xy ________

x 800y d (� x ____ 2000 ) 2 y2

7 x 2.4 105, y 9.6 106

Evaluate these expressions. Give your answer in standard form correct to 3 signifi cant fi gures where necessary.

a x2 __

y b

x2 y2

______ x y c

xy _____

x y

8 The distance of the Earth from the Sun is 1.5 108 km.The distance of the planet Neptune from the Sun is 4510 million km.Write in the form 1 : n the ratiodistance of the Earth from the sun : distance of the planet Neptune from the Sun

9 The mass of a uranium atom is 3.98 1022 grams. Work out the number of uranium atoms in 2.5 kilograms of uranium.

1 Work out the value of these expressions.

a 30 b 61 c 72 d 53

e (� 1 _ 9 ) 1

f (� 5 _ 2 ) 1

g (� 3 _ 4 ) 2

h (�1 1 _ 2 ) 3

2 Write these numbers in standard form.

a 45 000 b 0.000 62 c 894 d 0.007 21

e 100 000 000 f 0.000 000 000 007 g 90 h 0.16

3 Write these as ordinary numbers.

a 5.8 104 b 2 105 c 4.03 107 d 4 106

e 8.45 101 f 3.152 102 g 9.2 101 h 7 104

4 Write these in standard form.

a 278 104 b 0.087 109 c 89.2 104 d 5660 108

5 Work out and give your answer in standard form.

a (5 103) (7 109) b (9 107) (2 105) c (8.4 106) (2 105)

d (4.3 107) (8 103) e 800 000 0.000 000 02 f 0.000 002 4 5 000 000

Mixed exercise 25G

B

A

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619

25.3 Working with fractional indices

25.3 Working with fractional indices

You know the meaning of fractional indices.

Objective

Fractional indices are used when you model the rates at which things vibrate, such as your voice box.

Why do this?

Work out1. √

___ 100 2. 3 √

__ 8 3. 3 √

____ 27

Get Ready

Indices can be fractions. In general,

a 1 __ n n √

__ a

In particular, this means that

a 1 __ 2 √

__ a and a

1 __ 3 3 √

__ a

Key Points

Find the value of the following

a 2 5 1 __ 2 b (1000 )

1 __ 3 c 160.25

a 2 5 1 __ 2 √

___ 25

5

b (1000 ) 1 __ 3 3 √

_______ 1000

10

c 160.25 1 6 1 __ 4 1 ____

1 6 1 __ 4

1 _____ 4 √

___ 16

1 __ 2

Work out the value of a 8 2 _ 3 b 1 6 3 _ 4

a 8 2 __ 3 ( 8

1 __ 3 )2

22

4b 1 6 3 __ 4 1 ____

1 6 3 __ 4

1 ______ (1 6

1 __ 4 )3

1 ___ 23

1 __ 8

Example 11

Example 12

The square root of 25 is 5 because 5 5 25.

The cube root of 1000 is 10 because 10 10 10 1000.

Change the decimal into a fraction 0.25 1 __ 4 .Use the rule an 1 __ an .

1 6 1 __ 4 4 √

___ 16 2

because 24 16

Use the rule (am)n amn.Work out the cube root of 8 fi rst.Then square your answer.

Use an 1 __ an .Examiner’s Tip

It is easier to work out the root fi rst as this makes the numbers smaller and easier to manage.

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Chapter 25 Indices, standard form and surds

620 surd

1 Work out the value of the following.a 9

1 _ 2 b 4 9 1 _ 2 c 10 0

1 _ 2 d 4 1 _ 2 e (� 1 _ 4 )

1 _ 2

2 Work out the value ofa 2 7

1 _ 3 b 100 0 1 _ 3 c (64 )

1 _ 3 d 12 5 1 _ 3 e (� 1 _ 8 )

1 _ 3

3 Work out the value ofa 16

1 _ 4 b 4 1 _ 2 c 125

1 _ 3 d (� 1 __ 32 )

1 _ 5 e (� 4 _ 9 )

1 _ 2

4 Work out the value ofa 2 7

2 _ 3 b 100 0 2 _ 3 c 6 4

2 _ 3 d 1 6 3 _ 4 e 2 5

3 _ 2

5 Work out, as a single fraction, the value ofa 125

2 _ 3 b 10 000

3 _ 4 c 27

1 _ 3 d 8

2 _ 3 e 64

3 _ 2

f 12 5 2 _ 3 (� 1 _ 5 ) 2 g 8 1 _ 3 (� 2 _ 5 )

2

6 Find the value of n. a 1 _ 8 8n b 64 2n c 1 __

√__

5 5n d ( √

__ 7 )5 7n e ( 3 √

__ 2 )11 2n

Exercise 25H

25.4 Using surds

You can simplify surds. You can expand expressions involving surds. You can rationalise the denominator of a fraction.

Objectives

Surds occur in nature. The golden ratio 1 √__

5 _____ 2

occurs in the arrangement of branches along the stems of plants, as well as veins and nerves in animal skeletons.

Why do this?

1. Write down the fi rst 10 square numbers.2. Write down the value of a √

__ 36 b √

___ 100

3. Which of these has an exact answer: √__

5 , √__

9 , √__

37 , √__

64 ?

Get Ready

A number written exactly using square roots is called a surd. √

__ 2 and √

__ 3 are both surds.

2 √__

3 and 5 √__

2 are examples of numbers in surd form. √

__ 4 is not a surd as √

__ 4 2.

These two laws can be used to simplify surds.

√__

m √__

n √___

mn √__

m ___

√__

n √

__

m __ n

Simplifi ed surds should never have a surd in the denominator.

Key Points

B

A

A

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621rationalise the denominator

25.4 Using surds

To rationalise the denominator of a fraction means to get rid of any surds in the denominator.

To rationalise the denominator of a __ √

__ b you multiply the fraction by √

__ b __

√__

b . This ensures that the fi nal fraction has an

integer as the denominator.

a __ √

__ b a __

√__

b √

__ b __

√__

b

a √__

b ______

√__

b √__

b

a √__

b ____ b

Simplify √__

12 .

√___

12 √______

4 3

√__

4 √__

3

2 √__

3

Expand and simplify (2 √__

3 )(4 √__

3 ).

(2 √__

3 )(4 √__

3 ) 8 2 √__

3 4 √__

3 √__

3 √__

3

8 6 √__

3 3

11 6 √__

3

Example 13

Example 14

Use √__

m √__

n √____

mn . √

__ 4 2.

Multiply out the brackets.

Simplify the expression.

1 Find the value of the integer k.a √

__ 8 k √

__ 2 b √

__ 18 k √

__ 2 c √

__ 50 k √

__ 2 d √

__ 80 k √

__ 5

2 Simplifya √

___ 200 b √

__ 32 c √

__ 20 d √

__ 28

3 Solve the equation x2 30, leaving your answer in surd form.

4 Expand these expressions. Write your answers in the form a b √__

c where a, b and c are integers.a √

__ 3 (2 √

__ 3 ) b ( √

__ 3 1)(2 √

__ 3 ) c ( √

__ 5 1)(2 √

__ 5 )

d ( √__

7 1)(2 √__

7 ) e (2 √__

3 )2 f ( √__

2 5)2

5 The area of a square is 40 cm2. Find the length of one side of the square. Give your answer as a surd in its simplest form.

6 The lengths of the sides of a rectangle are (3 √__

5 ) cm and (3 √__

5 ) cm. Work out, in their simplifi ed forms:a the perimeter of the rectangle b the area of the rectangle.

7 The length of the side of a square is (1 √__

2 ) cm. Work out the area of the square. Give your answer in the form (a b √

__ 2 ) cm2 where a and b are integers.

Exercise 25I

A

A

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Chapter 25 Indices, standard form and surds

622

Rationalise the denominator of 2 ___ √

__ 3 .

2 ___ √

__ 3 2 ___

√__

3 √

__ 3 ___

√__

3

2 √__

3 ________ √

__ 3 √

__ 3

2 √__

3 ____ 3

Rationalise the denominator of 15 √__

5 _______ √

__ 5 and give your answer in the form a b √

__ 5 .

15 √__

5 ________ √

__ 5 15 √

__ 5 ________

√__

5 √

__ 5 ___

√__

5

15 √__

5 √__

5 √__

5 ________________ √

__ 5 √

__ 5

15 √__

5 5 _________ 5

1 3 √__

5

Example 15

Example 16

Multiply the fraction by √

__ 3 ___

√__

3 .

Simplify the denominator by using the fact that √

__ 3 √

__ 3 3.

Simplify the fraction by dividing both parts of the expression on the top of the fraction by 5.

1 Rationalise the denominators.a 1 ___

√__

2 b 1 ___

√__

5 c 2 ___

√__

7 d 5 ___

√__

3 e 5 ____

√__

11

2 Rationalise the denominators and simplify your answers.a 10 ___

√__

2 b 15 ___

√__

3 c 5 ____

√__

10 d 2 ___

√__

2 e 4 ____

√__

12

3 Rationalise the denominators and give your answers in the form a b √__

c where a, b and c are integers.

a 2 √__

2 ______ √

__ 2 b 6 √

__ 2 ______

√__

2 c 10 √

__ 5 _______

√__

5 d 12 √

__ 3 _______

√__

3 e 14 √

__ 7 _______

√__

7

4 The lengths of the two shorter sides of a right-angled triangle are √__

7 cm and 2 √__

3 cm. Find the length of the hypotenuse.

5 The diagram shows a right-angled triangle. The lengths are given in centimetres.Work out the area of the triangle.Give your answer in the form a b √

__ c where a, b and c are integers.

6 Solve these equations leaving your answers in surd form.a x2 6x 2 0 b x2 10x 14 0

Exercise 25J

2 3

9 2

Watch Out!

Remember to multiply both parts of the expression on the top of the fraction.

Watch Out!

A

A

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623

Chapter review

7 The diagram represents a right-angled triangle ABC.AB ( √

__ 7 2 ) cm AC ( √

__ 7 2 ) cm.

Work out, leaving any appropriate answers in surd form:a the area of triangle ABCb the length of BC.

( 7 � 2)

( 7 � 2)A

C

B

For non-zero values of aa0 1

For any number nan 1 __

an

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. 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 Indices can be fractions. In general,

a 1 __ n n √

__ a

A number written exactly using square roots is called a surd. These two laws can be used to simplify surds.

√__

m √__

n √___

mn √__

m ___

√__

n √

__

m __ n

Simplifi ed surds should never have a surd in the denominator. To rationalise the denominator of a fraction means to get rid of any surds in the denominator.

To rationalise the denominator of a __ √

__ b you multiply the fraction by √

__ b __

√__

b , this ensures that the fi nal fraction has an

integer as the denominator.

Chapter review

1 Work out the values ofa 40 b 41 c 20 d 23

2 Work out the values ofa 30 b (3)0 c 31 d (� 1 __ 3 ) 0

3 Work out the values ofa 1 ___ 31 b (� 1 __ 3 ) 1

c 2 41 d 2 ___ 41

4 Write as ordinary numbersa 3 104 b 1.67 103 c 2 104 d 3.8 105

5 Write in standard forma 5000 b 64 400 c 0.07 d 0.000 607

Review exercise

B

Page 15: 25 INDICES, STANDARD FORM AND SURDS...Chapter 25 Indices, standard form and surds 614 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

Chapter 25 Indices, standard form and surds

624

6 a Write 150 million in standard form. The distance of the Sun from the Earth is 150 million kilometres.b Change 150 million kilometres to metres. Give your answer in standard form.

7 The number of atoms in one kilogram of helium is 1.51 1026

Calculate the number of atoms in 20 kilograms of helium. Give your answer in standard form. June 2007

8 Work out

a 9 1 __ 2 b 10 0

1 __ 2 c 8

1 __ 3 d 6 4

1 __ 3

9 Work outa 90.5 b 49

1 __ 2 c 125

1 __ 3 d 8

1 __ 3

10 Work out

a 4 1 __ 2 b 8

1 __ 3

June 2009

11

The table above gives the average distance in kilometres of the nine major planets from the Sun.a Which planet is approximately 4 times further away than Mercury?b How far apart are the orbits of Neptune and Pluto?c Which planet is about half the distance from the Sun as Uranus?d Which planet is 40 times further away from the Sun than Venus?e A probe was sent from the Earth to Mars. If it took one year to reach Mars, what average speed would it have to travel? Give your answer in km/hr.

12 Estimate the value of each of the following using standard form.

a 672 000 0.003 42 b (0.0543 693)2 c 8700 0.000 198 ______________ 278 50

13 Work out (3.2 105) (4.5 104). Give your answer in standard form correct to 2 signifi cant fi gures. June 2005

14 a Write the number 40 000 000 in standard form.b Write 1.4 105 as an ordinary number.c Work out (5 105) (6 109). Give your answer in standard form. Nov 2009

B

A

AO2

Planet Average distance from the Sun in km

Mercury 5.8 107

Venus 1.1 108

Earth 1.5 108

Mars 2.3 108

Jupiter 7.8 109

Saturn 1.4 109

Uranus 2.9 109

Neptune 4.5 109

Pluto 5.9 109

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Page 16: 25 INDICES, STANDARD FORM AND SURDS...Chapter 25 Indices, standard form and surds 614 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

625

Chapter review

15 a i Write 7900 in standard form ii Write 0.000 35 in standard form.

b Work out 4 103 ________ 8 105 Give your answer in standard form.

16 In 2003 the population of Great Britain was 6.0 107.In 2003 the population of India was 9.9 108. Work out the difference between the population of India and the population of Great Britain in 2003.Give your answer in standard form. June 2007

17 8x 2y Express y in terms of x.

18 3 √__

27 3n Find the value of n. June 2006

19 a √__

54 k √__

6 Find the value of k. b √__

2 √__

8 p √__

2 Find the value of p.

20 8 √__

8 can be written in the form 8k.

a Find the value of k.

8 √__

8 can also be expressed in the form m √__

2 where m is a positive integer.

b Find the value of m.

c Rationalise the denominator of 1 ___

8 √__

8

Give your answer in the form √

__ 2 ___

p where p is a positive integer. June 2006

21 Work out

2 2.2 1012 1.5 1012 ______________________ 2.2 1012 1.5 1012

Give your answer in standard form correct to 3 signifi cant fi gures. Nov 2007

22 x √______

p q

_____ pq

p 4 108

q 3 106

Find the value of x.Give your answer in standard form correct to 2 signifi cant fi gures. Mar 2005

23 y2 ab _____ a b

a 3 108

b 2 107

Find y.Give your answer in standard form correct to 2 signifi cant fi gures. June 2003

24 A fl oppy disk can store 1 440 000 bytes of data.a Write the number 1 440 000 in standard form.

A hard disk can store 2.4 109 bytes of data.b Calculate the number of fl oppy disks needed to store the 2.4 109 bytes of data. Nov 2003

A

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Page 17: 25 INDICES, STANDARD FORM AND SURDS...Chapter 25 Indices, standard form and surds 614 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

Chapter 25 Indices, standard form and surds

626

25 a Write 5 720 000 in standard form. p 5 720 000q 4.5 105

b Find the value of p q _______ (p q)2

Give your answer in standard form, correct to 2 signifi cant fi gures. Winter 2005

26 A nanosecond is 0.000 000 001 second.a Write the number 0.000 000 001 in standard form.

A computer does a calculation in 5 nanoseconds.b How many of these calculations can the computer do in 1 second? Give your answer in standard form. Summer 2004

27 a Write 0.000 000 000 054 in standard form.

S 12.6 R2

R 0.000 000 000 054b Use the formula to calculate the value of S.

Give your answer in standard form, correct to 3 signifi cant fi gures. Winter 2005

28 Solvea 4x 1 __ 16 b 2x 1 __ 16 c 2 2x 1 __ 4 d 22x 1 __ 2

29 a Rationalise the denominator of 1 __

√__

3

b Expand (2 √__

3 ) (1 √__

3 ). Give your answer in the form a b √

__ 3 where a and b are integers. June 2008

30 The value of a car can be modelled by the equation: V 17 000 (0.9)t

where V the value of the car in £s and t age from new in years.a Find V when t 0.b Find V when t 4.c Find the age of the car when the price fi rst falls below £10 000.d Sketch a graph showing V against t.

31 Calculate 1 ______ √

_____ 2 1 1 ________

√__

3 √__

2 1 ________

√__

4 √__

3 ....... 1 ________

10 √__

99

A

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A

AO3

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