Download 25 indices, standard form and surds

25 INDICES, STANDARD FORM
AND SURDS
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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.
610
Objectives
Before you start
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.
You need to be able to:
use the index laws
round numbers to one significant figure.
25.1 Using zero and negative powers
25.1 Using zero and negative powers
Objectives
Why do this?
You know that n0  1 when n ≠ 0.
You know the meaning of negative indices.
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.
Get Ready
Work out
1. 32
2. 25
3. 43
Key Points
For non-zero values of a
a0  1
For any number n
1
an  __
an
Example 1
Work out the value of
a 30
b 5–1
c 6–2
–2
d __2
5
( )
a 30  1
Any number to the power of zero is 1.
1
b 51  __
5
1
Use the rule an  __
an
1
c 62  __
6
2
1
 __
36
d
(5 )
2
__
2
62  6  6  36.
Examiner’s Tip
1
2 2
__
5
5 2
 __
2
25
___

4
( )
( )
To work out the reciprocal of a fraction,
turn the fraction upside down.
Square the number on the top and the
number on the bottom of the fraction.
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.
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Chapter 25 Indices, standard form and surds
Exercise 25A
B
1
2
Questions in this chapter are targeted at the grades indicated.
Write down the value of these expressions.
a 70
b 81
3
e (2)
f 92
i (3)2
j (8)0
c 51
g 104
k 160
Work out the value of these expressions.
a (_13 )1
b (_27 )1
c
e (0.25)2
f
g
(1 )
j
i
_2 2
5
(_25 )3
(1 _13 )3
(_17 )2
(_53 )0
k (0.1)4
d 40
h 1450
l 106
d
h
(_14 )3
(_95 )1
l (0.2)3
25.2 Using standard form
Objectives
Why do this?
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.
Astronomers use standard form to record large
measurements. The Sun’s diameter is about
1.392  106 km. Biologists working with microorganisms sometimes use standard form to
record their very small sizes, like 2.1  104 cm.
Get Ready
1. Work out
a 103
b 10–2
2. Write 10 000 as a power of 10.
3. Work out 2.35  10 000.
Key Points
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.
2
8
These numbers are all in standard form – 4.5  10 , 9  10 , 1.2657  106.
These numbers are not in standard form – 67  109, 0.087  103 – because the first 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 .
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standard form
integer
25.2 Using standard form
Example 2
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
Use 3.46 not 34.6 or 346 as
3.46 is between 1 and 10.
Write as an ordinary number a 8.1  105
Example 3
b 6  108
a 8.1  105  8.1  100 000
 810 000
b 6  108  6  100 000 000
 600 000 000
Exercise 25B
1
2
3
4
B
Write these numbers in standard form.
a 700 000
b 600
c 2000
d 900 000 000
e 80 000
Write these as ordinary numbers.
a 6  105
b 1  104
c 8  105
d 3  108
e 7  101
Write these numbers in standard form.
a 43 000
b 561 000
c 56
d 34.7
e 60
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.
Example 4
Write in standard form
a 0.000 000 006
b 0.000 56
a 0.000 000 006  6  0.000 000 001
1
 6  __________
1 000 000 000
9
1
 6  __
10
1
0.000 000 001 is equivalent to ___________
.
1 000 000 000
1
Using an  __
an
 6  109
b 0.000 56  5.6  0.0001
1
 5.6  _____
10 000
4
1
 5.6  __
10
Use 5.6 rather than 56 as 5.6 is between 1 and 10.
 5.6  104
613
Chapter 25 Indices, standard form and surds
Example 5
Write as an ordinary number
a 3  106
b 1.5  103
3
a 3  106  ___
6
1.5
b 1.5  103  ___
3
10
10
3
 _______
1 000 000
15
 ____
1000
 0.000 003
 0.0015
Exercise 25C
B
1
2
3
4
5
6
Write these numbers in standard form.
a 0.005
b 0.04
c 0.000 007
d 0.9
e 0.0008
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
Write these numbers in standard form.
a 0.0047
b 0.987
c 0.000 803 4
d 0.000 15
e 0.601
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
Write these numbers in standard form.
a 457 000
b 0.0023
f 89 000
g 200
c 0.0003
h 0.005 26
d 2 356 000
i 6034
e 0.782
j 0.000 008 73
Write these as ordinary numbers.
a 4.12  104
b 3  103
1
f 7.5  10
g 1.5623  102
c 2.065  107
h 5.12  107
d 4  106
i 2.7  105
e 3.27  108
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.
Example 6
Write in standard form
a 40  102
b 0.008  102
Method 1
a 40  102  4  101  102
12
 4  10
 4  103
b 0.008  10–2  8  103  102
 8  103  2
 8  105
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Examiner’s Tip
Write 40 in standard form.
Use the rule am  an  amn.
Write 0.008 in standard form.
Use am  an  amn.
The power of 10 tells you how
many 0s there are.
102  100
2 zeros
102  0.01 2 zeros
25.2 Using standard form
Method 2
a 40  102  40  100
 4000
 4  103
1
b 0.008  102  0.008  ___
100
Work out the calculation.
Change the answer into standard form.
 0.008  100
 0.000 08
 8  105
1
Use the rule an  __
.
an
1
___
Multiplying by 100 is the same as dividing by 100.
Exercise 25D
1
2
B
Write these in standard form.
b 980  103
a 45  103
c 3400  102
d 186  1010
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
7
10
8
e 60 000  10
f 8.901  10
g 0.040 05  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
Example 7
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
Example 8
Rearrange the expression so the powers of 10 are
together.
Multiply the numbers.
Use am  an  amn to multiply the powers of 10.
12  109 is not in standard form.
Write your final answer in standard form.
By writing 760 000 000 and 0.000 19 in standard form correct to one significant figure, work
out an approximation for 760 000 000  0.000 19.
760 000 000  8  108 correct to one significant figure.
0.000 19  2  104 correct to one significant figure.
8
760 000 000  _________
8  10
_____________
0.000 19
2  10
4
8
10
8  _____
 __
2 10
 4  1084
4
 4  1012
Rearrange the expression so the powers of 10
are together.
Divide the numbers.
Use am  an  amn to divide the powers of 10.
615
Chapter 25 Indices, standard form and surds
Exercise 25E
A
1
2
3
AO3
AO2
AO3
Work out and give your answer in standard form.
a (4  108)  (2  103)
b (6  105)  (1.5  103)
7
6
d (6  10 )  (3  10 )
e (6  109)  (5  103)
c (4  107)  (3  105)
f (5  108)  (2  103)
Work out and give your answer in standard form.
a (4  108)  (2  103)
b (9  105)  (2  104)
8
13
d (8.6  10 )  (2  10 )
e (1  1012)  (4  103)
c (3  109)  (6  103)
f (7  109)  (7  105)
Express in standard form.
a (2  105)2
b (5  105)2
c (4  106)2
d (7  108)2
4
By writing these numbers in standard form correct to one significant figure, 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  103 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  105 seconds. It travels at a speed of 4.2  106 metres per
second. Calculate an approximation for the distance it travels in its lifetime.
Example 9
Use a calculator to work out
a (3.4  106)  (7.1  104 )
b (4.56  108)  (3.2  103)
a (3.4  106)  (7.1  104)
 2.414  1011
Use the EXP of 10x on your calculator.
b (4.56  108)  (3.2  103)  1.425  1011
Hint
In both of these cases the
brackets need not be used, but
in more complex expressions the
brackets must be used.
616
25.2 Using standard form
Example 10
x  3.1  1012, y  4.7  1011
xy
Use a calculator to work out the value of _____.
xy
Give your answer in standard form correct to 3 significant figures.
12
11
(3.1  10  4.7  10 )
_______________________
12
(3.1  10
Substitute the values into the expression.
 4.7  1011)
12
3.57  10
 _____________
1.457  1024
 2.4502…  1012
Write the number from your calculator correctly in
standard form showing more than 3 significant figures.
 2.45  1012
Give your answer correct to 3 significant figures.
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.
Exercise 25F
1
Evaluate these expressions, giving your answers in standard form.
0.08  480
a 500  600  700
b 0.006  0.004
c _________
180
8.82  5.007
65  120
f __________
g (12.8)4
e ________
1500
10000
2
Evaluate these expressions. Give your answers in standard form.
a (3.2  1010)  (6.5  106)
b (1.3  107)  (4.5  106)
10
6
c (2.46  10 )  (2.5  10 )
d (3.6  1020)  (3.75  106)
3
Express as a number in standard form.
a (3.2  108)  (6.5  106)
c (2.46  1010)  (2.5  106)
89000  0.0086
d _____________
48  0.25
h (36.4  24.2)3
b (1.3  107)  (4.5  106)
d (3.6  1020)  (3.75  106)
4
Evaluate these expressions. Give your answers in standard form correct to 3 significant figures.
a (3.5  1011)  (6.5  106)
b (1.33  1010)  (4.66  104)
8
6
d (3.24  108)  (6.4  104)
c (5.3  10 )  (6.45  10 )
5
Express as a number in standard form correct to 3 significant figures.
a (3.5  1011)  (6.5  106)
b (1.33  1010)  (4.66  104)
d (3.24  108)  (6.4  104)
c (5.3  108)  (6.45  106)
617
Chapter 25 Indices, standard form and surds
6
x  3.5  109, y  4.7  105
Work out the following. Give your answer in standard form correct to 3 significant figures.
xy
x
x 2  y2
a __
b x(x  800y)
c ________
d ____
y
x  800y
2000
(
7
)
x  2.4  105, y  9.6  106
Evaluate these expressions.
Give your answer in standard form correct to 3 significant figures where necessary.
x2  y2
xy
x2
a __
b ______
c _____
y
xy
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 ratio
distance of the Earth from the sun : distance of the planet Neptune from the Sun
9
The mass of a uranium atom is 3.98  1022 grams.
Work out the number of uranium atoms in 2.5 kilograms of uranium.
Mixed exercise 25G
B
1
Work out the value of these expressions.
a 30
e
2
3
4
b 61
(_19 )
1
618
5
(_52 )
1
g
d 53
(_34 )
2
h
(1 _12 )
3
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
Write these as ordinary numbers.
a 5.8  104
b 2  105
c 4.03  107
d 4  106
e 8.45  101
f 3.152  102
g 9.2  101
h 7  104
c 89.2  104
d 5660  108
Write these in standard form.
a 278  104
A
f
c 72
b 0.087  109
Work out and give your answer in standard form.
a (5  103)  (7  109)
b (9  107)  (2  105)
c (8.4  106)  (2  105)
d (4.3  107)  (8  103)
e 800 000  0.000 000 02
f 0.000 002 4  5 000 000
25.3 Working with fractional indices
25.3 Working with fractional indices
Objective
Why do this?
You know the meaning of fractional indices.
Fractional indices are used when you model the
rates at which things vibrate, such as your voice box.
Get Ready
Work___
out
1. √100
__
____
3
3. √ 27
3
2. √8
Key Points
Indices can be fractions. In general,
1
__
__
an  n√a
In particular, this means that
1
__
1
__
__
__
3
a2  √a and a3  √
a
Example 11
Find the value of the following
1
__
1
__
a 252
1
__
b (1000)3
c 160.25
___
a 25 2  √ 25
The square root of 25 is 5 because 5  5  25.
5
_______
1
__
The cube root of 1000 is 10 because
10  10  10  1000.
3
b (1000) 3  √ 1000
 10
__
c 160.25  16
1
4
1
 ____
1
__
164
1
 _____
___
4
√ 16
1
 __
2
Example 12
2
__
1
Change the decimal into a fraction 0.25  __
.
4
1
__
n
Use the rule a  an .
1
__
Work out the value of
1
__
a 8 3  (8 3 )2
 22
4
3
__
1
b 16 4  ____
3
__
164
1
 ______
1
__
(164)3
1
 ___
23
1
 __
8
4
___
164  √16  2
because 24  16
_2
a 83
_3
b 164
Use the rule (am)n  amn.
Work out the cube root of 8 first.
Then square your answer.
1
Use an  __
.
an
Examiner’s Tip
It is easier to work out the root
first as this makes the numbers
smaller and easier to manage.
619
Chapter 25 Indices, standard form and surds
Exercise 25H
B
1
2
3
A
4
5
Work out the value of the following.
_1
_1
a 92
b 492
c 1002
Work out the value of
_1
_1
a 273
b 10003
c (64)3
Work out the value of
_1
_1
a 16 4
b 4 2
c 125
Work out the value of
_2
_2
a 273
b 10003
c 643
6
_1
d 42
_1
_1
d 1253
_13
d
_2
_23
(_15 )
2
_13
g 8
(__321 )
_1
5
_3
d 164
Work out, as a single fraction, the value of
_2
_3
_1
a 125 3
b 10 000 4
c 27 3
f 125
A
_1
d 8
_23
e
(_14 )
2
e
(_18 )
3
e
(_49 )
_1
_1
_1
2
_3
e 252
e 64
_32
 (_25 )
Find the value of n.
a _18  8n
b 64  2n
2
__
1__
c __
 5n
√5
d (√7 )5  7n
3
__
e (√2 )11  2n
25.4 Using surds
Objectives
Why do this?
You can simplify surds.
You can expand expressions involving surds.
You can rationalise the denominator of a fraction.
1 5
Surds occur in nature. The golden ratio _____
2
occurs in the arrangement of branches along the
stems of plants, as well as veins and nerves in
animal skeletons.
__
Get Ready
1. Write down the first 10 square numbers.
__
___
√ 100
2. Write down the value of a √36
b
__
__
__
__
3. Which of these has an exact answer: √5 , √ 9 , √37 , √64 ?
Key Points
A__number__written exactly using square roots is called a surd.
and √3 are both surds.
√2
__
__
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
m
__
√ m  √ n  √ mn ___
__ 
√n
n
Simplified surds should never have a surd in the denominator.
√
620
surd
√
25.4 Using surds
To rationalise the denominator of a fraction means to get rid of any surds in the denominator.
__
b
a__
__ . This ensures that the final fraction has an
To rationalise the denominator of __
you multiply the fraction by __
√b
√b
integer as the denominator.
√
__
√
a__
a__
__
 __
 ____b
√b
√b
√b
__
a__ b__
 ______
√b  √b
√
__
a√b
 ____
b
__
Example 13
___
√ 12
Simplify √12 .
______
 √4  3
__
__
__
____
√ m  √ n  √ mn .
Use
__
√ 4  2.
__
 √4  √3
__
 2√3
Example 14
__
__
__
Expand and simplify (2  √3 )(4  √ 3 ).
__
__
__
__
__
Multiply out the brackets.
(2  √3 )(4  √3 )  8  2√ 3  4√3  √ 3  √3
__
 8  6√3  3
Simplify the expression.
__
 11  6√3
Exercise 25I
1
2
Find __the value
of the integer k.__
__
__
a √8  k√2
b √18  k√2
c √ 50  k√ 2
Simplify
___
a √200
c √ 20
__
__
__
__
__
b √32
__
__
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 )
2
√
√
√
√
d ( 7  1)(2  7 )
e (2  3 )
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 simplified 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.
__
__
A
d √ 80  k√ 5
A
__
__
rationalise the denominator
621
Chapter 25 Indices, standard form and surds
Example 15
__
2__ .
Rationalise the denominator of ___
√3
__
√3
2__  ___
2__  ___
___
__
√3
√3
√3
Multiply the fraction by ___
__ .
√3
√3
__
2  √3
 ________
__
__
√3
Simplify the denominator
by using
__
__
the fact that √3  √3  3.
 √3
__
2√3
 ____
3
__
Example 16
__
__
15 __√ 5
and give your answer in the form a  b√ 5 .
Rationalise the denominator of _______
√5
__
__
√5
15  √5  ________
15  √ 5  ___
________
__
__
__
√5
√5
__
√5
__
Watch Out!
__
Remember to multiply both parts
of the expression on the top of
the fraction.
15√5  √ 5  √5
 ________________
__
__
√5  √5
__
15√5  5
 _________
5
__
 1  3√5
Simplify the fraction by dividing both parts of
the expression on the top of the fraction by 5.
Exercise 25J
A
1
2
A
3
4
5
6
622
Rationalise the denominators.
5__
5
1__
2__
1__
__
b ___
c ___
d ___
e ____
a ___
√2
√7
√ 11
√3
√5
Rationalise the denominators and simplify your answers.
15__
5
10__
2__
4
__
__
b ___
c ____
d ___
e ____
a ___
√2
√2
√ 12
√3
√ 10
__
Rationalise
the denominators and
give your answers__in the form a  b√ c __where a, b and c are __integers.
__
__
12 __√3
10 __√ 5
6 __√2
14 __√ 7
2 __√2
e _______
b ______
c _______
d _______
a ______
√2
√2
√7
√3
√5
__
__
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.
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.
Solve these equations leaving your answers in surd form.
a x2  6x  2  0
b x2  10x  14  0
2
3
9
2
Chapter review
7
C
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 ABC
b the length of BC.
( 7 � 2)
A
( 7 � 2)
B
Chapter review
For non-zero values of a
a0  1
For any number n
1
an  __
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
Indices can be fractions. In general,
1
__
·
5 �10
7
__
an  n√a
A number written exactly using square roots is called a surd.
These two laws can be used to simplify surds.
__
__
__
__
___
√m
m
___
__
√ m  √ n  √ mn
__ 
√n
n
Simplified 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.
√
__
b
a__
__ , this ensures that the final fraction has an
To rationalise the denominator of __
you multiply the fraction by __
√b
√b
integer as the denominator.
√
Review exercise
1
2
3
4
5
B
Work out the values of
a 40
b 41
c 20
d 23
Work out the values of
a 30
b (3)0
c 31
d
c 2  41
2
d ___
41
c 2  104
d 3.8  105
c 0.07
d 0.000 607
Work out the values of
1
1
a ___
b __1
31
3
Write as ordinary numbers
a 3  104
b 1.67  103
( )
Write in standard form
a 5000
b 64 400
(__13 )
0
623
Chapter 25 Indices, standard form and surds
B
AO2
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.
8
9
10
Work out
1
__
b 1002
Work out
a 90.5
b 49
Work out
1
__
a 42
AO2
AO3
1
__
a 92
11
__1
2
1
__
1
__
c 83
c 125
June 2007
d 643
__1
3
__1
3
d 8
__1
3
b 8
Planet
June 2009
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
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
13
A
624
14
b (0.0543  693)2
8700  0.000 198
c ______________
278 50
Work out (3.2  105)  (4.5  104).
Give your answer in standard form correct to 2 significant figures.
a Write the number 40 000 000 in standard form.
b Write 1.4  105 as an ordinary number.
c Work out (5  105)  (6  109). Give your answer in standard form.
June 2005
Nov 2009
Chapter review
15
a i Write 7900 in standard form
A
ii Write 0.000 35 in standard form.
4  10 Give your answer in standard form.
b Work out ________
8  105
3
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
18
3  √27  3n
19
a √54  k√6
20
8√8 can be written in the form 8k.
AO2
Express y in terms of x.
__
__
Find the value of n.
__
Find the value of k.
June 2006
__
__
__
b √ 2  √ 8  p√ 2
Find the value of p.
__
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.
1__
c Rationalise the denominator of ___
8√ 8
__
√2
Give your answer in the form ___
where p is a positive integer.
p
21
June 2006
Work out
2  2.2  10  1.5  10
______________________
12
12
2.2  10  1.5  10
Give your answer in standard form correct to 3 significant figures.
12
12
Nov 2007
______
22
pq
x  _____
pq
√
p  4  108
q  3  106
Find the value of x.
Give your answer in standard form correct to 2 significant figures.
23
Mar 2005
ab
y2  _____
ab
a  3  108
b  2  107
Find y.
Give your answer in standard form correct to 2 significant figures.
24
June 2003
A floppy 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 floppy disks needed to store the 2.4  109 bytes of data.
AO3
Nov 2003
625
Chapter 25 Indices, standard form and surds
A
25
a Write 5 720 000 in standard form.
p  5 720 000
q  4.5  105
pq
b Find the value of _______2
(p  q)
Give your answer in standard form, correct to 2 significant figures.
AO2
26
Winter 2005
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.
27
Summer 2004
a Write 0.000 000 000 054 in standard form.
S  12.6 R2
R  0.000 000 000 054
b Use the formula to calculate the value of S.
Give your answer in standard form, correct to 3 significant figures.
A
28
Solve
1
a 4x  __
16
29
1__
a Rationalise the denominator of __
√3
1
b 2x  __
16
__
c 2  2x  __1
4
AO2
AO3
626
30
31
d 22x  __1
2
__
b Expand (2  √3 ) (1  √3 ).
__
Give your answer in the form a  b√ 3 where a and b are integers.
AO3
Winter 2005
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 first falls below £10 000.
d Sketch a graph showing V against t.
1 __  ________
1 __  ....... ________
1 __
1
_____  ________
__
__
Calculate ______
√2  1
√4  √3
√3  √2
10  √ 99
June 2008