SamplePages
Chapter 9 Standard form and surds All material labelled Core is also part of the Extended syllabus. Topics Level Key words 1 Standard form 2 Calculating with standard form 3 Standard form without a calculator 4 Surds 5 Rationalising the denominator In this chapter you will learn how to: CORE ● Use the standard form A × 10 n when n is a positive or negative integer and 1 ⩽ A < 10. (C1.7 and E1.7) CORE CORE standard form, index surd EXTENDED EXTENDED EXTENDED CORE AND EXTENDED EXTENDED ● Use the standard form 10×A n where n is a positive or negative integer, and 1 ⩽ A < 10. (C1.8 and E1.8) ● Convert numbers into and out of standard form. (C1.8 and E1.8) ● Calculate with values in standard form using a calculator. (C1.8 and E1.8) ● Calculate with values in standard form without using a calculator. (E1.8) ● Understand and use surds, including simplifying expressions. (E1.18) ● Rationalise the denominator. (E1.18) SamplePages
Why this chapter matters
Numbers
Scientists write large and small numbers
to do
The Earth is 149 600 000 km from the
is 1 427 000 000 km from the Sun.
different ways. This makes it easier to compare them
in standard form, these distances are 1.496
× km and 1.427 109× km.
see that the distance from Saturn to the Sun is approximately
you understand
times the distance from the Earth to the Sun.
mass of an electron in standard form is 9.110
kg.
Written out in full that is 0.000 000 000 000 000 000 000 000 000 000
kg. You cannot put the numbers of this mass in a calculator, but you can input the mass in standard form.
can be written in different ways. In this chapter you will look at two different ways.
in
and
calculations with them.
Sun. Saturn
Written
108
If
standard form, you can easily
10
The
31 ×
91
5 11 written as a decimal is 0.45454545... The decimal expansion does not terminate but there is a pattern to it. The digits 4 and 5 alternate. The 100th digit after the decimal point is 5. It is a rational number. Many square roots are irrational numbers. They cannot be written exactly as decimals. For example, 31.73205080... = and 34 5.83095189...= The decimal does not terminate and there is no pattern to the digits. We cannot work out the 100th digit of an irrational number. 3 m 5 m 34 m The hypotenuse of this triangle is 34 m and the perimeter is 834 + m. Expressions such as 34 and 834 + are called surds. You will learn how to manipulate surds and show surprising connections between them. For example, 32 42=× and 2() 31 () 31+− = Chapter 9: Standard form and surds 133 SamplePages
9.1 Standard form
Powers of ten:
100 = 10 × 10 = 102
1000 = 10 × 10 × 10 = 103
Extending this idea:
10 000 = 10 × 10 × 10 × 10 = 10 4
100 000 = 105
1 000 000 = 10 6 and so on.
The power of 10 is called the index
Standardform is a way of writing very large and very small numbers using powers of 10. In this form, a number is given a value between 1 and 10 multiplied by a power of 10. That is, A × 10 n where 1 ⩽ A < 10, and n is a whole number.
Follow through these examples to see how numbers are written in standard form.
52 = 5.2 × 10 = 5.2 × 101
73 = 7.3 × 10 = 7.3 × 101
625 = 6.25 × 100 = 6.25 × 10 2 The numbers in bold are in standard form.
389 = 3.89 × 100 = 3.89 × 10 2
3147 = 3.147 × 1000 = 3.147 × 10 3
When writing a number in this way, you must always follow two rules.
• The first part must be a number between 1 and 10 (1 is allowed but 10 isn’t).
• The second part must be a whole-number (negative or positive) power of 10. Note that you would not normally write the power 1.
Standard form on a calculator
A number such as 123 000 000 000 is obviously difficult to key into a calculator. Instead, you enter it in standard form (assuming you are using a scientific calculator):
123 000 000 000 = 1.23 × 1011
The key strokes to enter this into your calculator could be something like this:
1
• 23×10x 11
Your calculator display will display the number either as an ordinary number, if there is enough space, or in standard form. Make sure you know how to use standard form on your calculator. Some calculators may use a different symbol for ×10 x , for example, EXP
Chapter 9 . Topic 1
9.1 Standard form134 SamplePages
Standard
form of numbers less than 1 You will need to use a negative index for numbers between 0 and 1: 0.1 = 10 –1 0.01 = 10 –2 0.001 = 10 –3 0.0001 = 10 –4 and so on. For example: 0.000 729 = 7.29 × 0.0001 = 7.29 × 10 –4 in standard form These numbers are written in standard form. Make sure that you understand how they are formed. a 0.4 = 4 × 10 –1 b 0.05 = 5 × 10 –2 c 0.007 = 7 × 10 –3 d 0.123 = 1.23 × 10 –1 e 0.007 65 = 7.65 × 10 –3 f 0.9804 = 9.804 × 10 –1 g 0.0098 = 9.8 × 10 –3 h 0.000 0078 = 7.8 × 10 –6 On a calculator you would enter 1.23 × 10 –6 , for example, as: 1 • 23×10x (–)6 Try entering some of the numbers a to h (above) into your calculator for practice. EXERCISE 9A 1 These numbers are in standard form. Write them out in full. a 2.5 × 102 b 3.45 × 10 c 4.67 × 10 –3 d 3.46 × 10 e 2.0789 × 10 –2 f 5.678 × 103 g 2.46 × 102 h 7.6 × 103 i 8.97 × 105 j 8.65 × 10 –3 k 6 × 107 l 5.67 × 10 –4 2 Write these numbers in standard form. a 250 b 0.345 c 46 700 d 3 4 00 000 000 e 20 780 000 000 f 0.000 567 8 g 2460 h 0.076 i 0.000 76 j 0.999 k 234.56 l 98.7654 m 0.0006 n 0.005 67 o 56.0045 In questions 3 to 5, write the numbers given in each statement in standard form. 3 One year, 27 797 runners completed the New York marathon. 4 The largest number of dominoes ever toppled by one person is 321 197, although a team set up and toppled 4 491 863. CORE Chapter 9: Standard form and surds 9 . 1 135 SamplePages
5
Calculating with standard form
Chapter 9 . Topic 2
The asteroid Phaethon comes within 12 980 000 miles of the Sun. The asteroid Pholus, at its furthest point, is a distance of 2997 million miles from the Earth. The closest an asteroid ever came to Earth was 93 000 miles from the planet. 6 How many times bigger is 3.2 × 10 6 than 3.2 × 10 4? 7 The speed of sound (Mach 1) is 1236 kilometres per hour. A plane travelling at Mach 2 would be travelling at twice the speed of sound. How many kilometres would a plane travelling at Mach 3 cover in 1 minute? 8 Here are the distances of some planets from the Sun: Jupiter 778 million kilometres Mercury 58 million kilometres Pluto 5920 million kilometres Write these distances in standard form. 9.2
Calculations involving very large or very small numbers can be done more easily if you use standard form. You can enter numbers in a scientific calculator in standard form. The way you do this may be different for different models of calculator. Make sure you know how to do this with your calculator. Example 1 A pixel on a computer screen is 2 × 10 –2 cm long by 7 × 10 –3 cm wide. What is the area of the pixel? The area is calculated as length times width. Area in cm2 = 2 × 10 –2 × 7 × 10 –3 = 14 × 10 –5 (Use a calculator to do this.) The answer is not in standard form as the first part is not between 1 and 10, so you need to change it to standard form. 14 × 10 –5 = 0.000 14 = 1.4 × 10 –4 So area = 1.4 × 10 –4 cm2 If you use a calculator you can enter the numbers directly without any rearranging. Your calculator may give you the answer in standard form. CORE 136 9.2 Calculating with standard form SamplePages
0.06
0.07
2 ×
160
1.6 × 102
2 × 10
2 ×
(5
3
Example 2 The distance from the Sun to the Earth is 150 million kilometres. Light travels at 3.00 × 105 km/second. How long does light from the Sun take to reach the Earth? Distance = 150 000 000 = 1.5 × 10 8 km Time = distance ÷ speed = (1.5 × 10 8) ÷ (3.00 × 105) = 500 seconds (Use a calculator to work this out.) = 8 minutes 20 seconds EXERCISE 9B 1 These numbers are not in standard form. Write them in standard form. a 56.7 × 102 b
× 10 4 c 34.6 × 10 –2 d
× 10 –2 e 56 × 10 f
3 × 105 g 2 × 102 × 35 h
× 10 –2 i 23 million 2 Work these out. Give your answers in standard form. a 2 × 10 4 × 5.4 × 103 b
×
× 10 4 c 2 × 10 4 × 6 × 10 4 d
–4 × 5.4 × 103 e 1.6 × 10 –2 × 4 × 10 4 f
10 4 × 6 × 10 –4 g 7.2 × 10 –3 × 4 × 102 h
× 103)2 i (2 × 10 –2)3 3 Work these. Give your answers in standard form. a (5.4 × 10 4) ÷ (2 × 103) b (4.8 × 102) ÷ (3 × 10 4) c (1.2 × 10 4) ÷ (6 × 10 4) d (2 × 10 –4) ÷ (5 × 103) e (1.8 × 10 4) ÷ (9 × 10 –2) f √(36 × 10 –4) 4 A typical adult has about 20 000 000 000 000 red blood cells. Each red blood cell has a mass of about 0.000 000 000 1 g. Write both of these numbers in standard form and work out the total mass of red blood cells in a typical adult. CORE 137Chapter 9: Standard form and surds 9 . 2 SamplePages
5 A man puts one grain of rice on the first square of a chess board, two on the second square, four on the third, eight on the fourth and so on.
a How many grains of rice will he put on the 64th square of the board?
b How many grains of rice will there be altogether? Give your answers in standard form.
Advice and Tips
Compare powers of 2 with the running totals.
By the fourth square you have 15 grains altogether, and 24 = 16.
6 The surface area of the Earth is approximately 3.2 × 10 8 square kilometres. The area of the Earth’s surface that is covered by water is approximately 2.2 × 10 8 square kilometres.
a Calculate the area of the Earth’s surface not covered by water. Give your answer in standard form.
b What percentage of the Earth’s surface is not covered by water?
7 The Moon is a sphere with a radius of 1.74 × 103 kilometres. The formula for working out the surface area of a sphere is: surface area = 4 πr2
Calculate the surface area of the Moon.
8 Evaluate E M when E = 1.5 × 103 and M = 3 × 10 –2, giving your answer in standard form.
9 Work out the value of 3.2 × 107 1.4 × 102 giving your answer in standard form, correct to 2 significant figures.
10 In one year, British Airways carried 33 million passengers. Of these, 70% passed through Heathrow Airport. On average, each passenger carried 19.7 kg of luggage. Calculate the total mass of the luggage carried by these passengers.
11 In 2013 the world population was approximately 7.14 × 109. In 2014 the world population was approximately 7.24 × 109.
a By how much did the population rise? Give your answer as an ordinary number.
b What was the percentage increase?
12 Here are four numbers written in standard form.
1.6 × 10 4 4.8 × 10 6 3.2 × 102 6.4 × 103
a Work out the smallest answer when two of these numbers are multiplied together.
b Work out the largest answer when two of these numbers are added together. Give your answers in standard form.
13 The mass of Saturn is 5.686 × 1026 tonnes. The mass of the Earth is 6.04 × 1021 tonnes. How many times heavier is Saturn than the Earth? Give your answer in standard form to a suitable degree of accuracy.
14 A number is greater than 100 million and less than 1000 million. Write down a possible value of the number, in standard form.
CORE 138 9.2 Calculating with standard form SamplePages
15 Here are some population figures for some countries.
Country Population
Tunisia 1.10 × 107
Denmark 5.60 × 10 6
Senegal 1.39 × 107 Jamaica 2.71 × 10 6 Mexico 1.19 × 10 8 India 1.29 × 109
a Which country has the largest population?
b Which two countries have the largest difference in population?
c Find the total population of Senegal, Denmark and Jamaica. Give your answer in standard form to two significant figures.
d Complete this sentence:
The population of Mexico is approximately … times larger than the population of Denmark.
e Complete this sentence:
The population of India is approximately … times larger than the population of Jamaica.
16 This table shows the populations and the areas of five different countries.
Country Population Area
Russian Federation 1.44 × 10 8 1.71 × 107
Sri Lanka 2.07 × 107 6.56 × 10 4
Thailand 6.64 × 107 5.13 × 105 Togo 6.98 × 10 6 5.68 × 105 Iran 7.74 × 107 1.65 × 10 6
a Which country has the smallest population?
b Which country has the smallest area?
The population density is the population divided by the area.
c Which country has the largest population density?
d Which country has the smallest population density?
e What fraction of the area of the Russian Federation is the area of Sri Lanka?
Give your answer
in the form 1 N . CORE 139Chapter 9: Standard form and surds 9 . 2 SamplePages
9.3 Standard form without a calculator
Addition
Advice and
Chapter 9 . Topic 3
This section shows you how to do calculations with numbers in standard form without using a calculator.
and subtraction A = 7.5106 × and B = 4.3106 × and C = 9.2105 × A and B have the same index for 10. It is easy to add or subtract them. A + B = (7.5 10 )( 664.310)×+ × = (7.5 4.3) 106+× = 11.8101.1810 67 ×= × Just add the two numbers. Write the answer in standard form. Subtraction is similar. A – B = (7.5 10 )( 664.310)×− × = (7.5 4.3) 106−× = 3.2106 × Example 3 In 2020, the population of Australia was 2.49 107× . The population of New Zealand was 4.7106 × . a Find the total population of the two countries. b Find the difference between the two populations. a Total = (2.4910) (4.7 10 )65 ×+ × = (24.910) (4.7 10 )55 ×+ × = (24.94.7)105 +× = 29.6102.9610 56 ×= × in standard form. b Difference = (24.910) (4.7 10 )55 ×− × = (24.94.7)105 −× = 20.2102.0210 56 ×= × in standard form.
Tips Change one number so that both have the same index for 10. E 140 9.3 Standard form without a calculator SamplePages
Multiplication and division
9C
C = 9106 × and D = 1.5103 × To multiply C and D, you multiply the numbers and add the indices. (9 10 )(1.510) 63 ×= ×× ×CD = (9 1.5) 1063×× + = 13.5109 × = 1.35 1010× To divide C and D, you divide the numbers and subtract the indices. (9 10 )(1.510) 63 ÷= ×÷ ×CD = (9 1.5) 1063÷× = 6103 × You do not need to have the same indices when you multiply or divide. Check with a calculator that these answers are correct. Exercise
1 X = 2.4109 × Y = 7.2109 × Z = 8.1109 × Write the answers in standard form. a X + Y b Y + Z c Y – X d Z – Y e 2Y f 5Z 2 In 2020, the population of Pakistan was 2.12 108× . The population of Bangladesh was 1.61 108× . Write the answers in standard form. a The total population of Pakistan and Bangladesh b The difference between the two populations 3 The area of Spain is 5.06 105× km2 The area of France is 5.49 105× km2 a Write the total area of France and Spain in standard form. b Write the difference between the areas of France and Spain in standard form. 4 Work out the following. Write the answers in standard form. a (7 10 )(610) 89 ×+ × b (3.5 10 )(1.910) 76 ×+ × c (4.8 10 )(9.210) 87 ×− × d (1.2 10 )(8.210)11 10 ×− × EXTENDED 141Chapter 9: Standard form and surds 9 . 3 SamplePages
million
and
in
form.
P
number, P or Q, is equal to 0.00025?
Q in standard form.
C – Q in standard form.
walls of a room are painted.
total area of the walls is 2105 × cm2.
thickness of the paint is 510 3
cm.
volume of the paint is the area × the thickness.
out the volume of the paint.
speed of light is 3108 × m/s.
distance from the Sun to the Earth is 1.5108
km.
Write the distance from the Sun to the Earth in metres, in standard form.
distance speed
Work out the time for light to travel from the Sun to the Earth. Give the answer in minutes and seconds.
9106
and G
310
the following in standard form.
mass of an electron is 9.110
kg.
out the total mass of 200 million electrons.
Surds
An expression involving a square root is called a surd.
your answer in kg in standard form.
are 13 , 35 and 28 + Surds are irrational numbers. They cannot be written exactly as decimals.
can often be simplified.
5 S = 5
and N = 80 million a Write S
N
standard
b Write the answers in standard form. i S × N ii S2 iii N2 iv S3 6 P = 2.510 3 × and Q = 2.510 4 × a Which
b Write
+
c Write
7 The
The
The
×
The
Work
8 The
The
×
a
Time =
b
9 F =
×
=
4 × Write
a F × G b F ÷ G c F2 d G2 10 The
31 ×
Work
Write
9.4
Examples
Surds
E EXTENDED 142 9.4 Surds SamplePages
Advice and
Suppose 28=×x Then (2 8) (2 8)2 =× ××x = 228 8×× × 28 16=× = This shows that 28 28×= × It is true that ×=ab ab × where a and b are any two numbers. You can use this fact to simplify surds. Example 4 Simplify: a 35 × b 232 × c 20 5× a 35 35 15×= ×= b 2322 32 64 8×= ×= = c 20 5 100 10×= = You can simplify a square root if a square number is a factor of the number under the square root sign. Consider 54 . 9 is a square number and 9 is a factor of 54. So 54 96 96 36=× =×= Example 5 Simplify: a 28 b 75 c 82 + a 4 is a square number and it is a factor of 28. 28 47 47 27=× =× = b 25 is a square number and a factor of 75. 75 25 3253 53=× =× = c 4 is a factor of 8. 82 42 22 22 32+= ×+ =+ =
Tips Be careful. 82 + is not the same as 82 + 2(32 )+ is an expression with a bracket. Multiply each term in the bracket by 2 2(32 )( 23)( 22 )+=× +× = 32 2+ 143Chapter 9: Standard form and surds 9 . 3 SamplePages
Example 6 Simplify 8(52 ) 8(52 )( 85)( 82 )−= ×−)( × = 58 16 = 58 4 Exercise 9D 1 Simplify each answer as much as possible. a 11 3× b 75 × c 250 × d 218 × e 18 8× f 10 40× 2 Write each of the following as a multiple of 2 or 3 . For example, 300 10 3= a 8 b 12 c 32 d 75 e 200 f 147 3 Write each expression as the multiple of the square root of a prime number. a 128 b 20 c 63 d 44 e 45 f 405 4 Write each expression as the square root of an integer. For example, =53 75 . a 26 b 312 c 10 7 d 75 e 10 10 f 1 2 8 5 Simplify each answer as much as possible. a 232 + b 312 + c 200 2 d 80 45 e 75 147+ f 700 63 6 Multiply the brackets and simplify. a 2( 25) + b 2(38 )+ c 3(32 3)+ d 5( 20 3) e 2(15 72 ) f 2( 32 8) EXTENDED 144 9.4 Surds SamplePages
7 Match an expression on the left with an expression on the right. 2(22 )+ 33 3+ 3(33 )+ 32 3+ 2(33 )+ 22 2+ 3(23 )+ Which expression is the odd one out? 8 a Show that 81 818329 2++ = b 81832__172 ++ += Work out the missing number. 9 a Write 128 in the form ab where a and b are integers. How many ways can you do this? b Write 243 in the form ab in two different ways. 10 Show that (2 1)(2 (2 1)+− 1) is an integer. 9.5 Rationalising the denominator (2 3)(4 6) 51050++ =× = You could work this out as four separate multiplications. (2 3)(4 6) 2( 46)3(4 6)++ =+ ++ (2 4) (2 6) (3 4) (3 6)=× +× +× +× 812121850= +++ = The answer is the same. You can do this if each bracket contains a surd. (3 2)(4 2) 3( 42 )2(4 2)++ =+ ++ = (3 4) (3 2) (2 4) (2 2)×+ ×+× +× = 12 32 22 2+++ = 14 52+ You can also do this if there is a subtraction in the bracket. E EXTENDED 145Chapter 9: Standard form and surds 9 . 3 SamplePages
Example 7 Simplify: a (4 5)(2 5)−+ b (2 3)2 a (4 5)(2 5) 4(25 )5 (2 5)−+ =+ −+ = 84 52 55+− = 14 52+ b (2 3) (2 3)(2 3)2 −= = 2(23 )3(2 3) = 42 32 33−−+ = 74 3 Advice and Tips 33 3−× −= + Look at this fraction: 12 6 It has a surd in the denominator. You can remove it by multiplying the numerator and the denominator by 6 . 9 6 96 66 96 6 36 2 = × × == or 3 2 6 Removing the surd from the denominator of a fraction is called rationalisingthedenominator. This is a more complicated fraction: 42 32 + + The denominator is: 32 + We can multiply this by 32 (3 2)(3 2) 3(32 )2(3 2)+− =− +− = 93 23 22 7−+− = The answer is an integer. Multiply the numerator and the denominator of the fraction by 32 42 32 (4 2)(3 2) (3 2)(3 2) + + = +− +− = 12 42 32 2 7 −+ = 10 2 7 or 1 7 (102 ) Example 8 Rationalise the denominator of: a 5 7 b 5 47 a Multiply the numerator and the denominator by 7 . 5 7 57 77 57 7 = × × = or 5 7 7 146 9.5 Rationalising the denominator SamplePages
b Multiply the numerator and the denominator by 47 + . 5 47 5( 47 ) (4 7)(4 7) = + −+ = 5( 47 ) 16 7 + = 5( 47 ) 9 + or 5 9 (4 7)+ Exercise 9E 1 Multiply the brackets and simplify the answers. a (2 5)(3 5)++ b (4 3)(2 3)+− c (1 6)(4 6)++ d (3 2)(2 2)−+ e () 32 () 42 f 2() 72 + 2 Multiply the brackets and simplify the answers. a () 52 () 52+− b 04() 41 () 10−+ c 21() 12 () 22−+ d 11() 10 () 01−+ e () 32 () 32+− f 24() 42 () 22−+ 3 () 35−× n is an integer. Find a possible value for n. 4 Rationalise the denominator. a 8 2 b 10 3 c 18 12 d 20 5 e 8 2 f 5 20 5 Simplify each answer. a 25 5 + b 12 6 6 c 42 8 + d 27 2 3 6 Rationalise the denominator. a 2 32 + b 5 410 c 3 23 + d 5 35 7 The reciprocal of n is 1 n . a Show that the reciprocal of 23 + is 23 b What is the reciprocal of 23 ? c Find the reciprocal of 37 + . EXTENDED 147Chapter 9: Standard form and surds 9 . 3 SamplePages
Check
8 Rationalise the denominator. a 22 22 + b 13 33 + + c 45 35 d 12 6 12 3 9 Here is an equation: 212 −=xx a Show that 12=+x is a solution of the equation. b Show that 12=−x is another solution of the equation.
your progress Core • I can understand and write numbers in standard form • I can convert numbers into and out of standard form • I can calculate with values in standard form using a calculator Extended • I can calculate with values in standard form without using a calculator • I can understand and use surds, and simplify expressions • I can rationalise the denominator of an expression containing a surd EXTENDED 148 9.5 Rationalising the denominator SamplePages
