08
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January 2001
Number 08
Indices, Standard Form & Orders of Magnitude Indices is just another word for powers. This Factsheet will review the laws of indices and standard form, including their use on a calculator and explain what is meant by orders of magnitude.
Example 1. a) By using the relationship between work done, force and distance moved, show that 1 Joule is equivalent to 1 kgm 2s-2 b) Show that the equation P = Dv, where P = power, D= driving force and v = velocity, is homogeneous with respect to units
1. Indices There are three laws of indices:
a) Step 1. Write down the appropriate equation Work done (J) = Force (N) distance moved (m)
Step 2: Express everything on the right hand side of the equation in terms of kg, m and s (since these are what we want in the answer)
If you are multiplying, add the indices so y6 y2 = y6+2 = y8 If you are dividing, subtract the indices so y6 y2 = y6 – 2 = y4 If you are doing a power of a power, multiply the indices, so (y6)2 = y6 2 = y12
Since Force (N) = mass (kg) acceleration (ms-2), we have Work done (J) = Force (kgms-2) distance moved (m) Step 3: Simplify the powers on the left hand side So 1 J = 1 kgms-2 1m J = kg m s-2 m = kg m2 s-2
However, these only apply to powers of the same number. For example, 105 104 = 109, but you cannot simplify 105 24
b) Step 1: Express the units of everything in terms of kg, m and s. (use what we did in part a)!) energy kgm 2 s -2 Power = time s Force velocity = kgms-2 ms-1
In addition, you need to know the following about indices:
Anything to the power 0 is 1 so 60 = 1, y0 = 1, 27549.260 = 1 Anything to the power 1 is the number itself so 61 = 6, 0.31 = 0.3 In a power with fractions, the bottom of the fraction means a root, and the top of it means a power so y
1 2
= y
1
= y, y
2 3
=
3
y , y
3 2
=
y
Force velocity = kg m m s-2 s-1 = kg m2 s-2 + -1 = kgm2s-3
3
Negative powers mean “1 over” 1 1 1 1 so y-3 = 3 , y 2 = 1 = y y y2 You can expand brackets involving things multiplied or divided in the obvious way – (2y)2 = 22y2 = 4y2, (ab2)3 = a3b6,
2
Step 2: Simplify Power = kgm2s-2 s1 = kgm2s -2 1 = kgm2s-3
4a b2
4 a 2
2
Using your calculator for indices Your calculator will have a button that looks like xy, or yx, or ab (or on some graphical calculators, ^). To use this to find powers, you put in the number, then press this button, then the power.
a
eg To find 61.6, you press 6, then xy, then 1.6
b
b You cannot expand brackets involving things added or subtracted – you cannot simplify (y2 + 1), or (a + b)10
Your calculator will do indices before anything else except brackets, so if you were to put in 2 43, it will do the 43 first, then multiply by 2 (giving the answer 128). If you actually wanted it to do (2 4)3, you’d need to type in the brackets.
Most problems occur when students attempt to invent extra rules for indices! Before you attempt to simplify anything, check you are using one of the rules here – if you are not, you can’t do it!
There are two ways to find roots on your calculator (other than square roots, which have their own button): work out the power you need (cube root is the power one third, fourth root is the power one quarter etcetera) and treat it as finding the power Use the “root” function, which is almost always accessed by pressing “shift” or “2nd” or “inv” and the xy key.
One common use of indices is in determining units for a quantity, or checking that an equation is homogeneous with regard to units:
1
Indices, Standard Form and Orders of Magnitude Unfortunately, calculators can vary as to exactly how you use the “root” function.
Standard form on your calculator To enter a number in standard form into your calculator, you use the key labelled EXP or EE. eg 2.4 10-5 would be put in as 2.4 EXP –5
To find 3 8 , you’d do one of: 3 INV xy 8 8 INV xy 3. Find out which your calculator does NOW! You should get the answer 2 to the above calculation.
Standard form is an easy way of writing large or small numbers using powers of 10. For example, 200 = 2 100 = 2 102.
Your calculator will automatically display very large or very small numbers in standard form. The power is shown as small numbers to the right of the display. eg: 03 2.45 means 2.45 103 -12 9.5 means 9.5 10-12 14 -1.87 means –1.87 1014 You must NEVER write the number the way your calculator does! If you write 9.5-12, it looks like you mean 9.5 to the power –12, not 9.5 10-12.
To change a number into standard form: 1. Write the number down with a decimal point, if it hasn’t got one already. 2. “Jump” the decimal point to the left or right until you get a number between 1 and 9.999.... Count the number of jumps. 3. Write the number in standard form as:
The number between 1 and 9.999... you get after jumping the decimal point
10-5 is
You can practise this to check you’ve got it right – try entering 2 102 and 3 10-2. Your calculator should give you 200 and 0.03 respectively.
2. Standard Form
___ 10
Do NOT use the sign Do NOT use the power key xy Be careful that you put in the “-“ in the right place. 2.4 not the same as –2.4 105
___
You do calculations with numbers in standard form in the normal way – provided you enter them in the way described above, you don’t need to put brackets round them. eg to do 3.15 104 (2.1 103), you’d type in 3.15 EXP 4 2.1 EXP 3 (giving 15)
The number of times you jumped the point, with a minus sign if you jumped to the right
As a check – if the original number was big, the power of 10 is positive. If the original number was small, the power of 10 is negative.
3. Orders of magnitude Orders of magnitude are a way of saying roughly how big a number is without doing an accurage calculation – you can think of them as a special form of rounding. Sometimes it is more appropriate to give an order of magnitude estimate as the actual value can’t be calculated accurately.
Example 2. Write the following in standard form: i) 6234000 ii) 0.00578 i) 6234000. 6.234 with 6 jumps to the left So 6234000 = 6.234 106
ii) 0.00578 5.78 with 3 jumps to the right So 0.00578 = 5.78 10 –3
To change from standard form to ordinary numbers, you just reverse this: 1. Write down the number in front of the “10?” with a decimal point, if it hasn’t already got one. 2. Jump the decimal point by the number of spaces given by the power of 10. If the power is positive, jump it to the right, if it’s negative, jump it to the left. 3. Put zeroes in any gaps
eg 2450 is nearer 1000 than 10000, so our order of magnitude estimate is 1000 or 103 To find an order of magnitude estimate for a figure in standard form, use the following: If the number in front of the 10? is less than 5, use the power of 10 you have in the standard form If the number in front of the 10?is 5 or more, add one to the power of 10 you have in the standard form
As a check – if the original power was negative, you should end up with a small number; if it was positive, you’ll end up with a big number. As a really thorough check, you could try changing what you get back into standard form... Example 3. Write the following not in standard form: i) 5.92 10 2.49 105 i) ii) 2.49
-4
You give an order of magnitude estimate by giving the answer to the nearest power of 10 – in other words, deciding whether it’s nearer to 1000, 100, 10, 1, 0.1, 0.01..... These are often written in power form – so the above would be 103, 102, 101, 100, 10-1, 10-2 etc.
eg 6.8 10-5 would give 10-4. 2.49 1013 would give 1013 If you are asked to give an order of magnitude estimate for the answer to a calculation, you could, of course, do the calculation normally then look at the answer, but that isn’t really the way to do it if you have to show your working (because you’re not really using orders of magnitude) and sometimes you aren’t actually given all the figures.
ii)
5.92 gives 0.000592
A better way to do it is to put an order of magnitude estimate for each figure involved in the calculation, and use the laws of powers to work out the answer.
gives 249000
2
Indices, Standard Form and Orders of Magnitude Answers
Example 4. Obtain an order of magnitude estimate for the factor by which the speed of light is faster than the speed of sound. 8
1. a) 1013 e) ¼
-1
Orders of magnitude: speed of light ~ 10 ms speed of sound ~ 102 ms-1 6 8 2 So 10 10 = 10
i) 3x
GMm r2
2x 3y 2
d) 1000 (or 103) h) y1.5
k) doesn’t simplify
2. a) 2088.27064576 b) 1.71 (3SF) c) 3.30 (3 SF) d) 0.00703 (3 SF) ,
where G = 6.7 10-11 Nm2kg-2. Use this to estimate the gravitational force of attraction between two adult human beings in the same room, giving your answer as an order of magnitude.
3.
2
gives
Nm 2 kg 2 kgkg 2
m r which is the units of force
Nm2 m
5. a) i) 2.36 101 ii) 9.8 100 b) i) 0.000278 ii) 906 c) 1.69 10-3 (3SF)
10 10 10 2 10 2 = 10-6 N 2 100
GMm
2
=N,
4. kgm2 s-2K-1
M ~ 102 kg, m ~ 102 kg. G ~ 10-10 Nm2kg-2. r ~ 100
So F ~
j)
c) 101.26 g) LM-1T-1
l) (x + y)2 ( = x2 + 2xy + y2)
Example 5. The gravitational force (in newtons) between two bodies of masses M and m kilograms a distance r metres apart is given by F=
b) 10-1 f) 1/3
6. a) i) 103 b) 106 ms-1 c) 107 J
Tip: Make sure you put everything in SI units for this sort of calculation.
ii) 10-1
iii) 106
iii) 4.057 10-4iv) 6.547108 iii) 6.3 iv) 9 900 000 iv) 10-3
Questions 1. Without using a calculator, simplify the following as far as possible: 3
a) 105 108
b) 1017 1018
c) (100.63)2
e) 2-2
f) 9- ½
g)
i) (9x) ½
j)
3
8x 3 27 y
6
d) 100 2
L2 M 3
h) y y
LM 4 T
k) (x2 + 4)
l)
3
( x y) 6
2
2. Use your calculator to find a) 2.68 b)
3
5
c) 6 3
d) 10.6 –2.1
3. Show that the equation for gravitational force between two objects given in example 5 above is homogeneous with respect to its units. 4. In the equation PV = RT, T is temperature (in kelvin), V is volume (in m3) and P is pressure (in Nm-2). Find the units of R, giving your answer in terms of kg, m and K. 5. a) Convert the following to standard form: i) 23.6 ii) 9.8 iii) 0.0004057
iv) 654700000
b) Write the following as normal numbers, not in standard form ii) 9.06 102 iii) 6.3 100 iv) 9.9 106 i) 2.78 10-4 c) Use your calculator to find
5.23 10 4 7.98 10 4 3.1 10 7
6. a) Give each of the following figures to the nearest order of magnitude i) 2456 ii) 0.067 iii) 1259000 iv) 0.0045
Acknowledgements: This Factsheet was researched and written by Cath Brown Curriculum Press, Unit 305B The Big Peg, 120 Vyse Street, Birmingham B18 6NF. Physics Factsheets may be copied free of charge by teaching staff or students, provided that their school is a registered subscriber. They may be networked for use within the school. No part of these Factsheets may be reproduced, stored in a retrieval system or transmitted in any other form or by any other means without the prior permission of the publisher. ISSN 1351-5136
b) An object of mass 36 grammes has kinetic energy 4.2 109J. Without using a calculator, obtain an order of magnitude estimate for its velocity. c) Obtain an order of magnitude estimate for the kinetic energy of a car travelling on a motorway.
3
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