Write each number in standard form (scientific notation).
(a)Write 4300000 in standard form.(1)
(b)Write 0.00072 in standard form.(1)
(c)Write 12500 in standard form.(1)
(d)Write 0.03 in standard form.(1)
2
Write these standard form numbers as ordinary numbers.
(a)Write 6.5 x 103 as an ordinary number.(1)
(b)Write 9.1 x 10-5 as an ordinary number.(1)
3
Order these numbers from smallest to largest. Express each comparison using their standard form where helpful: 3.2 x 10-3, 0.00045, 1.1 x 10-2.
(3)
4
Multiply and give your answer in standard form.
(a)Work out (3 x 104) x (2 x 103).(2)
(b)Work out (4.5 x 10-2) x (2 x 103).(2)
5
Divide and give your answer in standard form.
(a)Work out (6 x 105) / (2 x 102).(2)
(b)Work out (8.4 x 10-1) / (2 x 102).(1)
6
A star is 4.2 x 1016 metres from Earth. A light wave from the star travels at 3 x 108 m/s. Estimate how many seconds the light takes to reach Earth. Give your answer in standard form, to 2 significant figures.
(2)
7
Express each of these in standard form, using integers for powers of ten and one decimal place for the coefficient if needed.
(a)0.0009(1)
(b)2500000(1)
8
Calculate and give your answer in standard form, simplifying coefficients to be between 1 and 10 if needed: (7 x 104) x (5 x 10-7) + (3 x 10-2).
(3)
9
The mass of a bacterium is about 2.5 x 10-12 kg. Write the mass of 1 million such bacteria in standard form.
(2)
10
A chemical container holds 4.8 x 102 cm3 of liquid. Another holds 1.2 x 10-1 cm3. What is the total volume in standard form?
(3)
11
Convert 0.0064 to standard form and give the coefficient to 2 significant figures.
(2)
12
A physics exam question: The charge on an electron is about 1.6 x 10-19 coulombs. How many electrons are needed to make a charge of 3.2 x 10-16 C? Give your answer in standard form, to 2 significant figures.
(3)
13
State whether each of these numbers is greater than, equal to, or less than 1 in standard form. Give an explanation for each.
(a)5.0 x 10-1(1)
(b)9.1 x 100(1)
14
A comet travels 2.5 x 109 metres in 5.0 x 104 seconds. Find its average speed in metres per second in standard form.
(3)
15
Show your working: Simplify (1.2 x 106) + (3.8 x 105) and give the answer in standard form.
(4)
16
A radio signal has wavelength 0.0025 metres. Express this wavelength in standard form and then calculate how many wavelengths fit into 1 kilometre. Give the final answer in standard form to 2 significant figures.
(3)
17
Explain why a number written as 10.2 x 103 is not in standard form and rewrite it correctly.
(2)
18
Stretch question: The brightness of a distant star is 2.4 x 10-8 units. After a change it is measured as 6.0 x 10-9 units. By what factor did the brightness change? Give your answer in standard form and explain whether it increased or decreased.
(3)
Mark scheme · KS3.M-N13 Standard Form
Question 1
(a) B1 4.3 x 106 cao
(a) Answer: 4.3 x 106
(b) B1 7.2 x 10-4 cao
(b) Answer: 7.2 x 10-4
(c) B1 1.25 x 104 cao
(c) Answer: 1.25 x 104
(d) B1 3 x 10-2 cao
(d) Answer: 3 x 10-2
Question 2
(a) B1 6500 cao
(a) Answer: 6500
(b) B1 0.000091 cao
(b) Answer: 0.000091
Question 3
M1 convert to same form or decimals, e.g. 3.2 x 10-3 = 0.0032, 1.1 x 10-2 = 0.011
A1 correct order: 0.00045, 3.2 x 10-3, 1.1 x 10-2 cao
B1 one correct comparison shown (allow follow through if order given correctly)
Answer: 0.00045, 3.2 x 10-3, 1.1 x 10-2
Question 4
(a) M1 multiply coefficients and add powers: 3 x 2 = 6 and 104 x 103 = 107
(a) A1 6 x 107 cao
(a) Answer: 6 x 107
(b) M1 multiply coefficients and add powers: 4.5 x 2 = 9.0 and 10-2 x 103 = 101
(b) A1 9 x 101 cao
(b) Answer: 9 x 101
Question 5
(a) M1 divide coefficients and subtract powers: 6/2 = 3 and 105-2 = 103
(a) A1 3 x 103 cao
(a) Answer: 3 x 103
(b) B1 4.2 x 10-3 cao
(b) Answer: 4.2 x 10-3
Question 6
M1 distance divided by speed: 4.2 x 1016 / 3 x 108 or equivalent method
A1 1.4 x 108 s cao to 2 significant figures
Answer: 1.4 x 108 s
Question 7
(a) B1 9.0 x 10-4 cao
(a) Answer: 9.0 x 10-4
(b) B1 2.5 x 106 cao
(b) Answer: 2.5 x 106
Question 8
M1 multiply first two terms: 7 x 5 = 35 and 104-7 = 10-3 giving 35 x 10-3
M1 convert 35 x 10-3 to 3.5 x 10-2 and add 3 x 10-2 to get 6.5 x 10-2
A1 6.5 x 10-2 cao
Answer: 6.5 x 10-2
Question 9
M1 1 million = 106 so multiply coefficients: 2.5 x 10-12 x 106
A1 2.5 x 10-6 kg cao
Answer: 2.5 x 10-6 kg
Question 10
M1 convert to same power or adjust: 4.8 x 102 = 480, 1.2 x 10-1 = 0.12 then add to get 480.12
M1 express 480.12 in standard form: 4.8012 x 102 or round if required
A1 4.8012 x 102 cao
Answer: 4.8012 x 102 cm3
Question 11
M1 write 0.0064 as 6.4 x 10-3 then round coefficient to 2 s.f
A1 6.4 x 10-3 cao
Answer: 6.4 x 10-3
Question 12
M1 divide total charge by charge per electron: 3.2 x 10-16 / 1.6 x 10-19
M1 compute 3.2/1.6 = 2 and subtract powers: 10-16 - (-19) = 103 giving 2 x 103
A1 2.0 x 103 cao to 2 s.f
Answer: 2.0 x 103
Question 13
(a) B1 less than 1 with reason: coefficient 5.0 x 10-1 = 0.5 < 1
(a) Answer: Less than 1 because 5.0 x 10-1 = 0.5
(b) B1 greater than 1 with reason: 9.1 x 100 = 9.1 > 1
(b) Answer: Greater than 1 because 9.1 x 100 = 9.1
Question 14
M1 divide coefficients and subtract powers: 2.5/5.0 = 0.5 and 109-4 = 105 giving 0.5 x 105
M1 adjust to standard form 5.0 x 104
A1 5.0 x 104 m/s cao
Answer: 5.0 x 104 m/s
Question 15
M1 write 3.8 x 105 as 0.38 x 106 or convert 1.2 x 106 to same power and add coefficients
M1 add coefficients: 1.2 + 0.38 = 1.58
M1 express as standard form 1.58 x 106
A1 1.58 x 106 cao
Answer: 1.58 x 106
Question 16
M1 convert 0.0025 to 2.5 x 10-3 and 1 km = 1000 m = 1.0 x 103 m
M1 divide: (1.0 x 103) / (2.5 x 10-3) = 0.4 x 106 or equivalent
A1 4.0 x 105 cao to 2 s.f
Answer: 4.0 x 105
Question 17
M1 explain that coefficient must be between 1 and 10, and 10.2 is ≥10 so not standard form
A1 correct rewrite 1.02 x 104 cao
Answer: 1.02 x 104
Question 18
M1 find ratio 6.0 x 10-9 / 2.4 x 10-8 or 2.4 x 10-8 / 6.0 x 10-9 depending on wording
M1 compute ratio = 0.25 = 2.5 x 10-1 or inverse 4.0 x 100; identify decrease
A1 brightness decreased by a factor of 2.5 x 10-1 (ie it is 0.25 of original) cao