Standard Form (Copy)
Practice Questions — 1.8 Standard Form
Question 1
Write the following numbers in standard form:
(a) 72 000
(b) 0.00053
(c) 8 950 000
Question 2
Write the following numbers in ordinary form:
(a) 6.4 × 10³
(b) 3.25 × 10⁻⁴
(c) 9.81 × 10⁵
Question 3
Express 0.0000072 in standard form.
Question 4
Express 4.89 × 10⁶ as an ordinary number.
Question 5
The speed of light is about 3 × 10⁸ m/s.
Express this in ordinary form.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 6
Multiply: (2.5 × 10³) × (4 × 10⁵).
Question 7
Divide: (6.4 × 10⁶) ÷ (8 × 10²).
Question 8
Simplify: (3 × 10⁴) × (2 × 10⁻³).
Question 9
Simplify: (7.2 × 10⁻⁵) ÷ (2.4 × 10⁻²).
Question 10
A population is estimated at 5.46 × 10⁷.
Write this in ordinary form.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 11
A grain of sand has a mass of 3.6 × 10⁻⁴ g.
Write this mass in ordinary form.
Question 12
A galaxy is 4.2 × 10⁶ light years away.
Express this in ordinary form.
Question 13
Convert into standard form:
(a) 0.00000081
(b) 12 300 000
(c) 0.0045
Question 14
Simplify: (5 × 10⁷) ÷ (2.5 × 10⁻³).
Question 15
Multiply: (9 × 10⁻⁴) × (7 × 10⁻²).
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 16
Divide: (8 × 10¹²) ÷ (4 × 10⁶).
Question 17
Express in standard form:
(a) 0.00032
(b) 56 700 000
(c) 0.00000000091
Question 18
Simplify: (1.2 × 10³) × (2.5 × 10⁻⁵).
Question 19
Simplify: (6 × 10⁻⁸) ÷ (3 × 10⁻¹¹).
Question 20
The diameter of the Earth is about 1.27 × 10⁷ m, while the diameter of the Moon is about 3.48 × 10⁶ m.
(a) Express both in ordinary form.
(b) Find the ratio of Earth’s diameter to Moon’s diameter in simplest form.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Answer key and explanations — 1.8 Standard Form
1.
(a) 72 000 = 7.2 × 10⁴
(b) 0.00053 = 5.3 × 10⁻⁴
(c) 8 950 000 = 8.95 × 10⁶
Explanation: move the decimal point so that 1 ≤ A < 10, count shifts to set the power of 10.
2.
(a) 6.4 × 10³ = 6400
(b) 3.25 × 10⁻⁴ = 0.000325
(c) 9.81 × 10⁵ = 981 000
Explanation: multiply the decimal part by powers of 10. Negative index moves decimal left, positive moves right.
3.
0.0000072 = 7.2 × 10⁻⁶
Explanation: decimal moved 6 places right to get 7.2.
4.
4.89 × 10⁶ = 4 890 000
Explanation: positive index moves decimal right 6 places.
5.
3 × 10⁸ = 300 000 000
Explanation: speed of light written in full form.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
6.
(2.5 × 10³) × (4 × 10⁵) = (2.5 × 4) × 10^(3+5) = 10 × 10⁸ = 1.0 × 10⁹
Explanation: multiply decimals separately, add powers.
7.
(6.4 × 10⁶) ÷ (8 × 10²) = (6.4 ÷ 8) × 10^(6−2) = 0.8 × 10⁴ = 8 × 10³
Explanation: divide decimals, subtract powers, adjust to 1 ≤ A < 10.
8.
(3 × 10⁴) × (2 × 10⁻³) = 6 × 10¹
Explanation: multiply 3 × 2 = 6; add indices: 4 + (−3) = 1.
9.
(7.2 × 10⁻⁵) ÷ (2.4 × 10⁻²) = (7.2 ÷ 2.4) × 10^(−5−(−2)) = 3 × 10⁻³
Explanation: divide decimals = 3; subtract powers = −5+2 = −3.
10.
5.46 × 10⁷ = 54 600 000
Explanation: move decimal right 7 places.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
11.
3.6 × 10⁻⁴ = 0.00036 g
Explanation: negative power shifts decimal left.
12.
4.2 × 10⁶ = 4 200 000 light years
Explanation: positive power shifts decimal right.
13.
(a) 0.00000081 = 8.1 × 10⁻⁷
(b) 12 300 000 = 1.23 × 10⁷
(c) 0.0045 = 4.5 × 10⁻³
14.
(5 × 10⁷) ÷ (2.5 × 10⁻³) = (5 ÷ 2.5) × 10^(7−(−3)) = 2 × 10¹⁰
Explanation: divide coefficients, subtract indices.
15.
(9 × 10⁻⁴) × (7 × 10⁻²) = 63 × 10⁻⁶ = 6.3 × 10⁻⁵
Explanation: multiply decimals, then adjust to standard form.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
16.
(8 × 10¹²) ÷ (4 × 10⁶) = (8 ÷ 4) × 10^(12−6) = 2 × 10⁶
17.
(a) 0.00032 = 3.2 × 10⁻⁴
(b) 56 700 000 = 5.67 × 10⁷
(c) 0.00000000091 = 9.1 × 10⁻¹⁰
18.
(1.2 × 10³) × (2.5 × 10⁻⁵) = 3.0 × 10^(3−5) = 3.0 × 10⁻²
19.
(6 × 10⁻⁸) ÷ (3 × 10⁻¹¹) = (6 ÷ 3) × 10^(−8−(−11)) = 2 × 10³
20.
Earth: 1.27 × 10⁷ = 12 700 000 m
Moon: 3.48 × 10⁶ = 3 480 000 m
Ratio = 12 700 000 ÷ 3 480 000 = 127 ÷ 34.8 ≈ 3.65 : 1
Explanation: Earth’s diameter is about 3.65 times larger than Moon’s.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course