N9: Standard Form
Calculate with numbers in standard form A × 10^n where 1 ≤ A < 10
📋 Key Concepts
Standard Form: A way of writing very large or very small numbers. The format is A × 10ⁿ where 1 ≤ A < 10 and n is an integer.
Standard Form Rules:
• A must be between 1 and 10 (1 ≤ A < 10)
• n is positive for large numbers, negative for small numbers
• The × 10ⁿ part tells you how far to move the decimal point
• A must be between 1 and 10 (1 ≤ A < 10)
• n is positive for large numbers, negative for small numbers
• The × 10ⁿ part tells you how far to move the decimal point
| Number | Standard Form | n value |
|---|---|---|
| 3000 | 3 × 10³ | Positive (large) |
| 52,000 | 5.2 × 10⁴ | Positive (large) |
| 0.004 | 4 × 10⁻³ | Negative (small) |
| 0.00052 | 5.2 × 10⁻⁴ | Negative (small) |
📝 Converting to Standard Form
Method: Count how many places the decimal point moves to get A between 1 and 10.
Example 1
Write 45,000 in standard form
Solution:
Move decimal point left: 4.5 (need A between 1 and 10)
Moved 4 places: 4.5 × 10⁴
Example 2
Write 0.00073 in standard form
Solution:
Move decimal point right: 7.3
Moved 4 places: 7.3 × 10⁻⁴
📝 Converting from Standard Form
Method: Positive n → move decimal right. Negative n → move decimal left.
Example 3
Write 5.4 × 10⁵ as an ordinary number
Solution:
Positive exponent → large number
Move decimal 5 places right: 540,000
Example 4
Write 3.2 × 10⁻³ as an ordinary number
Solution:
Negative exponent → small number
Move decimal 3 places left: 0.0032
📝 Multiplying in Standard Form
Method: Multiply the A parts, add the powers of 10, then adjust to proper standard form.
Example 5
Calculate: (2 × 10⁴) × (3 × 10⁵)
Solution:
2 × 3 = 6
10⁴ × 10⁵ = 10⁹
Answer: 6 × 10⁹
Example 6
Calculate: (4 × 10³) × (2 × 10⁴)
Solution:
4 × 2 = 8
10³ × 10⁴ = 10⁷
Answer: 8 × 10⁷
Example 7
Calculate: (5 × 10³) × (4 × 10²)
Solution:
5 × 4 = 20
10³ × 10² = 10⁵
20 × 10⁵ = 2 × 10⁶ (adjust to standard form)
📝 Dividing in Standard Form
Method: Divide the A parts, subtract the powers of 10, then adjust to proper standard form.
Example 8
Calculate: (6 × 10⁸) ÷ (2 × 10³)
Solution:
6 ÷ 2 = 3
10⁸ ÷ 10³ = 10⁵
Answer: 3 × 10⁵
Example 9
Calculate: (8 × 10⁶) ÷ (4 × 10⁻²)
Solution:
8 ÷ 4 = 2
10⁶ ÷ 10⁻² = 10⁸
Answer: 2 × 10⁸
📝 Adding and Subtracting
Method: Convert to ordinary numbers, calculate, then convert back to standard form. Or make the powers of 10 the same first.
Example 10
Calculate: (3.2 × 10⁴) + (5.1 × 10³)
Solution:
Convert: 32,000 + 5,100 = 37,100
Back to standard form: 3.71 × 10⁴
❓ Practice Questions
Q1: Write 7,500,000 in standard form
Q2: Write 0.000082 in standard form
Q3: Write 4.7 × 10⁻² as an ordinary number
Q4: Calculate: (3 × 10⁵) × (4 × 10⁶)
Q5: Calculate: (9 × 10⁸) ÷ (3 × 10²)
✅ Answers
- 7.5 × 10⁶
- 8.2 × 10⁻⁵
- 0.047
- 12 × 10¹¹ = 1.2 × 10¹²
- 3 × 10⁶
🎯 Exam Tips
- Check A is between 1 and 10 (not 0.something and not ≥ 10)
- Large numbers → positive n, Small numbers → negative n
- Use your calculator's EXP or ×10ˣ button for standard form
- When multiplying: multiply A's, add powers
- When dividing: divide A's, subtract powers
🧠 Problem-Solving Strategies
Problem-Solving
For standard form calculations, multiply/divide the A parts and powers of 10 separately, then adjust so A is between 1 and 10. For addition/subtraction, convert to ordinary numbers first or equalise the powers. Always check your final answer is in proper standard form (1 ≤ A < 10).
Multi-Step Problem
The population of Country A is 6.2 × 10⁷ and Country B is 8.4 × 10⁶. How many times larger is Country A's population? Give your answer to 2 significant figures.
Solution: Ratio = (6.2 × 10⁷) ÷ (8.4 × 10⁶) = (6.2 ÷ 8.4) × 10¹ = 0.738... × 10 = 7.38... ≈ 7.4 times.
⚠️ Common Errors
Watch Out!
1. Wrong: 12 × 10³ is in standard form Correct: A must be between 1 and 10, so 12 × 10³ = 1.2 × 10⁴
2. Wrong: 0.3 × 10⁵ is in standard form Correct: A must be ≥ 1, so 0.3 × 10⁵ = 3 × 10⁴
3. Wrong: When adding (3 × 10⁴) + (2 × 10³) = 5 × 10⁷ Correct: Convert first: 30,000 + 2,000 = 32,000 = 3.2 × 10⁴
✍️ 6-Mark Exam Question
Extended Answer
6 marks: The distance from Earth to the Sun is 1.5 × 10⁸ km. Light travels at 3.0 × 10⁵ km/s. (a) Calculate how many seconds it takes light to travel from the Sun to Earth. Give your answer in standard form. (b) Convert your answer to minutes. (c) A star is 4.2 × 10¹³ km from Earth. How many years does light from this star take to reach Earth? (1 year = 3.15 × 10⁷ seconds)
(a) Time = distance ÷ speed = (1.5 × 10⁸) ÷ (3.0 × 10⁵) = 0.5 × 10³ = 5.0 × 10² seconds.
(b) 5.0 × 10² ÷ 60 = 500 ÷ 60 = 8.33 minutes.
(c) Time = (4.2 × 10¹³) ÷ (3.0 × 10⁵) = 1.4 × 10⁸ seconds. Years = (1.4 × 10⁸) ÷ (3.15 × 10⁷) = 1.4/3.15 × 10 = 4.44 years.
Mark scheme: 2 marks for (a) including standard form adjustment, 1 mark for (b), 3 marks for (c) with clear working and unit conversion
📊 AO3: Reason & Interpret
Reasoning and Interpretation
The mass of the Earth is 5.97 × 10²⁴ kg. The mass of Jupiter is 1.9 × 10²⁷ kg.
(a) How many times more massive is Jupiter than Earth? Give your answer to 1 significant figure.
(b) A student says "Jupiter is about 3000 times more massive than Earth." Is this a reasonable claim?
(c) The Sun's mass is 1.99 × 10³⁰ kg. Express the combined mass of Earth and Jupiter as a fraction of the Sun's mass.
Answers: (a) (1.9 × 10²⁷) ÷ (5.97 × 10²⁴) = (1.9/5.97) × 10³ ≈ 0.318 × 1000 ≈ 318 ≈ 300 (1 s.f.). (b) The student said 3000, but the answer is about 300. The student is off by a factor of 10 — they likely miscalculated the power of 10. (c) Combined = 5.97 × 10²⁴ + 1.9 × 10²⁷ ≈ 1.906 × 10²⁷ kg. Fraction = (1.906 × 10²⁷)/(1.99 × 10³⁰) ≈ 9.58 × 10⁻⁴ ≈ 0.001.
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