Standard form
Standard form is a short way to write very large and very small numbers, such as the mass of the Earth or the width of a cell. It always has the shape A × 10ⁿ with 1 ≤ A < 10.
Part 1 of 3: Learn it
In short
- A number in standard form is A × 10ⁿ, where 1 ≤ A < 10 and n is a whole number.
- Big numbers have a positive power; numbers between 0 and 1 have a negative power.
- To multiply, multiply the A parts and add the powers. To divide, divide the A parts and subtract the powers.
Where this is in your specification
Spec points: DfE N9 (with N7 for the index laws)
| Board | Topic: Powers, roots and standard form |
|---|---|
| DfE content | N6, N7, N9 (and A4 for indices with letters) |
| AQA 8300 | Number |
| Edexcel 1MA1 | Number |
| OCR J560 | 3 Indices and surds; 1 Number operations and integers |
| Eduqas C300 | HN6, HN7, HN9 |
| Cambridge IGCSE 0580 | E1.3, E1.7, E1.8, E2.4 |
| National 5 C847 75 | Numerical skills: indices, scientific notation |
Writing numbers in standard form
- 4 700 000 = 4.7 × 10⁶ (the decimal point moves 6 places).
- 0.000 52 = 5.2 × 10⁻⁴ (the point moves 4 places the other way).
- 8.1 × 10³ = 8100 and 3 × 10⁻² = 0.03.
Multiplying and dividing
Group the number parts and the powers of 10, then use the index laws.
If the number part ends up outside 1 to 10, adjust it: 15 × 10⁴ = 1.5 × 10⁵ and 0.4 × 10⁶ = 4 × 10⁵.
Adding and subtracting
You cannot just add the powers. Either write both numbers out in full, or rewrite them with the same power of 10 first: 6 × 10⁴ + 3 × 10³ = 60 000 + 3000 = 63 000 = 6.3 × 10⁴.
On a calculator
Use the ×10ˣ (or EXP) key rather than typing × 1 0 ^. A display such as 2.4E-05 means 2.4 × 10⁻⁵, and you must write it that way in your answer.
Write 0.000 61 in standard form.
Show the answer
6.1 × 10⁻⁴.
Part 2 of 3: See it worked
Worked examples
Example 1
Work out (3 × 10⁵) × (6 × 10⁻²). Give your answer in standard form.
- 3 × 6 = 18
- 10⁵ × 10⁻² = 10³
- 18 × 10³ = 1.8 × 10⁴
Answer: 1.8 × 10⁴.
Example 2
Work out (8.4 × 10⁷) ÷ (2.1 × 10³).
- 8.4 ÷ 2.1 = 4
- 10⁷ ÷ 10³ = 10⁴
Answer: 4 × 10⁴.
Example 3
Work out 7.2 × 10⁵ − 4 × 10⁴.
- 720 000 − 40 000
- = 680 000
- = 6.8 × 10⁵
Answer: 6.8 × 10⁵.
Common mistakes
- Leaving the number part as 18 or 0.5, which is not standard form.
- Giving 0.0003 as 3 × 10³. Numbers less than 1 need a negative power.
- Adding the powers when adding two numbers in standard form.
- Copying the calculator display, such as 4.6E8, as the final answer.
Work out (5 × 10³) × (4 × 10⁶) in standard form.
Show the answer
20 × 10⁹ = 2 × 10¹⁰.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Write 0.000 61 in standard form.
6.1 × 10⁻⁴.
Work out (5 × 10³) × (4 × 10⁶) in standard form.
20 × 10⁹ = 2 × 10¹⁰.
Write 3.08 × 10⁵ as an ordinary number.
308 000.
Jobs that use this
- Astronomer (s'ouvre dans un nouvel onglet)
- Physicist (s'ouvre dans un nouvel onglet)
- Microbiologist (s'ouvre dans un nouvel onglet)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Ces fiches sont en anglais car elles suivent les programmes d'examen britanniques.
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses