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
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).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses