NumberFoundation + Higher
Standard Form.
a × 10n.
Standard form writes very big and very small numbers compactly. Format: a × 10n where 1 ≤ a < 10 and n is an integer. Multiplying and dividing becomes a game of index laws.
I · Key definitions
Standard form
a × 10n where 1 ≤ a < 10.
3.45 × 10⁵
Big numbers
Positive power of 10.
7,500 = 7.5 × 10³
Small numbers
Negative power of 10.
0.00042 = 4.2 × 10⁻⁴
The value a
Must be at least 1 and less than 10.
3.2 ✓ · 32 ✗ · 0.3 ✗
The power n
Counts place shifts of the decimal point.
5 places right = 10⁵
Ordinary form
The 'normal' way to write it.
3.5 × 10³ = 3500
II · The decimal-point game
Standard form is how scientists write every size — from atoms to galaxies — on one scale. The power of 10 tells you the order of magnitude.
III · Conversions · the rules
Ordinary → standard
Move decimal until 1 ≤ a < 10
Count moves → that is n
Left moves → positive
Right moves → negative
42,000 → 4.2 × 10⁴
0.00056 → 5.6 × 10⁻⁴
Standard → ordinary
Move decimal n places
Positive n → right
Negative n → left
6.3 × 10⁵ → 630,000
2.5 × 10⁻³ → 0.0025
Multiplication
(a × 10ᵐ)(b × 10ⁿ) = ab × 10⁽ᵐ⁺ⁿ⁾
Multiply as and bs, add indices
(3×10⁵)(2×10⁻³) = 6 × 10²
Division
(a×10ᵐ)/(b×10ⁿ) = (a/b) × 10⁽ᵐ⁻ⁿ⁾
Divide as and bs, subtract indices
(6×10⁸)/(3×10⁵) = 2 × 10³
IV · Worked examples
Convert to standard form
2 marks
Write these in standard form: (a) 3,420,000 (b) 0.000047
(a)Move decimal 6 places left: 3.42 × 10⁶
(b)Move decimal 5 places right: 4.7 × 10⁻⁵
3.42 × 10⁶ · 4.7 × 10⁻⁵
Multiply in standard form
3 marks
Work out (6 × 10⁷) × (4 × 10⁻³), giving your answer in standard form.
i.Multiply the a parts: 6 × 4 = 24
ii.Add the indices: 10⁷ × 10⁻³ = 10⁴
iii.So far: 24 × 10⁴. But a must be < 10.
iv.Adjust: 24 = 2.4 × 10¹, so 2.4 × 10⁵
(6 × 10⁷) × (4 × 10⁻³) = 2.4 × 10⁵
Compare in standard form
2 marks
Which is bigger: 3.2 × 10⁵ or 4.1 × 10⁴?
i.Compare powers of 10 first: 10⁵ > 10⁴
ii.Larger power wins — regardless of a part
3.2 × 10⁵ is bigger (= 320,000 vs 41,000)
V · Common mistakes & examiner tips
Common mistakes
a value outside 1 ≤ a < 10. 24 × 10⁴ is not standard form — adjust to 2.4 × 10⁵.
Direction of decimal move. 42,000: move left → positive power. 0.00042: move right → negative.
Adding instead of multiplying the a's (or vice versa).
Comparing 3.2 × 10⁵ with 9.9 × 10⁴ — look at the power first, not the a.
Forgetting to convert back to standard form at the end.
Examiner tips
Use calculator's EXP or ×10ˣ button to enter standard form directly.
Always compare powers first — they dominate.
Count decimal moves carefully — common source of 1-place errors.
Write out the full workings with index laws — method marks are generous.
Final check: is 1 ≤ a < 10? If not, adjust.