Key Idea: Standard form is the IB's way of writing very large and very small numbers compactly. It usually appears as the last line of a question — give your answer in the form a × 10ᵏ — on both papers.
🔢 The structure: a × 10ᵏ
- the coefficient — one non-zero digit before the decimal point
- the exponent — a positive or negative whole number
Big number (≥ 10) → exponent positive, point moves left. e.g. 52 000 = 5.2 × 10⁴. Small number (< 1) → exponent negative, point moves right. e.g. 0.0007 = 7 × 10⁻⁴. Count place moves, not digits — and check 1 ≤ a < 10 every time.
🖐️ Calculating by hand (Paper 1)
✏️ IB-style worked examples
IB-style question — write a small number in standard form
Write 0.000 805 in standard form.
Step by step:
Put the point just after the first non-zero digit (8).
The point moved 4 places right, so the exponent is negative.
Write them together.
0.000 805 = 8.05 × 10⁻⁴
IB-style question — compute, then express (Paper 2)
A sphere has radius 9.4 cm. Find its volume in the form a × 10ᵏ cm³, where 1 ≤ a < 10.
Step by step:
Use the volume formula and your GDC to work out the value.
The GDC does the arithmetic.
Rewrite in standard form — move the point 3 places.
V = 3.48 × 10³ cm³ (3 s.f.)
IB-style question — cube and re-normalise (Paper 1)
A cube has edge length 4 × 10² cm. Find its volume in the form a × 10ᵏ cm³, where 1 ≤ a < 10, without a calculator.
Step by step:
Volume of a cube = edge³. Cube the coefficient and multiply the exponent.
Work each part out.
64 is not between 1 and 10 — re-normalise: 64 = 6.4 × 10¹.
V = 6.4 × 10⁷ cm³
Important: The final answer must have exactly one non-zero digit before the decimal point (1 ≤ a < 10). After a calculation, re-normalise: 27 × 10⁶ → 2.7 × 10⁷, 0.5 × 10⁻³ → 5 × 10⁻⁴. With powers, raise the coefficient too: (4 × 10²)³ = 64 × 10⁶, not 4 × 10⁶.
Tap each card to reveal the answer.
Exam Tips
- Check the coefficient a is between 1 and 10 — every time, including after a calculation.
- Big number → positive exponent; small number → negative exponent. Count place moves, not digits.
- Multiply → add powers. Divide → subtract. Power → multiply (and raise the coefficient too).
- Re-normalise an untidy answer: 27 × 10⁶ → 2.7 × 10⁷.
- On Paper 2, the GDC's ᴇ means × 10 — rewrite it; never leave ᴇ in your final answer.