Three rules for the powers of ten: Multiply → add the exponents.
Divide → subtract the exponents.
Power → multiply the exponents.
These rules only work when the base is the same.
See the rules in action
- Multiply: 10² × 10⁵ = 102 + 5 = 10⁷
- Divide: 10⁸ ÷ 10³ = 108 − 3 = 10⁵
- Power: (10⁴)² = 104 × 2 = 10⁸
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
Deal with the two parts separately: Handle the coefficients (the a's) and the powers of ten separately, then re-normalise.
For example, (3 × 10⁴) × (2 × 10³) = 6 × 10⁷.
IB-style question — multiply
Work out (4 × 10⁵) × (2 × 10³), giving your answer in standard form.
Step by step
- Multiply the coefficients.
- Add the powers of ten.
- Put them together — 8 is already between 1 and 10.
Final answer
(4 × 10⁵) × (2 × 10³) = 8 × 10⁸
IB-style question — divide and re-normalise
Work out (3 × 10⁴) ÷ (6 × 10⁷), giving your answer in standard form.
Step by step
- Divide the coefficients.
- Subtract the powers of ten.
- 0.5 is less than 1 — re-normalise: 0.5 = 5 × 10⁻¹.
Final answer
(3 × 10⁴) ÷ (6 × 10⁷) = 5 × 10⁻⁴
Stop wasting time on topics you know
Our AI identifies your weak areas and focuses your study time where it matters. No more overstudying easy topics.
Cube or square each part: To raise a standard-form number to a power, raise the coefficient to that power and multiply the exponent.
For example, (2 × 10³)² = 4 × 10⁶.
This is a Paper 1 favourite: A common no-calculator question gives a length in standard form and asks for a volume in the form a × 10ᵏ.
You cube the value, then re-normalise.
IB-style question — cube and re-normalise
A cube has edge length 3 × 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.
- 27 is not between 1 and 10 — re-normalise: 27 = 2.7 × 10¹.
Final answer
V = 2.7 × 10⁷ cm³
Exam tip: Write the un-tidy line (27 × 10⁶) first, then the final answer.
IB awards a mark for showing the powers of ten combine, even before you re-normalise.