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
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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
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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.