Multiplying and dividing in standard form
Big idea: For multiplication and division, work with the front numbers first and the powers of 10 second.
Then check whether the final answer still needs re-writing into valid standard form.
Multiplication
Worked example — multiplication
Evaluate (3.2 × 10⁶)(1.8 × 10³).
Step by step
- Multiply the front numbers.
- Add the exponents on the powers of 10.
- Combine the two parts.
Final answer
5.76 × 10⁹
Division
Worked example — division
Evaluate (3.0 × 10⁻²) ÷ (6.0 × 10⁻⁴).
Step by step
- Divide the front numbers.
- Subtract the exponents on the powers of 10.
- First write the combined answer.
- 0.5 isn't between 1 and 10, so rewrite into valid standard form: shift the decimal one place right and drop the exponent by 1.
Final answer
5 × 10¹
Exam traps
- Forgetting to rewrite 0.5 × 10² as 5 × 10¹
- Adding exponents when dividing
- Subtracting exponents in the wrong order
- Leaving the final answer without units or context
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Adding and subtracting in standard form
Addition and subtraction are different from multiplication and division.
You cannot combine the powers straight away unless the powers of 10 already match.
Core rule:
- Rewrite the numbers so they use the same power of 10.
- Then add or subtract the coefficients.
- Finally, re-normalise if the coefficient is no longer between 1 and 10.
| Question | What to do first | Answer |
|---|---|---|
| 1.8 × 10⁵ - 3.0 × 10⁴ | Rewrite 3.0 × 10⁴ as 0.3 × 10⁵ | 1.5 × 10⁵ |
| 6.3 × 10⁴ - 2.1 × 10⁴ | Powers already match | 4.2 × 10⁴ |
| 3.2 × 10⁶ + 4.5 × 10⁵ | Rewrite 4.5 × 10⁵ as 0.45 × 10⁶ | 3.65 × 10⁶ |
Common mistake
- 1.8 × 10⁵ - 3.0 × 10⁴ = -1.2 × 10⁹
- Subtracts coefficients and exponents separately
- Treats subtraction like multiplication
Correct method
- Match the powers first
- Then work only with the coefficients
- Keep the shared power of 10 until the end
Exam tips
- If the powers are not the same, rewrite one number first.
- Choose the common power that makes the arithmetic easiest.
- After subtracting, check whether the answer still needs re-normalising.
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Applications, calculator outputs, and full-mark working
IB questions usually hide standard form inside a context.
Do the calculation, then give the answer in the form asked.
If you see this, do this
- Population × amount per person -> multiply coefficients, add exponents.
- Mass or size comparison -> divide coefficients, subtract exponents.
- How many more / how many remain -> make powers match, then subtract.
- Calculator output not in valid standard form -> rewrite before the final line.
1-minute micro-practice:
- (2.4 × 10⁷)(3.0 × 10²)
- (8.0 × 10⁶) ÷ (4.0 × 10²)
- 6.3 × 10⁴ - 2.1 × 10⁴
Reveal answer 1
(2.4 × 10⁷)(3.0 × 10²) = (2.4 × 3.0) × 107+2 = 7.2 × 10⁹
Reveal answer 2
(8.0 × 10⁶) ÷ (4.0 × 10²) = (8.0 ÷ 4.0) × 106-2 = 2.0 × 10⁴
Reveal answer 3
6.3 × 10⁴ - 2.1 × 10⁴ = (6.3 - 2.1) × 10⁴ = 4.2 × 10⁴