To multiply or divide numbers in standard form, treat the A parts and the powers of 10 separately. To add or subtract, make the powers match first. Then check that your answer is still in standard form.
What is the small gap?
Standard form calculations chain three small skills: arithmetic on the A parts, index rules on the powers, and re-writing the answer. Students usually have each one, but skip the last step or use the multiply rule on an addition.
How do you calculate, step by step?
Multiply or divide.
- Multiply or divide the A values.
- Add the powers (multiply) or subtract them (divide).
- If A is now 10 or more, or less than 1, adjust A and n.
Add or subtract.
- Write both numbers with the same power of 10.
- Add or subtract the A values, keep the power.
- Adjust if needed.
A scaffolded example
An original problem: a cell has mass 4 × 10⁻¹² g. A sample contains 5 × 10⁸ cells. Find the total mass in standard form.
Multiply the A parts: 4 × 5 = 20. Add the powers: −12 + 8 = −4. So the product is 20 × 10⁻⁴ g.
20 is not between 1 and 10. Rewrite 20 as 2 × 10¹, so the answer is 2 × 10¹ × 10⁻⁴ = 2 × 10⁻³ g.
Check: 10⁻¹² × 10⁸ is 10⁻⁴, about 0.0001, and 4 × 5 = 20 of those gives about 0.002. It matches 2 × 10⁻³.
What does the common mistake look like?
A student adds 3 × 10⁵ + 4 × 10⁴ and gets 7 × 10⁹, adding both the A values and the powers. The size check fails at once: the sum is about 300 000 + 40 000, which is nowhere near 7 billion.
The correct method: write 4 × 10⁴ as 0.4 × 10⁵. Then 3 + 0.4 = 3.4, so the sum is 3.4 × 10⁵.
Self-check
Work out (6 × 10⁷) ÷ (1.5 × 10³), then 2.5 × 10⁴ + 8 × 10³.
Answer
Division: 6 ÷ 1.5 = 4, and 7 − 3 = 4, so the answer is 4 × 10⁴. Addition: write 8 × 10³ as 0.8 × 10⁴.
Then 2.5 + 0.8 = 3.3, so 3.3 × 10⁴. Size check: 25 000 + 8000 = 33 000 = 3.3 × 10⁴.
Where does this show up in SPM?
In SPM Mathematics, it appears in standard form questions and in estimation. In Physics, quantities such as charge and wavelength are multiplied in this form, and the final power of 10 is part of the answer.
If the form itself is unsteady, review writing standard form, and revise the rules in applying basic index rules. The hub has the full sequence.
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