Multiplication and division use index laws.
(a×10m)×(b×10n)=(a×b)×10m+n
(a×10m)÷(b×10n)=(a/b)×10m−n
Example. (3×106)×(4×10−2).
- Coefficients: 3×4=12.
- Powers: 106×10−2=104.
- Result: 12×104.
- ADJUST: 12 is too big. Divide by 10, multiply n by another 10: 1.2×105.
Example. (8×106)÷(2×104).
- Coefficients: 8/2=4.
- Powers: 106/104=102.
- Result: 4×102.
Addition and subtraction.
You CANNOT just add/subtract coefficients if the powers differ. Convert to the SAME power first.
Example. (4.5×106)+(2.7×105).
Convert smaller to larger power: 2.7×105=0.27×106.
Now: (4.5+0.27)×106=4.77×106.
Adjust at the end.
ALWAYS check: is the coefficient in [1,10)?
- Too big (≥10): divide by 10, increase n. 12×104→1.2×105.
- Too small (<1): multiply by 10, decrease n. 0.5×104→5×103.
Worked qualitative. Why use standard form if calculator does the work?
- Numbers like 6,023,000,000,000,000,000,000,000 (Avogadro's number) are unwieldy.
- Standard form: 6.023×1023. Compact, readable, comparable.
- Used universally in physics, chemistry, biology, engineering.
Edexcel tip. Always do the adjustment step. Mark schemes deduct for answers like 12×104 even if the value is correct.