Scientific notation

Your approach to each question

1. Split every number into its two parts: the number in front, between 1 and 10, and the power of ten. Work the two parts separately.
2. For × and ÷, do the fronts first: multiply them, or divide them. Then the powers: add them for ×, subtract them for ÷.
3. Tidy the front: if it came out 10 or more, divide it by 10 and add one to the power; if it came out under 1, multiply it by 10 and take one off.
4. For the conversions, count how many places the decimal point moves, and which way: left for a big number means a positive power; right for a small one means a negative power.

Techniques and tricks

• Fronts multiply, powers add

  Multiply the numbers in front, add the powers, then move one power across if the front reached 10.

  e.g. (5 × 10³) × (9 × 10⁻²): 5 × 9 = 45 and 3 + (−2) = 1, so 45 × 10¹ = 4.5 × 10²

• Fronts divide, powers subtract

  Divide the numbers in front, subtract the second power from the first, then borrow a power if the front fell below 1.

  e.g. (3.168 × 10⁴) ÷ (8.8 × 10²): 3.168 ÷ 8.8 = 0.36 and 4 − 2 = 2, so 0.36 × 10² = 3.6 × 10¹

• Spot the carry before you start

  If the two fronts multiply to 10 or more — any pair like 3.6 and 8, or 5 and 2 — the power will go up by one. Decide that first, then work out the front.

  e.g. 3.6 × 8 = 28.8, so (3.6 × 10⁰) × (8 × 10¹) = 2.88 × 10², one power above the 10¹ the exponents give

• Minus signs in the powers

  Adding a negative power is taking away; subtracting a negative power is adding. Say it in words before you write it.

  e.g. 10⁵ × 10⁻² = 10³; 10⁴ ÷ 10⁻³ = 10⁷

• Count moves, not zeros

  To write a number in, count the places the point moves until one non-zero digit sits in front of it. For a small number, that count is one more than the zeros after the point.

  e.g. 45,200,000 = 4.52 × 10⁷ (seven moves left); 0.000452 = 4.52 × 10⁻⁴ (three zeros, four moves right)

• Writing out: pad as you move

  A positive power moves the point that many places right, filling with zeros. A negative power moves it left, filling with zeros in front.

  e.g. 3.07 × 10⁵ = 307,000; 3.07 × 10⁻³ = 0.00307

• The digit count check

  For a power of 0 or more, a × 10ⁿ written out has n + 1 digits before the point. Count the digits of an option and you know its power.

  e.g. 307,000 has 6 digits, so it is a × 10⁵

How to practise

• Use “Questions” to practise one kind at a time — “Divide” is the one most worth drilling alone, because of the borrow.
• Switch “Answer by” to “Typing it” once choosing is easy. For most questions you type n, the power, which isolates the part people get wrong; for “Write it out” you type the whole number, as long as its power is no more than 15 either way.
• Raise “Options shown” to 6 or 8: the options are mostly the same front at different powers, so the power decides, and every slip is on show.