Key ideas
What it looks like
a × 10n
a is between 1 and 10 (it can be 1, but not 10).
n is a whole number: how many places the digits move.
Large numbers
The power is positive.
45 000 = 4.5 × 104
The digits moved 4 places.
Small numbers
The power is negative.
0.0032 = 3.2 × 10−3
The digits moved 3 places.
Back to normal form
Positive power: the number gets bigger.
Negative power: the number gets smaller.
Fill any gaps with zeros.
On a calculator: use the ×10x button (or EXP). For 3.2 × 105 type 3.2 ×10x 5.
Large numbers to standard form
Write 386 000 in standard form.(2 marks)
Make a number between 1 and 10.3.86 put the point after the first digit
In plain EnglishKeep the digits 3, 8, 6 and put the decimal point after the first one, so the number is between 1 and 10.
Count the places.386 000 = 3.86 × 100 000= 3.86 × 105✓1Check: 3.86 × 105 → move 5 places → 386 000 ✓✓2
In plain EnglishTo get from 3.86 back to 386 000, the digits move 5 places to the left (the number gets bigger). So the power is 5.
Small numbers to standard form
Write 0.000 47 in standard form.(2 marks)
Make a number between 1 and 10.4.7 first non-zero digit, then the point
In plain EnglishStart at the first digit that isn't zero. 4.7 is between 1 and 10.
Count the places.0.000 47 = 4.7 × 10−4✓1Check: 4.7 → 0.47 → 0.047 → 0.0047 → 0.000 47 (4 places) ✓✓2
In plain EnglishTo get from 4.7 to 0.000 47 the digits move 4 places to the right: the number gets smaller, so the power is negative: −4.
⚠ A common mistake: forgetting the minus sign. A small number (less than 1) always has a negative power.
Standard form to normal form
Write (a) 5.02 × 106 (b) 8.1 × 10−3 as normal numbers.(2 marks)
Positive power: bigger.5.02 × 106: move 6 places so it gets bigger= 5 020 000✓1
In plain EnglishMultiply by 10 six times. The digits move 6 places left, and we fill the gaps with zeros.
Negative power: smaller.8.1 × 10−3: move 3 places so it gets smaller= 0.0081✓2
In plain EnglishDivide by 10 three times. The 8 moves to the third decimal place.
Standard form on a calculator
Calculate (2.4 × 107) × (3.5 × 104). Give your answer in standard form.(2 marks)
Type it in.2.4 ×10x 7 × 3.5 ×10x 4 =✓1
In plain EnglishUse the ×10ˣ button, not × 10 ^. That keeps each number together.
Read the answer.Calculator shows 8.4 × 1011= 8.4 × 1011✓2
In plain EnglishIf your calculator shows an ordinary number, press the button that changes it to standard form, or convert it yourself.
Standard form in context
Light travels 3 × 108 metres in one second. How far does it travel in one minute? Give your answer in standard form.(2 marks)
Multiply by 60.3 × 108 × 60 = 18 000 000 000✓1
In plain EnglishOne minute is 60 seconds, so multiply the distance by 60.
Back into standard form.= 1.8 × 1010 metres✓2
In plain English18 000 000 000 has the point after the 1 in 1.8, then 10 places.
0 of 0 complete
Your turn
For standard form, type the number between 1 and 10 in the first box and the power of 10 in the second (use a minus sign for small numbers). Calculator allowed for the calculator and context sections. Stuck? Tap Need a hint?