Place value and rounding
Multiplying and dividing by powers of 10
Multiplying by 10³ does not change any digit — it only moves the decimal point. Count the zeros and you know how far.
Powers of 10
10² = 100, 10³ = 1000, 10⁴ = 10 000, 10⁵ = 100 000. The INDEX tells you how many zeros: 10⁵ has five zeros.
Multiplying
To multiply by a power of 10, move the decimal point to the RIGHT by the number of zeros. Fill any empty spaces with zeros. 4.56 × 10⁵ There are five zeros, so move the point five places right: 4 5 6 _ _ _ Three spaces are empty, so fill them with zeros. 4.56 × 10⁵ = 456 000
Dividing
To divide by a power of 10, move the decimal point to the LEFT by the number of zeros. 2 350 000 ÷ 10⁵ Five zeros, so move the point five places left. It stops between the 3 and the 5. 2 350 000 ÷ 10⁵ = 23.5
Why 'just cross off the zeros' is risky
Some people say: to divide a whole number by 10⁴, cross four zeros off the end. That works when the number HAS enough zeros — 80 000 ÷ 10⁴ = 8. But it fails as soon as it does not. 2 350 000 ÷ 10⁵ has only four zeros to cross off and needs five. Moving the decimal point always works; crossing off zeros only sometimes does.
Worked examples
4.56 × 10⁵
- 10⁵ has five zeros, so move the point five places right.
- Fill the empty places with zeros.
Answer: 456 000
2 350 000 ÷ 10⁵
- Five zeros, so move the point five places left.
- It stops between the 3 and the 5.
Answer: 23.5
How many zeros in 10⁴?
- The index tells you the number of zeros.
Answer: Four — 10 000
80 000 ÷ 10⁴
- Move the point four places left.
- Here you could also cross off four zeros.
Answer: 8
Why crossing off zeros is risky
- 2 350 000 ÷ 10⁵ needs five zeros crossed off.
- Only four are there to cross.
Answer: Moving the decimal point always works
0.7 × 10³
- Three zeros, so move the point three places right.
- Fill the gaps with zeros.
Answer: 700
Where you see this in real life
- Converting 4.56 km to metres is × 10³ = 4560 m — the digits never change, only where the point sits.
- A population of 2 350 000 written in hundred-thousands is 23.5 — the same digits, read at a different scale.
Key words
- Power of 10
- 10 multiplied by itself a number of times, such as 10³ = 1000
- Index
- the small raised number showing the power
- Place value
- what a digit is worth because of its position
- Decimal point
- the dot separating whole numbers from parts
- Multiply
- make larger by a given factor
- Divide
- share into equal parts