Multiplying and Dividing by Powers of 10
Theory
Multiplying by \(10\), \(100\), \(1000\), \(\ldots\) moves the decimal point to the right. Dividing moves it to the left. The point moves one place for each zero. This Year 7 maths page shows how to use powers of \(10\) in index form, fill places with zeros, change units and find a missing power.
In \(10^3\) the base is \(10\) and the index (or power) is \(3\). In expanded form \(10^3 = 10 \times 10 \times 10 = 1000\). The index tells you how many zeros.
Multiplying by \(10\), \(100\), \(1000\), \(\ldots\) makes a number bigger: the point moves one place right for each zero. Fill empty places with zeros: \(4.2 \times 1000 = 4200\).
Dividing by \(10\), \(100\), \(1000\), \(\ldots\) makes a number smaller: the point moves one place left for each zero. Write zeros in front when you run out of digits: \(3.6 \div 100 = 0.036\).
\(0.1\) is one tenth, so multiplying by \(0.1\) is the same as dividing by \(10\), and multiplying by \(0.01\) is the same as dividing by \(100\).
One place for each zero:
| Multiply by | Point moves | Divide by | Point moves |
|---|---|---|---|
| \(10 = 10^1\) | 1 place right | \(10\) | 1 place left |
| \(100 = 10^2\) | 2 places right | \(100\) | 2 places left |
| \(1000 = 10^3\) | 3 places right | \(1000\) | 3 places left |
How to multiply or divide by a power of 10
- Count the zeros (or read the index): that is the number of places.
- Multiply: move the point right. Divide: move the point left.
- Fill any empty places with zeros.
- Check the size: multiplying by \(10\), \(100\), \(\ldots\) makes it bigger; dividing makes it smaller.
(i) \(11.22 \times 100\)
(ii) \(35.3 \times 1000\)
(iii) \(6.8 \times 10\)
| \(11.22 \times 100\) | \(=\) | \(1122\) |
| \(35.3 \times 1000\) | \(=\) | \(35\,300\) |
| \(6.8 \times 10\) | \(=\) | \(68\) |
(i) \(2\) places right
(ii) \(3\) places right: add two zeros to the end
(iii) \(1\) place right
(i) \(169.1 \div 100\)
(ii) \(5.7 \div 1000\)
(iii) \(93.1 \div 10\)
| \(169.1 \div 100\) | \(=\) | \(1.691\) |
| \(5.7 \div 1000\) | \(=\) | \(0.0057\) |
| \(93.1 \div 10\) | \(=\) | \(9.31\) |
(i) \(2\) places left
(ii) \(3\) places left: write zeros in front
(iii) \(1\) place left
(ii) Calculate \(0.0617 \times 10^3\).
(iii) Calculate \(52 \div 10^3\).
(ii) \(10^3 = 1000\): \(0.0617 \rightarrow 0.617 \rightarrow 6.17 \rightarrow 61.7\)
(iii) \(52 \rightarrow 5.2 \rightarrow 0.52 \rightarrow 0.052\)
(i) \(10\,000\)
(ii) \(61.7\)
(iii) \(0.052\)
(ii) Ruby is \(154\) cm tall. What is her height in metres?
| \(1.25 \times 1000\) | \(=\) | \(1250\) |
| \(154 \div 100\) | \(=\) | \(1.54\) |
(i) \(1250\) mL
(ii) \(1.54\) m
(ii) \(9.3 \div \square = 0.0093\)
(i) \(0.806 \rightarrow 8.06 \rightarrow 80.6\): \(2\) places right.
(ii) \(9.3 \rightarrow 0.93 \rightarrow 0.093 \rightarrow 0.0093\): \(3\) places left.
(i) \(100\)
(ii) \(1000\)
| \(64 \times 0.1\) | \(=\) | \(64 \div 10 = 6.4\) |
| \(64 \div 100\) | \(=\) | \(0.64\) |
\(64 \times 0.1\) is larger.
Common pitfalls
Frequently asked questions
Which way does the point move when you multiply by 100?
Two places to the right. \(11.22 \times 100 = 1122\).
Which way does the point move when you divide by 1000?
Three places to the left. \(5.7 \div 1000 = 0.0057\).
What does 10 to the power 3 mean?
\(10 \times 10 \times 10 = 1000\). The index \(3\) is the number of zeros.
Is multiplying by 0.1 the same as dividing by 10?
Yes. \(0.1\) is one tenth, so \(64 \times 0.1 = 6.4\).
How do you change litres to millilitres?
Multiply by \(1000\). \(1.25\) L \(= 1250\) mL.