Powers and logarithms provide a powerful way of representing large and small quantities, and performing complex calculations. Understanding powers will allow you to make better 'back-of-the-envelope' calculations or to quality-check results from your calculator.
You can also download the complete web document as a printable PDF file by clicking here.
First, look at the number sequence:
10
100
1 000
10 000
100 000
1 000 000
Each of these numbers is the previous number multiplied by 10. This list can be re-written as:
| 10 | = | 10 | 1 |
| 100 | = | 10 × 10 | 2 |
| 1 000 | = | 10 × 10 × 10 | 3 |
| 10 000 | = | 10 × 10 × 10 × 10 | 4 |
| 100 000 | = | 10 × 10 × 10 × 10 × 10 | 5 |
| 1 000 000 | = | 10 × 10 × 10 × 10 × 10 × 10 | 6 |
The numbers in the right-hand column are the number of multiples of 10 for each number. This is the power of 10 and can be added as a superscript to 10 to represent each number:
| 10 | = | 10 | 101 |
| 100 | = | 10 × 10 | 102 |
| 1 000 | = | 10 × 10 × 10 | 103 |
| 10 000 | = | 10 × 10 × 10 × 10 | 104 |
| 100 000 | = | 10 × 10 × 10 × 10 × 10 | 105 |
| 1 000 000 | = | 10 × 10 × 10 × 10 × 10 × 10 | 106 |
HINT: Count the zeros after the 'one' in the left-hand column - this equals the power
This pattern continues below 10:
| 1 | = | 10 ÷ 10 | 100 |
| 0.1 | = | 10 ÷ 10 ÷ 10 | 10-1 |
| 0.01 | = | 10 ÷ 10 ÷ 10 ÷ 10 | 10-2 |
| 0.001 | = | 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 | 10-3 |
| 0.0001 | = | 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 | 10-4 |
| 0.00001 | = | 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 | 10-5 |
HINT: Count the number of zeros after the decimal place and before the first 'one' in the left-hand column, then add one for the decimal point - this equals the 'minus' power
Note that minus power indicates a value that is the reciprocal of the positive power, thus:
10-3 = 0.001 = 1 ÷ 1000 = 1 ÷ 103