## 1. Introduction

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.

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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 = 10 100 0.1 = 10 ÷ 10 10-1 0.01 = 10 ÷ 10 ÷ 10 10-2 0.001 = 10 ÷ 10 ÷ 10 ÷ 10 10-3 0.0001 = 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 10-4 1e-05 = 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:

Note that minus power indicates a value that is the reciprocal of the positive power, thus:

10-3 = 0.001 = 1 ÷ 1000 = 1 ÷ 103