🔭 Powers: Numbers That Grow by Multiplying

Let’s fold a piece of paper in half. It is now 2 layers thick. Fold it again: 4 layers. Then 8, then 16. Every fold doubles the thickness. There is a short way to write this fast growth: a power.

Instead of 2 × 2 × 2 × 2 × 2 we write 25. The bottom number is the base: the number we multiply. The small raised number is the index, or power: how many times the base is multiplied. 25 = 2 × 2 × 2 × 2 × 2 = 32.

Careful, a common mistake! 25 is not 2 × 5 = 10. The index tells you how many 2s to multiply: 2 × 2 × 2 × 2 × 2 = 32.

Powers of 10 are easy. 101 = 10, 102 = 100, 103 = 1,000. Whatever the index is, that is how many zeros come after the 1. 106 is one million: 6 zeros.

Let’s take a journey through distances. A room is about 101 metres across, a neighbourhood is 103 metres, and the diameter of the Earth is about 107 metres. From the Earth to the Sun is about 1011 metres! As the index grows, the number grows very fast.

Now let’s shrink! A grain of sand is about 10−3 metres, that is 0.001 m. A bacterium is about 10−6 metres, that is 0.000001 m. A negative index means getting smaller: 10−n = 1 / 10n.

To write big numbers in a short way, we use standard form: a × 10n, where 1 ≤ a < 10. For 45,000, the decimal point hops 4 places to the left: 4.5. The number of hops becomes the index: 45,000 = 4.5 × 104.

The Earth is about 150,000,000 km from the Sun. When the decimal point hops 8 places to the left, we get 1.5. So the distance is 1.5 × 108 km. In standard form, a must be between 1 and 10: 1 ≤ 1.5 < 10.

Very small numbers are written in the same way. For 0.000003 m, the decimal point hops 6 places to the right: 3. For a small number, the index is negative: 3 × 10−6 m. A big number has a positive index, and a small number has a negative one.

Summary: a power shows how many times we multiply the same number. Powers of 10 have as many zeros as the index. A negative index means getting smaller. Standard form is written a × 10n, with 1 ≤ a < 10. Now try it in Power Telescope!