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Squares, roots and powers

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2 Powers

2.1 The impact of a power

Here is a tale based on an ancient Eastern legend, which gives an idea of the impact of raising a number to a power.

Example 6

A long time ago there lived a very rich king whose son's life was saved by a poor old beggar woman. The king was naturally very grateful to the woman, so he offered to give her anything that she wanted. Much to the king's surprise the old lady just asked for two bags of rice. When asked why she specifically wanted the rice, she explained that she would like to have her son to stay and that if she had some rice she would be able to feed him. The king felt that this was far too small a reward so he asked if there were any other members of her family that she might like to have to stay. She replied that she would like to have her two daughters the following week, so four bags of rice then would be a great help. The king agreed to this and suggested that she might like to think of having more of her family to stay. He said he would be prepared to double the number of bags he gave her each week. At this point his accountant got very agitated and asked him to consider very carefully what he was offering! What was he worried about?

In order to decide if the king was being exceptionally generous, you need to look at exactly what he was proposing. In the beginning the numbers are quite small, 2 bags of rice in the first week, 4 bags in the second, then 8 in the third, 16 in the fourth, 32 in the fifth and so on.

Week 1: 2 bags of rice = 2

Week 2: 2 × 2 bags of rice = 4

Week 3: 2 × 2 × 2 bags of rice = 8

Week 4: 2 × 2 × 2 × 2 bags of rice = 16

Week 5: 2 × 2 × 2 × 2 × 2 bags of rice = 32

However, to get a feel for how the numbers are growing, look at them in another way.

It is not particularly convenient to write out all these 2s, particularly if you want to go on and look at the situation in week 52, say, i.e. a year later. However, there is a short and neater way of expressing these numbers.

You can write 2 × 2 = 22 = 4 and 2 × 2 × 2 = 23 = 8.

In a similar manner, in week 4 you can write 2 × 2 × 2 × 2 = 24 = 16, … and in week 10, when there are ten twos multiplied together, you can write 210. Sometimes you may find it written 2 ^ 10 (^ is the ‘power of’ sign).

In week 1, when the number of bags is just 2, you can write this as 21.

These numbers can be worked out on your calculator. Check each of 24, 25 and 210 using your calculator.

You should find that 210 = 1024 and so see what the accountant was worried about.