Index Notation

Writing two times two times two times two times two takes a while, and it is easy to lose count of how many twos you wrote. Index notation is the shorthand for it. You write the number being multiplied once, then a small raised number saying how many of them there are. So five twos multiplied together is written two to the power five, and it works out to thirty-two. The big number at the bottom is called the base — that is the thing being multiplied. The small raised number is called the index, or the power, and its only job is to count the factors. The whole expression is called a power of two.

Now learn to say them. An index of two is read as 'squared', so five to the power two is 'five squared'. An index of three is read as 'cubed', so two to the power three is 'two cubed'. Every other index is read as 'to the power of' — ten to the power of four, for instance. Those two special names come from shapes, and you will meet those shapes in a moment. You also need to go backwards. If you meet seven squared, unfold it into seven times seven and you get forty-nine. If you meet ten to the power of four, that is ten times ten times ten times ten, which is ten thousand. Unfolding is always safe when you are unsure.

Why is an index of two called 'squared'? Because of the picture. Take five rows of five tiles and you have built a square, five along and five down, holding twenty-five tiles altogether. So five squared is twenty-five, and the word simply describes the shape you would build. The square numbers are the ones you get this way: one, four, nine, sixteen, twenty-five, thirty-six, forty-nine, sixty-four, eighty-one, one hundred, one hundred and twenty-one, and one hundred and forty-four. Learning that list by heart up to twelve squared is the single most useful thing in this chapter. You will meet these numbers constantly — in roots, in areas, and in factorising.

'Cubed' comes from a shape too. Stack three layers, each one a three-by-three square, and you have built a cube that is three along, three across and three high. It holds twenty-seven small blocks, so three cubed is twenty-seven. The cube numbers grow far faster than the squares: one, eight, twenty-seven, sixty-four, one hundred and twenty-five, two hundred and sixteen. Compare six squared, which is thirty-six, with six cubed, which is two hundred and sixteen. Powers of ten are worth noticing separately: ten squared is one hundred, ten cubed is one thousand, and ten to the power of four is ten thousand. For a power of ten the index simply counts the zeros that follow the one.

Four things trip people up, and all four are avoidable. One: the index counts the factors, so two to the power five is thirty-two, not ten and not seven. Two: the base and the index are not interchangeable — two to the power five is thirty-two, but five squared is twenty-five, and those are different numbers. Three: an index of one means a single factor, so three to the power one is just three. Four: in the order of operations, powers come before adding and multiplying, so three plus four squared is three plus sixteen, which is nineteen, not seven squared. Work out every power first and then do the rest, and all four traps disappear at once.