Exponential notation: repeated multiplication written once

0:00Here is how thick that folded sheet is after seven folds. Zero point zero zero one, times two, times two, times two, times two, times two, times two, times two. It is correct, and nobody wants to write it. Worse, nobody wants to read it. You have to count the twos to know what you are looking at, and if I had written six of them just now, you would not have noticed.

0:28So we write the count down instead. Two to the seventh. And here is the thing about that seven. It was already in the problem. It was sitting in the length of that line, doing nothing, where nobody could see it. You have met this twice already, without the general idea. Two of something multiplied together is that thing squared. Three of them is that thing cubed. And there is no reason at all to stop there.

0:59Four of them is n to the fourth. Seven of them is n to the seventh. And if you do not know how many there are, call the count a, and write n to the a. Now be careful about what that small number counts. It counts the factors — how many n's are standing in the line. It does not count how many times you multiplied. Multiplying happens one time fewer than that.

1:26Which is exactly why n to the first is n, and not n times n. Take five to the fourth. Written out, that is five times five times five times five. Six hundred and twenty-five. The five underneath has a name: it is the base, and it names what is being repeated. The four on the shoulder is the exponent, and it counts how often. Six hundred and twenty-five, written that way, is in exponential form.

1:59You can read it out loud in more than one way — five raised to the power four, five to the power four, or the fourth power of five. They all mean four fives, multiplied. And a number we have already met comes straight back. Ten twos multiplied is two to the tenth, which is one thousand and twenty-four. Here is a question worth getting wrong first. Call the starting thickness v.

2:29Which of these gives the thickness after ten folds? Ten v. Ten plus v. Two times ten times v. Two to the tenth. Two to the tenth, times v. Ten squared, times v. With our sheet, where v is one thousandth of a centimetre, those six come out as ten, eleven, twenty, one thousand and twenty-four, one thousand and twenty-four, and one hundred. Two of them land on the right answer, so our sheet cannot decide between them.

3:04So change the sheet. Make it three thousandths instead. Now they give thirty, thirteen, sixty, one thousand and twenty-four, three thousand and seventy-two, and three hundred. And only one is still right. Two to the tenth on its own is a bare number, in a place where the answer has to be a thickness. One confusion is behind half of those wrong answers. Three fours added come to twelve. Three fours multiplied come to sixty-four.

3:39The same two numbers, and the answers are not even close. n times a is not n to the a. And if you check every small pair, they agree in only two situations. When the exponent is one, which is hardly a case at all. And for two squared, where two twos added and two twos multiplied both give four. That one coincidence is probably where the confusion starts. It is a coincidence.

4:09There are no others. What if the base is negative? Take minus four, three factors of it. Minus four times minus four is sixteen, and times minus four again is minus sixty-four. So a negative base can certainly give a negative answer. But not always — and what decides is not the base. Minus one to the fifth is minus one. Minus one to the fifty-sixth is plus one. It is whether the exponent is odd or even that settles the sign, every time, for any negative base you like.

4:49And the brackets are doing real work here. Bracket minus two, to the fourth, is sixteen. Minus, two to the fourth, is minus sixteen, because that minus sign never joined in the repeating. Zero is the easy one: any number of zeros multiplied is zero. The notation really earns its keep on prime factors. Take thirty-two thousand four hundred, and divide the primes out of it. Halve it four times.

5:22Then take out the threes, four of them. Then the fives, two of them, and you land on one. Ten prime factors. Written flat, that is a long line to read. Counted, it is two to the fourth, times three to the fourth, times five squared. And the order you pull them out in makes no difference to the answer. Take the fives before the threes and you walk down through different numbers — four thousand and fifty, two thousand and twenty-five, four hundred and five, eighty-one.

5:56A different route. The same ten primes spent, and the same three counts at the end. Letters take the notation unchanged. Three a's and two b's, all multiplied together, is written a cubed b squared. That is five factors altogether: three of one thing and two of another. The two counts stay separate, because the two bases are different things. Put numbers in and you can watch it not be anything else.

6:28With a as two and b as five, a cubed b squared is two hundred. Swap the counts over — a squared, b to the fourth — and the same two letters give two thousand five hundred. Same letters, different counts, and the counts were the whole content. Now the second thing a power is, and it is not obviously the same thing at all. There was a king with three daughters.

6:55Each daughter had three baskets. Each basket held three keys. Each key opened three rooms. Each room had three tables, each table three necklaces, and each necklace three diamonds. How many diamonds? Do not multiply anything yet. Count the levels instead. Three daughters, then three baskets each, so nine baskets. Twenty-seven keys. Eighty-one rooms. Every level multiplies by three, and the thing you are really counting is the levels. Seven of them, so three to the seventh: two thousand one hundred and eighty-seven diamonds.

7:38That is worth stopping on, because nothing here was doubled. Nothing grew. We counted the ways a repeated choice can come out. Every diamond is one path down that tree, and no two diamonds share a path. Three choices, taken seven times over. And notice the trap sitting in it. Four levels of three-way branching is not three times four. Three times four is twelve. Three to the fourth is eighty-one.

8:10When someone hands you a branching story, the number of levels is an exponent, not a multiplier. The same counting runs a lock. A two-digit lock: ten choices for the first digit, ten for the second, so one hundred settings. Three digits: every one of those hundred takes ten third digits, so a thousand. Five digits is ten to the fifth — one hundred thousand — which is just every string from all zeros to all nines, and nothing else.

8:46Now make the slots letters. Six slots of twenty-six letters is twenty-six to the sixth: three hundred and eight million, nine hundred and fifteen thousand, seven hundred and seventy-six. Over three thousand times as many. Though more slots is not automatically the better move — five letter slots gives eleven million, and six digit slots gives one million. And one caution. Seven dresses, two hats and three pairs of shoes is forty-two ways to dress, and that is not a power of anything.

9:25A power is the special case where every choice offers the same number of options. So a power is two things, and they turn out to be one thing. It is what you get when you multiply a fixed number by itself a fixed number of times. And it is the count of ways a repeated choice can come out. The folded sheet and the king's diamonds are the same arithmetic, seen from opposite ends.

9:54What makes both of them work is the split. The base names what is repeated. The exponent counts how often. And once that count is a symbol sitting where you can see it, every question about repeated multiplication turns into a question about a number you can point at. That is the whole of it. Every rule that comes later is a rule about that count.