The laws of exponents, and where each one comes from

0:00A king's diamonds, counted last time. Seven levels, three ways at every level, so three to the seventh — two thousand one hundred and eighty-seven. Now forget the tree, and look at what that power actually is. It is a row. Seven 3s, side by side, waiting to be multiplied. Three, three, three, three, three, three, three. Nothing else. And here is the whole of today in one sentence. You are allowed to put brackets anywhere in that row.

0:34Multiplication does not care how you group things, or what order you take them in. Every rule we are about to meet is just a place to put a bracket. So put one in. Bracket the first four 3s, and the last three. The first bracket is three to the fourth, which is eighty-one. The second is three to the third, which is twenty-seven. And eighty-one times twenty-seven is two thousand one hundred and eighty-seven.

1:06Which is what we already had — because we did not add or remove a single three. Now look at the counts. Four, and three. Seven. The bracket cannot change the answer, because it did not change the row, and all it did to the exponents was split the seven in two. So put the bracket somewhere else. Two 3s, then five. Nine times two hundred and forty-three. Two thousand one hundred and eighty-seven, again.

1:39There are six places to cut a row of seven, and here they all are. One and six: three times seven hundred and twenty-nine. Two and five: nine times two hundred and forty-three. Three and four: twenty-seven times eighty-one. And then the same three pairs, read the other way round. Six different-looking multiplications, one answer every time, because there was only ever one row. Letters make it impossible to miss.

2:12Write out p to the fourth: p, p, p, p. Now p to the sixth: p, p, p, p, p, p. Push them together and count what you have got. Ten p's. p to the tenth. You did not need to know what p is, and you did not do any arithmetic at all. And notice what you did not get. You did not get p to the twenty-fourth. Four next to six is ten, not twenty-four.

2:42So here is the first law. n to the a, times n to the b, is n to the a plus b. And the reason is not a rule you have to take on trust. It is that a row of a n's, followed by a row of b n's, is a row of a plus b n's. That is the entire argument. The commonest mistake in this whole topic is to multiply those two counts instead of adding them.

3:10Three to the seventh is two thousand one hundred and eighty-seven. Three to the twelfth is five hundred and thirty-one thousand, four hundred and forty-one. Expanding the row settles that in about four seconds. Now cut the row a different way. Take six 4s. Instead of one cut, chop the row into equal blocks. Blocks of three: that gives two blocks, each worth sixty-four, and sixty-four times sixty-four is four thousand and ninety-six.

3:46Blocks of two: three blocks, each worth sixteen, and sixteen times sixteen times sixteen is four thousand and ninety-six as well. The same six 4s. The same answer. All we changed was where the walls went. Blocks of blocks get a notation of their own. Two blocks of three 4s is four cubed, squared. Three blocks of two 4s is four squared, cubed. And here is the thing to be careful about.

4:20Four cubed, squared, does not mean four cubed times two. That would be a hundred and twenty-eight. It means four cubed, multiplied by itself. Write the six 4s out once and that confusion cannot survive. As for the counts: two blocks of three is six factors, three blocks of two is six factors. Blocks, times block size. The counts multiply. Which is why the two counts can trade places. Take ten 2s.

4:54Cut them into five pairs, and you have five copies of two squared. Cut the same ten into two blocks of five, and you have two copies of two to the fifth. Both come to one thousand and twenty-four, and of course they do — the row never moved. So n to the a, all raised to the b, equals n to the b, all raised to the a, and both equal n to the a times b.

5:21It also tells you every way of rewriting a power so that its base is a power. For an exponent of six: one and six, two and three, three and two, six and one. The ways to split the exponent are the ways to factorise it. Now a pond. It holds one lotus, the flowers double every day, and on day thirty the pond is completely covered. On which day was it half covered?

5:51Not day fifteen. Day twenty-nine — because whatever was there on day twenty-nine doubled overnight, and doubling the half gives you the whole. On day fifteen the pond is one part in thirty-two thousand, seven hundred and sixty-eight of the way there. Now put a second pond beside it, one that triples every day. Start a single flower in the doubling pond and wait four days: two to the fourth, sixteen flowers.

6:23Carry all sixteen across to the tripling pond, wait four more days, and the count gets multiplied by four 3s. Sixteen times eighty-one. One thousand two hundred and ninety-six. Now do it the other way round: tripling pond first, which is eighty-one, then four days of doubling. Eighty-one times sixteen. The same one thousand two hundred and ninety-six. Of course it is the same. But look at what the second version lets you see.

6:58You have a row of four 3s, and a row of four 2s. Pair them off — first with first, second with second, and so on. Each pair is a three times a two. A six. Four pairs, so four 6s. Six to the fourth. And that is why two to the fourth times three to the fourth is six to the fourth, and not six to the eighth. There were four pairings, not eight.

7:26Six to the eighth is one million, six hundred and seventy-nine thousand, six hundred and sixteen — nowhere near. So the third law. m to the a, times n to the a, is m times n, all to the a — when the two counts match. Two to the fifth times five to the fifth is ten to the fifth, one hundred thousand, which is the lock we counted last time.

7:54And it runs backwards for division: ten to the fourth over five to the fourth is two to the fourth. Sixteen, exactly, with nothing left over. But watch the condition. Two cubed times three to the fourth is six hundred and forty-eight, and nothing at all combines. The bases are different and the counts are different, so there is no pairing to make and no row to join. Six hundred and forty-eight is not a power of six, or of anything else.

8:28Step back and count what we actually proved. Nothing. Not one of those three laws is a new fact about numbers. Each one is a sentence about where you may put a bracket in a row of identical factors, and you were always allowed to put it anywhere. That is why they cannot fail, and why you do not have to memorise them. Expand the row, and the rule falls out in seconds.

8:55One warning, though. Every count in this video was a counting number — one, two, three. A row can have four factors in it. It cannot have none of them, or minus two of them, or half of one. What those would mean is a real question, and it is not answered yet.