Cube roots, and what successive differences expose

0:00Two cubed is eight. Now run it the other way. You are handed eight, and the question is what got cubed. That number is called the cube root of eight, and it is two. There is a symbol for it: the root sign, with a small three sitting in the notch. The three is doing real work there. And once you write it that way, the cube root of any number cubed is just the number you started with.

0:29Here is the first place cubes behave better. Six squared is thirty-six. But minus six squared is also thirty-six. Two different numbers, one answer. Squaring pressed them together, and that is why the square root sign needs a rule about which of the two it means. Now cube them. Six cubed is two hundred and sixteen. Minus six cubed is minus two hundred and sixteen. Different answers, because cubing keeps the sign.

1:02So nothing gets pressed together, and there is no choice to make. Cube every whole number from minus fifty to fifty and you get a hundred and one different answers. Square them and you get fifty-one. Squaring loses things. Cubing does not, and almost everything that follows is that one sentence again. So how do you find one. Break the number into primes. Three thousand three hundred and seventy-five is three, times three, times three, times five, times five, times five.

1:39Now deal those into three piles, and insist they come out identical. One three and one five into each. They do, so the number is a cube — and look what is in your hand. One pile is three times five, which is fifteen. The root is not something you go on to work out. It is the pile. Try it on five hundred: two, times two, times five, times five, times five.

2:08The fives deal. The twos do not — there are only two of them. So five hundred is not a cube, and the same test said so. Why should dealing into three piles work at all? Look at what cubing does to a factorisation. Twelve is two, times two, times three. Twelve cubed is one thousand seven hundred and twenty-eight, and that is six twos and three threes. The two twos became six.

2:39The one three became three. Every prime turned up three times over, because the cube is three copies of the number multiplied together. So a cube can never have a prime appearing a number of times that is not a multiple of three. Which is the test, and undoing it — taking a third of each pile — is the root. Now something you can use without factorising anything. Cube the ten digits and watch only the last digit of each answer.

3:12Two cubed is eight, so two ends in eight. Three cubed is twenty-seven, so three ends in seven. Then five, six, three, two, nine. Read that list of endings: zero, one, eight, seven, four, five, six, three, two, nine. That is all ten digits, each one used exactly once. Ten went in and ten different digits came out. And this is not a spot check that might fail later — there are only ten digits, so that list is the whole of it.

3:46There is a tidier way to hold it. Two goes to eight. And eight goes back to two. Three goes to seven, and seven goes back to three. Every other digit does not move at all: zero, one, four, five, six and nine each cube to an ending of themselves. So the whole map is one swap, another swap, and six things standing still. Which means it undoes itself. Apply it twice and you are back where you started.

4:18That is worth more than a memory trick, because a map that undoes itself cannot send two digits to the same place. If it did, you could not get back — which one would you return to? Squares cannot do this. Square the ten digits and take the endings: zero, one, four, nine, six, five, six, nine, four, one. Only six different endings, out of ten possible. And four of those six get used twice.

4:50So suppose a square ends in six. Its root ends in four, or in six, and nothing in front of you says which. The arrows collide. For cubes, no two arrows ever land in the same place, so a cube's last digit names its root's last digit outright. So here is four thousand nine hundred and thirteen, and you are told it is a cube. Name the root, without factorising.

5:19It ends in three, and only seven cubes to an ending of three, so the root ends in seven. Now cover the last three digits, and four is left. Four sits between one cubed and two cubed — between one and eight. So the tens digit is one. Seventeen. And notice what just happened, because it was not a guess. Ending in seven, on its own, leaves ten possible roots below a hundred.

5:49A tens digit of one, on its own, leaves ten. Put them together and exactly one number survives. You did not guess it; there was nothing else it could be. Once more, quickly. Twelve thousand one hundred and sixty-seven. It ends in seven, so the root ends in three. Cover the last three digits: twelve is left, and it sits between eight and twenty-seven. So the tens digit is two, and the root is twenty-three.

6:22Thirty-two thousand seven hundred and sixty-eight. Ends in eight, so the root ends in two. Thirty-two is left, and it sits between twenty-seven and sixty-four. Tens digit three, root thirty-two. And one thousand three hundred and thirty-one gives eleven the same way. This is not a trick for four lucky numbers. It reads off every cube up to ninety-nine cubed, which is nine hundred and seventy thousand, two hundred and ninety-nine.

6:56Now change the subject completely. Write the squares in a row: one, four, nine, sixteen, twenty-five, thirty-six. Take the gap between each pair. Three, five, seven, nine, eleven — the odd numbers, which you may have met before. Not constant, so do it again to that new row. Two, two, two, two. Flat. It took two rounds to go flat, and the level it flattened at is worth remembering. Run exactly the same machine on the cubes.

7:31One, eight, twenty-seven, sixty-four, a hundred and twenty-five, two hundred and sixteen. The gaps are seven, nineteen, thirty-seven, sixty-one, ninety-one. Nowhere near flat. Difference those. Twelve, eighteen, twenty-four, thirty. Still not flat — but look at it, it is climbing by a fixed step now. So one more round. Six, six, six. The cubes went flat at level three, one level further down than the squares. And the flat value is six.

8:07Almost everybody predicts three, because the power is three. It is six. Both numbers have a reason. Each round of differencing knocks the power down by one, and hands you a factor of the power on the way past. Start with cubes. One round leaves something built on squares, carrying a three. The next leaves something built on the plain counting numbers, carrying a two as well. The next has nothing left to vary, so it is flat — at three times two times one.

8:43Six. The squares did the same over two rounds and came out at two times one, which is why their flat row was twos. So the level counts the power, and the flat value multiplies everything down from it. Test that on the fourth powers: one, sixteen, eighty-one, two hundred and fifty-six. Four levels, and the flat value is twenty-four. Four times three times two times one. One last thing.

9:15Which is bigger: sixty-seven squared minus sixty-six squared, or sixty-seven cubed minus sixty-six cubed? The square one is a gap between neighbouring squares, so it is one of those odd numbers, and it is simply sixty-six plus sixty-seven. A hundred and thirty-three. The cube one is an entry in that first row under the cubes — the row that still had two levels to fall. It is three times sixty-six times sixty-seven, plus one.

9:50Thirteen thousand two hundred and sixty-seven. The square difference is a sum. The cube difference is a product, tripled. Ninety-nine times bigger — not remotely comparable. And there is the whole topic. Cubing folds nothing — not the sign, not the last digit — so nothing is lost and a root can be read straight off. And a cube holds three copies of its root, so the primes deal into three piles and the differences need three levels.

10:24The same three, twice over.