Square roots, cube roots and higher roots

A pallet of tiles lands on a driveway. A hundred and forty-four of them. They have to go down as one perfect square: the same number along the top edge as down the side. The instinct is to divide. But divide a hundred and forty-four by what? The number along the edge is the thing nobody knows yet. Division needs that answer before it can even start. So ask a different question. What number, multiplied by itself, gives a hundred and forty-four? Twelve. Twelve times twelve. So twelve tiles run along each edge. That question has a name. It is the square root.

So, what a root actually is. One idea, and the rest of this video is that idea with different numbers on it.

A power runs forwards. In the video on square numbers and cube numbers, four times four gives sixteen. A root runs backwards. It starts at sixteen and asks what was multiplied to make it. Four. So the square root of sixteen is four.

Now the trap. Dividing a hundred and forty-four by twelve does give twelve. That is not luck, it is what a square is. But that division only runs if twelve is already in your hand. Dividing can check a root. It can never find one.

Cube roots work the same way, with one more copy. Five times five times five is a hundred and twenty-five. Go backwards, and the number multiplied three times was five. So the cube root of a hundred and twenty-five is five.

Here is the phrase to carry into the exam hall. The little number counts the copies. A square root wants two copies. A cube root wants three. A fourth root wants four.

Your turn, and it is one step. Which of these three is the square root of sixty-four? Eight, thirty-two, or four.

Have a think. I'll wait.

The answer is eight. Eight times eight is sixty-four. Thirty-two is sixty-four halved, and thirty-two times thirty-two is over a thousand, so dividing has missed again. Four is worth a look though. Four times four times four is sixty-four, so four is the cube root of sixty-four. Same number, two different roots.

Now, roots past the square and the cube. Fourth roots. Fifth roots. Same job, bigger count.

Work out the fourth root of eighty-one. Four copies of the same number, multiplied together, have to give eighty-one. Try three. Three times three is nine. Nine times three is twenty-seven. Twenty-seven times three is eighty-one. Four threes. So the fourth root of eighty-one is three.

Run the count on another one. The cube root of a thousand. Three copies. Ten times ten is a hundred, and a hundred times ten is a thousand. So the cube root of a thousand is ten.

There is a shortcut going round for fourth roots: take the square root twice. It does work here, and only because four is two twos. A fifth root will not split like that. Counting copies works every time.

Examiners report this one directly. Their words, from a Foundation paper report:

Part (b) was not well answered, with most students not able to process the fourth root of a number.

Not able to process it. Not able to start. A fourth root is not a new operation to learn. It is the same question with a four in front of it.

Now, where a root shows up at the end of a longer calculation. This is the one that gets abandoned.

A ladder leans against a wall. The foot of the ladder is seven metres from the wall. The ladder itself is twenty-five metres long. How high up the wall does the ladder reach?

Getting from those two lengths to the height is right-angled triangle work, and the full method lives in the video on Pythagoras' theorem. Borrow it for a moment. Square both known lengths, and here, subtract.

Twenty-five squared is six hundred and twenty-five. Seven squared is forty-nine. Six hundred and twenty-five take away forty-nine leaves five hundred and seventy-six.

Stop there. Five hundred and seventy-six. Is that the height of the wall, or is a step still missing?

Take a moment. I'll wait.

A step is still missing. Five hundred and seventy-six is not a length. It is a length squared. The whole calculation was done in squares, so the answer comes back out through a square root. Twenty-four times twenty-four is five hundred and seventy-six. The ladder reaches twenty-four metres up the wall.

Finding that twenty-four by hand is worth a look. Twenty squared is four hundred, too small. Thirty squared is nine hundred, too big, so the answer is in the twenties. Five hundred and seventy-six ends in a six, and only a four or a six squared ends in six. Twenty-four fits.

Examiners watch that last step go missing. Their words:

Some did not realise they needed to square root their answer or did not know how and divided by 2 or 10.

Two ways to lose it, in one sentence. Some stopped at five hundred and seventy-six, one step short of a length. Others saw a step was missing and reached for division, halving it or dividing by ten. The squaring was right. The subtracting was right. The last move had to be a root.

Now the case where the root does not come out whole.

Find the square root of fifty. No whole number multiplied by itself gives fifty. Seven gives forty-nine. Eight gives sixty-four. Fifty is stuck in the gap between them.

The tempting move is to round fifty down to forty-nine, root that, and write seven. That is a true answer to a different question. The square root of forty-nine is seven. The square root of fifty is not.

So trap it instead. Seven squared is forty-nine. Eight squared is sixty-four. Fifty sits between those two, so its square root sits between seven and eight. And fifty is barely past forty-nine, so the answer is only just above seven. Write that down as the answer: the square root of fifty is between seven and eight, and much nearer seven. That sentence is the finished answer, not a step on the way to one. Nothing has been rounded away.

Your turn. Between which two whole numbers does the square root of forty sit?

Have a go at this one. I'll wait.

The answer is between six and seven. Six squared is thirty-six. Seven squared is forty-nine. Forty sits between those two, so its square root sits between six and seven. Forty is near the bottom of that gap, so it is closer to six.

Now the exam side, and one instruction that changes what has to go on the page.

Show that the square root of eighty-one is nine. Read that carefully. The answer is already printed in the question. Nine is not the thing being asked for. The evidence is.

So write the multiplication. Nine times nine is eighty-one, so the square root of eighty-one is nine. One line, and it is done. A number sitting on its own shows nothing.

This is exactly where the marks go. Their words, from a Foundation paper report:

Quite a few students calculated the square root by dividing by 12 which was not accepted in this 'Show that' question.

That question was a hundred and forty-four, and dividing by twelve does give twelve. The number was right. It was not accepted, because dividing by twelve needs twelve already known. Twelve times twelve is a hundred and forty-four. That is the line that earns the mark.

Right. Backwards along a power, all in one go.

A root runs backwards along a power, and the little number counts the copies. The square root of sixteen is four, two copies. The cube root of a hundred and twenty-five is five, three copies. The fourth root of eighty-one is three, four copies. A root is never found by dividing. It is found by asking what multiplies to make the number, and proved by writing that multiplication down. The square root of a hundred is ten, because ten times ten is a hundred. When a calculation squares things, the root at the end is the answer, not an extra. And when a root is not whole, trap it between the two squares either side and say where it sits.

Next in this chapter: index laws, where powers of the same number get multiplied and divided without working either one out first.

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