Square Roots and Cube Roots

A root asks a power question backwards. Seven squared equals forty-nine asks: what do I get when I multiply seven by itself? The square root of forty-nine asks the opposite: what number, multiplied by itself, gives forty-nine? The answer is seven. Picture forty-nine tiles pushed into a square. Squaring is being told the side and asked for the number of tiles. Taking the square root is being told the number of tiles and asked for the side. It is the same relationship, read in the other direction. So because six times six is thirty-six, the square root of thirty-six is six. The list of square numbers you already learned is now a list of square roots as well.

Here is the symbol. A root sign with nothing written on it always means square root — mathematicians leave the little two off because square roots come up so often. So the square root of eighty-one is nine, because nine nines are eighty-one, and the square root of one hundred and forty-four is twelve. Because you already know the square numbers, you already know all of these roots: just read your list backwards. One gives one, four gives two, nine gives three, sixteen gives four, and so on all the way to one hundred and forty-four giving twelve. These are called exact roots, because the answer comes out as a whole number.

Cube roots work the same way, one dimension up. Three cubed is twenty-seven, so the cube root of twenty-seven is three. Picture twenty-seven blocks stacked into a cube, three along, three across and three high — the cube root is the edge length. The symbol carries a small three tucked into the root sign, and that small three is the only thing telling you it is a cube root rather than a square root, so look for it. The cube roots you should recognise on sight come straight off the cube numbers: one gives one, eight gives two, twenty-seven gives three, sixty-four gives four, one hundred and twenty-five gives five, two hundred and sixteen gives six, and one thousand gives ten.

This is where marks go missing. Sixty-four is both a square number and a cube number, so it has both kinds of root — and they are not the same. The square root of sixty-four is eight, because eight eights are sixty-four. The cube root of sixty-four is four, because four times four times four is sixty-four. So read the symbol before you answer. There is one more idea worth having. A root and a power undo each other. Take the square root of twenty-five to get five, then square that five and you are back at twenty-five. That gives you a free check on every single answer: square your square root, or cube your cube root, and see whether you land back where you started.

Roots are not just a symbol game — they answer real questions. A square lawn covers eighty-one square metres. How long is each side? The square root of eighty-one is nine, so the sides are nine metres. A cube-shaped box holds one hundred and twenty-five cubic centimetres. How long is each edge? The cube root of one hundred and twenty-five is five, so five centimetres. That is exactly what these operations are for: multiplying takes you from a side to a space, and the root takes you back again. One warning. Only numbers on your squares and cubes lists have whole-number roots. The square root of twenty is not a whole number, and finding it is a job for next year.