Estimating Square Roots

Before we can estimate square roots, we need to know our perfect squares. These are the numbers you get when you square a whole number: one, four, nine, sixteen, twenty-five, thirty-six, forty-nine, sixty-four, eighty-one, one hundred. If you know these by heart, estimating square roots becomes easy.

Let’s place the square root of fifty on a number line. By the way — instead of saying ‘square root of fifty’ every time, we can just write it as ‘root fifty’. It’s quicker. Now, root fifty isn’t a whole number, and we can only ever estimate it — its decimal goes on forever, without any repeating pattern, so we can’t write it down exactly. Numbers like this are called irrational. But we can still place it very closely using perfect squares. Forty-nine is a perfect square, and root forty-nine is seven. Since fifty is just one more than forty-nine, root fifty is just a tiny bit more than seven.

Now let’s try root thirty. Thirty isn’t a perfect square either, but it sits between twenty-five and thirty-six. Root twenty-five is five, and root thirty-six is six. So root thirty must be between five and six. Thirty is roughly halfway between twenty-five and thirty-six, so a good estimate is about five and a half.

Let’s put these four values in order from smallest to largest. To compare square roots, we estimate each one using perfect squares. Root ten is between three and four. Root fifty is just past seven. Six is already a whole number. And root ninety-nine is just under ten. Now we can line them up from smallest to biggest.