A student and I were slaying a tricky geometry problem from an old AMC. She hadn’t taken geometry yet, but she had excellent insights, and we were making good progress. We were close to the end and we needed to simplify . Since is a multiple of a square (), and since the square root of a product is the product of the square roots, we can rewrite as .
But she couldn’t do it. I thought maybe she hadn’t learn how to simplify square roots, but she admitted she had learned them, but hadn’t solved a square root problem for a long time and was out of practice. Fair enough.
As a student progresses through a math curriculum over the years, they are gradually adding tools to their problem solving tool box. Simplifying square roots is another tool. The textbook teaches you how to use it, you practice using it on a bunch of squares, then you throw into your tool box for some unspecified time when you will need it later. Then they go on to learn another tool. And another tool.
The trouble is, like my student, you can forget how to use your tools if you don’t practice.
That’s another reason why practicing math contests can be so helpful in keeping those skills fresh. Math contests typically have all kinds of problems: geometry, probability, number theory, and yes solving square roots. Sometimes, like in that geometry problem, you need to deploy several of your tools to arrive at a solution.