\(\sqrt x = x^{1/2}\)

Ok …

But why?

This is surprisingly hard to answer, for an equation that both looks so simple and is used in such basic math courses. This must be why the usual high school teacher’s answer is “It just is, don’t worry about it.” I sympathize: A high school teacher is tasked with getting a large group of uncooperative teenagers through a state-dictated curriculum within too small a window of time.

Even then, a very brave and diligent high school teacher will sometimes give this explanation: Notice that \(\sqrt x^2 \) simplifies to x, and so does \((x^{1/2})^2\). In short, if you square both of these, you get the same thing. So the things you had before squaring must have been the same also. I mean, if you take \(2^2\) you get 4, and if you take \(3^2\) you get 9. They’re different because the things you squared were different—so conversely, if you square two things and get the same thing then what you had before the square must have been the same.

For most high school students, I think that will be an adequate explanation—mostly because I think the average high school student doesn’t really care that much about the reasons, they just want to get their homework done. And I’m fine with this. If someone really wants a deeper answer, they can take the college courses where you learn the deeper answers.

But it’s worth recognizing that this cannot be the full answer. There are a few ways you could tell. For one thing the argument’s principles are obviously false, because this principle would tell you that because \(1^2\) is 1 and also \((-1)^2\) is 1, then therefore the two numbers before taking squares were equal. But we know -1 does not equal 1.

But also, somewhat obviously, a full answer must come to terms with the idea of continuation. We know what \(x^2\) means and we know what \(x^3\) means. It’s repeated multiplication. But for any exponent not an integer greater than 1, this definition does not make sense. Even for an exponent of 1 this does not make sense, even though very often educators stomp their foot and insist that it does. It doesn’t.

And I’m fine with this too! It’s ok to admit that \(x^1\) doesn’t make any sense … so long as you emphasize that it doesn’t make sense, when your definition of an exponent is in terms of repeated multiplication. By the time you start using powers other than integers greater than 1, though, you simply cannot use this as your definition any longer. You must continue the definition from integers greater than 1, to a larger class of numbers, and the only way to do that is to redefine what the exponent means.

But this isn’t the wild west. We have laws. You can’t just continue it in any way that you want. For one thing, however you continue the definition of the exponent, you cannot give a definition which would infringe on the old definition. You could not redefine exponents and get the result that \(2^3=7\) or something like that. Your continuation must be consistent with the narrow definition.

There are other principles that a continuation must obey. I have a busy day, so I may write about these another day.