I think of "square root" as meaning "do half of something".
For example, A^2 means do the operation A twice, A^1 = A means do the operation A once, A^0.5 = sqrt(A) means do half of A. What's half of A? Something that when done twice gives A.
For example, take a number line. We have integers going from zero to infinity. Now lets add a "negation operation", and call it -1, so we can make the numbers from zero to negative infinity. On paper, this is equivalent to rotating the number line by 180 degrees. Now the number line runs from negative infinity to positive infinity. Now let's do the operation sqrt(-1), which means "do half of a negation". On paper, this means we rotate by 90 degrees instead of 180 degrees. We now have a new number line at right angles to the original one, and the original number line has turned into a number plane. (Feel free, at this point, to launch into an explanation of complex numbers, with i=sqrt(-1) meaning "move in an orthogonal direction".)
Similarly, a cube root means "do a third", and so on.
Spoiler alert: When people talk about "group theory," this is the sort of thing that it does. The operations of addition, multiplication, and functional composition share certain symmetries that group theory makes formally sound. Once you grok the relationships, and how multiplication is "the same type of thing as" rotations, then you can see how square roots are similar to fractional rotations.
Less abstractly... the Exponent operation lets you turn addition into multiplication. It takes a little while to wrap your head around that, but once you do, it's straightforward to see how square roots correspond to fractions.
This is also concretely visible in Matrix algebra. Some matrices literally are rotations, and the square roots of such matrices are rotations by half as much. Matrices are nice to study in group theory, because they bridge the gap between numeric operations and functional composition.
Reminds me of category theory, which I tried to understand a little of yesterday, based on a discussion of Haskell. I don't know enough of either to know what the difference is between the two.
Nice, I didn't realise the relationship between rotations and the matrix square root: multiplying a vector by the sqrt of the pi/2 rotation matrix rotates it by pi/4!
I don't like this explanation at all. This fuzzy thinking works out because you knew it to work out based on your much less fuzzy understanding of the complex numbers. Similarly fuzzy thinking in a new problem domain won't work out 90% of the time.
That's another interesting way to do it. When I read the blog post and saw the two legs of what I assumed would be a right pythag triple triangle forming, I thought I was about to see the classic Pythagorean theory and its geometric proof whipped out, which might be a little too much all at the same time for a kid (or, maybe not?). Hard to describe in text but its pretty much what you get with a 3x3, 4x4, and 5x5 gridded squares connected to each other giving a 3,4,5 right triangle.
The problem is this rapidly devolves into "very nice anecdote but why can't you use any random lengths" and "why does this only work in 2 dimensions" and next thing you know its gettin pretty deep which is a bit much to start with WRT whats a sqrt.
For example, A^2 means do the operation A twice, A^1 = A means do the operation A once, A^0.5 = sqrt(A) means do half of A. What's half of A? Something that when done twice gives A.
For example, take a number line. We have integers going from zero to infinity. Now lets add a "negation operation", and call it -1, so we can make the numbers from zero to negative infinity. On paper, this is equivalent to rotating the number line by 180 degrees. Now the number line runs from negative infinity to positive infinity. Now let's do the operation sqrt(-1), which means "do half of a negation". On paper, this means we rotate by 90 degrees instead of 180 degrees. We now have a new number line at right angles to the original one, and the original number line has turned into a number plane. (Feel free, at this point, to launch into an explanation of complex numbers, with i=sqrt(-1) meaning "move in an orthogonal direction".)
Similarly, a cube root means "do a third", and so on.