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.