Convention for Expressions Involving Exponents

Question

Is there a reason why positive exponents are preferred in some settings over negative? Further, I've noticed if there's a positive rational exponent, then it is sometimes expressed as a product with factors: base to an integral power and the base to a fractional power less than one. E.g., $ x^2\sqrt{x} $ being preferred over $ xx^{3/2} $ or $ x^{5/2} $. Why would the first be preferred over the others?

Answer 1

Usually, $x^2x^{1/2}$ is not preferred over $x^{5/2}$, simply because the latter is much easier to deal with (e.g. integrate) and to write.

If I had to choose between $x^2x^{1/2}$ and $xx^{3/2}$ then I would definitely go for the former; writing $x^{5/2}$ like that at least has a purpose (i.e. to show that integer and fractional parts of the power).