When given exercises for my coursework, I often encounter the problem of not knowing how pedantic to be in my proofs. Some seem to be statements so trivial that I'm forced to question whether or not the exercise is allowing me to use basic operations that I take for granted. For instance, I have the following exercise tonight:
Given $1 < b\in\mathbb{R}$, and $r = \frac{m}{n} = \frac{p}{q}$ prove that: $$(b^m)^\frac{1}{n} = (b^p)^\frac{1}{q}$$
I dont' know if I'm allowed to use a statement like $(x^a)^b = (x^{ab})$ in this context.
I've had this situation go both ways. I've taken the option of writing a page long proof, and had the professor tell me that I expanded on it too much. I've also gone the route of using statements like the above to make it a few scant lines, only to have the professor tell me I wasn't allowed to use a statement like that.
To be clear, I don't want help with this particular exercise. I'd like to know if anybody has general guidelines for knowing which theorems one is allowed to use for a given exercise.