What is the basic method for proving the irrationality of a number?

I know to prove the irrationality of we simply minipulate the following:

until we hit the contradiction of not being in reduced terms. I also know that we must take into account that all number can be written in the form or for ; and the fact that if is an even number then is an even number. Heres my question, how do you prove the irrationality of a certain opertation on numbers in general? And for odd square roots?

Two examples of what I mean:

[1] How would you prove that if that is always irrational?

[2] How would you prove is irrational? Would you find a way to define numbers in terms of 3's ? Just like with the ? So all numbers would be of the form , , and ; and would you some how use that fact to prove the irrationality of ?

Thanks in advance