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