I'll just toss in my 2 cents worth here and say that proving

is irrational is

*subtlely* more difficulty than proving

is irrational ......

If you're assuming the usual thing,

where m,n are relatively prime integers, then at some stage you must prove that if m^2 is divisible by 3, then m is also divisible by 3.

This is slightly more difficult than proving that if m^2 is divisible by 2 then m is divisible by 2 .... (but can nevertheless be easily done by using the fact that every positive integer has a unique prime factorisation).