If the question is simply can "irrational plus irrational be irrational", then you only need to show an example to answer "yes": . If the question is must "irrational plus irrational be irrational", a counter example shows that the answer is "no": .
To show that is irrational for all n, use a proof "by contradiction": If were rational, we would have for integers a and b. Then showing that is rational, a contradiction.
Since , for n> 2. If x and y are rational, x< y, y- x is a positive number and is positive and less than y- x. Finally, is larger than x but less than y.