How do I prove that is irrational?

We are assuming, of course, that n is a positive integer.

If

\displaystyle x=\sqrt{n+\sqrt{n}}

then

\displaystyle x^2=n+\sqrt{n}

\displaystyle x^2-n=\sqrt{n}

\displaystyle (x^2-n)^2=n

\displaystyle x^4-2nx^2+(n^2-n)=0

This is a monic polynomial (leading coefficient 1) with integer coefficients. Rational roots of such polynomials must actually be integers (proof: clear the denominators. This is sometimes called the “rational root theorem”).

So we already know that if \sqrt{n+\sqrt{n}} wants to be rational then it must actually be an integer. Can it?

If it is, then n+\sqrt{n} is also an integer, in fact a perfect square. But then \sqrt{n} is an integer, so n=m^2 for some m, and n+\sqrt{n}=m^2+m.

This is not a perfect square: it sits strictly between m^2 and (m+1)^2=m^2+2m+1. Therefore, \sqrt{n+\sqrt{n}} is not an integer, so it’s irrational. QED