We will need this lemma in our proof.
Let \(x = \sqrt{2} + \sqrt{3}\). Firstly, \(x\) cannot be an integer because \(3 \lt x \lt 4\). Secondly:
$$\begin{gather} x^2 = 2 + 2 \sqrt{2}\sqrt{3} + 3 \\ x^2 = 5 + 2 \sqrt{6} \\ x^2 - 5 = 2\sqrt{6} \\ x^4 - 10x^2 + 25 = 24 \\ x^4 - 10x^2 +1=0 \end{gather}$$
This lemma tells us that \(√2 + √3\) is irrational.