Alternative way to prove that $\mathbb{Q}(a,b)=\mathbb{Q}(a + b)$ for certain $a, b$

Question

I want to know if there is an alternative method to prove that $\mathbb{Q}(a,b)=\mathbb{Q}(a + b)$ (I know this doesn't happen for any $a, b$). For example, to prove that $\mathbb{Q}(\sqrt{2},\sqrt{3})=\mathbb{Q}(\sqrt{2}+ \sqrt{3})$ I would write the explicit expression of any element in each body, and then realize that they are the same.
This is easy to do here because $(\sqrt{2})^2\in \mathbb{Q}$ (the same for $\sqrt{3}$). However, if I have roots of a higher degree $n$ this process is very long. Is there an alternative way to do this?