Non-existence of an element in a splitting field -- obvious, but hard to prove

Question

I've been working through some problems in Dummit and Foote and have in general been struggling with rigorously finding the degree of a field extension. Intuitively the answer is usually obvious because the adjoined roots are some how "independent" of one another. For instance, how can I find the degree of $\mathbb{Q}(\sqrt{2}, \sqrt[8]{5})$? It's obviously 16 but my issue is that I can't prove that $\sqrt{2} \notin \mathbb{Q}(\sqrt[8]{5})$. Or perhaps there is another approach?