How to compare Dehn Invariants

Question

I just learned about tensor products in the context of Hilbert's Third Problem. I think I understand what a tensor product is, at least what it was used for in the Dehn Invariant. However, I get stuck when I want to compare two tensor products (for example the Dehn Invariant of a regular tetrahedron and that of a regular octahedron). In the example, I get $6\sqrt[3]{6\sqrt{2}}\otimes \arccos\left({\frac{1}{3}}\right)$ for the tetrahedron and $8\sqrt[3]{12\sqrt{2}}\otimes \arccos\left({\frac{\sqrt{3}}{3}}\right)$ for the octahedron. How do I compare the two? Both are in $\mathbb{R}\otimes\mathbb{R}/\pi\mathbb{Q},$ so it isn't obvious if they are actually different. I guess what I am asking is how to actually compute tensor products in the context of Dehn invariants.