Let $a,b,c$ be odd primes. In particular, $ab, ac, bc$ are all odd numbers. We can use this to our advantage, since then $\sqrt[ac]{x} : \Bbb{R} \to \Bbb{R}$ is well-defined and a bijection.
Let $V = \Bbb{R}^3$ as sets and define scalar multiplication on $V$ by $\lambda \in \Bbb{R}, \ v=(x,y,z) \in V \implies \lambda\cdot v = (\lambda^{bc} x, \lambda^{ac} y, \lambda^{ab} z)$. Define addition of vectors by: $$ u,v \in V, u = (x,y,z), v = (x', y', z') \implies \\ u + v = ((\sqrt[bc]{x} + \sqrt[bc]{x'})^{bc}, (\sqrt[ac]{y} + \sqrt[ac]{y'})^{ac}, (\sqrt[ab]{z} + \sqrt[ab]{z'}))^{ab}) $$
What vector space is this?
Does it actually form a ring too?