$V$ is a vector space of positive real numbers with vector addition being defined as regular multiplication and scalar multiplication defined with the following
$$a \times v = v^a$$
The answer key says that this is a vector space but I can't seem to prove the following axiom
$$a(v+w) = av + aw$$
which turns to the following
$$(v+w)^a = v^a + w^a$$
I don't know if there is an identity that makes this true for all positive $v$ and $w$ but I can't seem to prove this holds. Is there something I'm missing?
Edit: I got it $v+w$ becomes $vw$ so $$a(v+w) = av + aw$$
becomes
$$(vw)^a = (v^a)(w^a)$$
and it hold, all good now, I'm just blind