My question is whether all manifolds that can be embedded in $\mathbb R^n$ are homeomorphic to a smooth manifold?

I know that every smooth manifold can be triangulated which I think is a result of Whitehead and I think every manifold in $\mathbb R^n$ can be triangulated so this lends plausibility I think. (If the dimension 4 E8 manifold can be embedded in $\mathbb R^n$ then we have a counterexample but I'm not sure if it can be. All manifolds of dimension up to 3 can be triangulated).

I appreciate if this is a basic question but it doesn't seem to be spelt out explicitly anywhere. Most textbooks define a manifold and then a smooth manifold but don't say in which cases these concepts may be equivalent.

Anymanifold can be embedded in $\mathbb R^n$ for some $n$, see e.g. here $\endgroup$ – Wojowu Aug 30 '19 at 9:17