For a group G, let \(g\rightarrow g^k\) be the k-th power map \(P_k\) on G. The purpose of this article is two-fold. First, we consider G as an algebraic group defined over \({\mathbb {R}}\) and characterise the density of the images of the power map \(P_k\) on \(G({\mathbb {R}})\) in terms of Cartan subgroups. Next, we consider the linear algebraic group G over a non-Archimedean local field \({\mathbb {F}}\) with any characteristic. If the residual characteristic of \({\mathbb {F}}\) is p, and an element admits \(p^k\) -th root in \(G({\mathbb {F}})\) for each k, then we prove that some power of the element is unipotent. In particular, we prove that an element \(g\in G({\mathbb {F}})\) admits roots of all orders if and only if g is contained in a one-parameter subgroup in \(G({\mathbb {F}})\) . Also, we extend these results to all linear algebraic groups over global fields.
