Dense images of the power maps for a disconnected real algebraic group

Let $G$ be a complex algebraic group defined over $\mathbb R$, which is not necessarily Zariski connected. In this article, we study the density of the images of the power maps $g\to g^k$, $k\in\mathbb N$, on real points of $G$, i.e., $G(\mathbb R)$ equipped with the real topology. As a result, we extend a theorem of P. Chatterjee on surjectivity of the power map for the set of semisimple elements of $G(\mathbb R)$. We also characterize surjectivity of the power map for a disconnected group $G(\mathbb R)$. The results are applied in particular to describe the image of the exponential map of $G(\mathbb R).$