Assume $k_1,\dots,k_{n+1}$ are coprime, which we can always do. The $n$-dimensional weighted projective space $X = \mathbb{P}(k_1,\dots,k_{n+1})$ is a toric variety, hence is automatically rational since it contains an isomorphic copy of $\mathbb{G}_m^n$ as a dense open subvariety. Indeed, let $T$ be the torus $\{(t_1,\dots,t_{n+1}) \mid t_i \in \mathbb{G}_m\}/\{(t,\dots,t)\mid t\in \mathbb{G}_m \}$. Then $T$ acts on $X$ via $$(t_1,\dots,t_{n+1}) \cdot [x_1 : \dots :x_{n+1}] = \left[t_1^{k_1}x_1 :\dots :t_{n+1}^{k_{n+1}}x_{n+1} \right]$$ and the orbit of $[1: \dots : 1]$ is isomorphic to $T$ (since the stabilizer is trivial since $k_1,\dots,k_{n+1}$ are coprime.

The $n$-dimensional weighted projective space $X = \mathbb{P}(k_1,\dots,k_{n+1})$ is a toric variety, hence is automatically rational since it contains an isomorphic copy of $\mathbb{G}_m^n$ as a dense open subvariety. Indeed, let $T$ be the torus $\{(t_1,\dots,t_{n+1}) \mid t_i \in \mathbb{G}_m\}/\{(t,\dots,t)\mid t\in \mathbb{G}_m \}$. Then $T$ acts on $X$ via $$(t_1,\dots,t_{n+1}) \cdot [x_1 : \dots :x_{n+1}] = \left[t_1^{k_1}x_1 :\dots :t_{n+1}^{k_{n+1}}x_{n+1} \right]$$ and the orbit of $[1: \dots : 1]$ is isomorphic to $T$, since its stabilizer is a product of cyclic groups.