How to understand $k \otimes_R \alpha \cong k \oplus k$?

Question

Let $R = k[x, y]$, where $k$ is field. Then we have a projective resolution of $k$ :

$$0 \longrightarrow R \stackrel{f}\longrightarrow R\oplus R \stackrel{g}\longrightarrow R \longrightarrow k \to 0 $$

where $f(r)=(-ry, rx), \,\,g(r_1, r_2)=r_1x + r_2 y$.

Take $\alpha = (x, y)$, which is an ideal of $R$. Then we can use the projective resolution and the short sequence $0 \rightarrow \alpha \rightarrow R \rightarrow k \rightarrow 0$ computing $\operatorname{Tor}_i^R(k, \alpha)$. Hence we have

$\operatorname{Tor}_{i-1}^R(k, \alpha) \cong \operatorname{Tor}_i^R(k, k)$.

In particular we have $\operatorname{Tor}_0^R(k, \alpha) \cong \operatorname{Tor}_1^R(k, k) \cong k \oplus k$.

Question:

1. How to understand $k \otimes_R \alpha \cong k \oplus k$?
2. How to compute $\operatorname{Tor}_i^R(k, \alpha^n)$

I appreciate all your help.