The operations of addition and scalar multiplication have been defined as: $$(a_1, a_2) + (b_1, b_2) = (a_1 + b_2, a_2b_2)$$ and $$k \cdot (a1, a2) = (k\,a1, a_2^k).$$
This was a problem on one of my tests, and I missed all possible points. When we went over the test, I paid attention to how it was supposed to be done, but did not write it on my test for reference. I tried looking for examples in my textbook, but there aren't any. Any help would be appreciated. My attempt:
zero vector: (a1, a2) + (0,0) = (a1 + 0, a2(0)) = (a1, 0) additive inverse: (a1, 0) + (0,0) = (a1, 0)
This is clearly wrong.