Can someone help me understand changing between bases?

For example, if we have

(34)_5 + (23)_5, I think of it this way:

We have 5 5's and then 7 1's so we have 6 5's and 2 1's which means we have:

1 (5^2) .. 1 (5^1) .. 2 (5^0)

So our answer then would be (112)_5?

Alternatively, can we just say (34)_5 + (23)_5 = (57)_5? I'm assuming not, since we're using base 10 rules in adding.

Instead, I think it would be (34)_5 + (23)_5 = (62)_5

Some advice?