Data Sufficiency

I answered "C" to the following question since

a^b=1 could be that b is 0 and a is any number but with the added info that b is not zero there is the possibility of a^1=1 which would mean a=1. Does this logic make sense?

Here is the question:

a^b=c

If a, b and c are integers, what is the value of a?

(1) c=1

(2) b≠0


statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;

statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;

BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked;

EACH statement ALONE is sufficient to answer the question asked;

statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.