Hmm.. I dont agree with the answer

Z^n can only be 1 if either n is 0 (in which case Z can be anything except 0), or if n = z = 1

1)

n is non zero means z has to be 1 for Z^n = 1

Another (non GMAT) way is to take logarithm

n*log(Z) = log 1
n * log(Z) = 0
log(Z) = 0
This means Z = 1

2)
No information given about n, but we can safely assume n=0 for a non-zero Z. However, not sufficient to find Z itself.

Should be (A) in my opinion.

Z^n=1

z=?

stmt1,

n is a non zero number.

Z=1,n=1,Z^n=1

Z=-1,n=2,Z^n=1

Insuff

stmt2,

Z>0

Z=2,n=0,Z^n=1

Z=1,n=1,Z^n=1

Insuff

Combining 1 and 2,

Z>0 and n is non zero number.

-->Z=1

Suff

Pick C

Nice explanation, selango - and just to reaffirm, the catch here is that Z could be -1 as long as n is even, or 1 regardless of n.

I mentioned this in another thread today - when a statement is clearly not sufficient on its own, as statement 2 is here, you should ask yourself:

Why Are You Here?

It's either to provide you with information that you think you need but don't, or to provide information that you actually do need. But either way you should know that you have to make a decision:

Do I need to know that z is positive?

That should force you to consider negative numbers (as well as 0). And -1 does allow for Z^n to still be 1, so that's why Statement 1 alone is not sufficient.