All you know in your case is that squaring both terms makes them equal, thus up to sign, they indeed are the same. Unless carefully defining everything, nothing more can be said. So this is mostly a question of definitions. Until you very accurately define what x^y, i and sqrt are meaning, it is quite hard to compare them. But you _could_ define (-x)^(1/2) to mean i·sqrt(x) for positive reals x, which obviously makes it true.
Also consider that i and -i are completely indistinguishable in the following sense: if you swap any occurence of one with the other, including all the definitions, it wont make any difference on mathematics. This is _not_ the same as saying that they are equal, though, just that there is no algebraic nor analytic way to really say one is "better" than the other. Or still another way, you could always define stuff with -i instead and nothing would change, so setting (-x)^(1/2) to mean -i·sqrt(x) is fine, too.
To give a real world example of this effect that is even related: assume I replace absolutely every occurance of left with right and vice versa in language (but not meaning); then you wouldn't be able to tell the difference at all, as there is no special property the one has but the other doesn't. This is different with up/down or front/back.
(@possible nitpickers: yeah, I know that unless I also change charges and some other physical properties along CPT symmetry, you actually could tell right and left apart; which by the way is exactly according to the i<->-i symmetry)