Why is $(-1)^3=(-1)^{6/2}=((-1)^6)^{1/2}=1^{1/2}=1$ wrong? Why is this wrong? 
$$(-1)^3=(-1)^{6/2}=((-1)^6)^{1/2}=1^{1/2}=1$$
 It seems logical but I know it's wrong.
Hint: $1$ has more than one possible square root, and you need to check them all when you do a calculation like you are doing.
Hint:

$$\sqrt{9}=3 \land -3$$

People tend to forget these simple things...