text: what is wrong with this logic? (-1)(-1)=1 sqrt[(-1)(-1)]=sqrt[1] sqrt[(-1)]sqrt[(-1)]= sqrt[1] so  i x i =-1 =1 what's wrong with it is that every number has two square roots. the validity of the manipulation √(xy) →√x√y is derived from the equality (xy)² = x²y². however, this equality also holds for (-xy)², so all you can really say about √x√y is that it is equal to ±√(xy). thus, all you have managed to prove is that -1=±1, which is true, since -1=-1.