Starting with $\frac{-1}{1}=\frac{1}{-1}$ and taking square root: proves $1=-1$ In this blog post, RJ. Lipton mentions an example of common mathematical traps. In particular, that ``square root is not a function''. He shows the following trap:
Start with:
$\frac{-1}{1}=\frac{1}{-1}$, then take the square root of both sides:
$$
\frac{\sqrt{-1}}{\sqrt{1}}=\frac{\sqrt{1}}{\sqrt{-1}}
$$
hence
$$
\frac{i}{1} = \frac{1}{i} \\
i^2=1 \enspace ,
$$
which contradicts the definition that $i^2=-1$.
Question 1: I know that the square root is not a function because it is multi-valued, but I still can not wrap my head around this example. Where was the problem exactly? Was is that we 


*

*can not convert $\sqrt{1/-1}$ to $\sqrt{1}/\sqrt{-1}$? 

*both the RHS and LHS are unordered sets?

*both?


Question 2: Also, does this problem only arise in equalities or in general algebraic manipulation? Because it would be a nightmare when manipulating an expression with fractional powers! Are there easy rules to determine what is safe to do with fractional powers? To see what I mean, there is another example of a similar trap:
One might easily think that $\sqrt[4]{16x^2y^7}$ is equivalent to $2x^{1/2}y^{7/4}$, which is not true for $x=-1$ and $y=1$.

at real analysis
$$\sqrt{\frac{x}{y}}\ = \frac{\sqrt{x}}{\sqrt{y}}$$
only for $y,x>0$
