Possible Duplicate:
What's the thing with $\sqrt{-1} = i$
What is wrong with the following?:
$$ \frac{1}{\text{i}}=\frac{1}{\sqrt{-1}}=\frac{\sqrt{1}}{\sqrt{-1}}=\sqrt{\frac{1}{-1}}=\sqrt{-1}=\text{i} $$
There have been a few posts here with similar queries as above saying that
$$\frac{\sqrt{1}}{\sqrt{-1}}=\sqrt{\frac{1}{-1}}$$ does not hold.
But if we are simplifying an expression containing two variables $a$ and $b$ for example in
$$\sqrt{\frac{\left (a+b \right )^2}{-\left( a+ b \right )^2}}$$
we do get
$$\sqrt{\frac{1}{-1}}=\frac{\sqrt{1}}{\sqrt{-1}}$$
and $\sqrt{-1}$ eventually.
Something must be fundamentally wrong here. What is it?
Thanks a lot...
Update: How would we simplify
$$\sqrt{\frac{\left (a+b \right )^2}{-\left( a+ b \right )^2}}$$ then?
We can say
$$\sqrt{\frac{\left (a+b \right )^2}{-\left( a+ b \right )^2}} = \sqrt{\frac{1}{-1}}$$ and then I'm stuck!
Thanks