Question

What is the mistake in computation of $1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1}\times \sqrt{-1}=i\times i=-1$

Answer 1

The formula $\sqrt{a}\sqrt{b}=\sqrt{ab}$ is applicable only when at least one $a$ and $b$ is positive.
Otherwise $\sqrt{-1}\sqrt{-b}=-\sqrt{ab}$ is $a,b>0$
$\therefore \sqrt{(-1)(-1)}=-\sqrt{1\times 1}=-1$
But here $\sqrt{-1}\sqrt{-1}=\sqrt{1}=1$