Square root multiplication

Question

$\sqrt {-3}$ multiplied by $\sqrt{-3}$ is $-3$. But this can also be written as $\sqrt {-3} \cdot \sqrt{-3} = \sqrt {(-3).(-3)}= \sqrt{9} =3$
So my question is why is this not possible?

Answer 1

Because $\sqrt{ab}=\sqrt{a} \sqrt{b}$ if and only if $a,b\geq 0$

Answer 2

The reason is the notation $\sqrt{\phantom{0}}$ can be used with two different meanings:

• by abuse of language, $\sqrt{-3}$ denotes any complex number with square equal to $-3$. There are two of them.
• the ‘normal’ $\sqrt{9}$ denotes, of the two (real) numbers with square equal to $9$, the positive one.

Answer 3

While teaching complex numbers, my teacher mentioned it as a theorem:

If a and b are positive real numbers, then $\sqrt{-a}\times\sqrt{-b}=-ab$.