Is the product of 2 square roots of negative numbers a negative or positive value???

Question

I couldn't quite tell if the product of two square roots of negative numbers in ℂ (complex numbers) should be a positive real number or a negative one.. and this is because of the following two possibilities of solving that.

For example:

√−4×√−9=√(2i)²×√(3i)²
   =|2i|×|3i|
   =|6i²|
   =|−6|
   =6

Or

√−4×√−9=√(2i)²×√(3i)²
   =2i×3i
   =6i²
   =−6

So which one is correct and why?

Answer 1

Your dilemma comes from using the square root symbol as if it singled out a particular complex number, but that's not the case.

When $a$ is real and positive the equation $$ x^2 = a $$ has two real solutions. Just one of those is positive, and we call that one $\sqrt{a}$. The other one is $-\sqrt{a}$.

When $a$ is negative it has two complex square roots but there is no reasonable consistent way to call one of them $\sqrt{a}$, so that expression isn't used.

So for example both $2i$ and $-2i$ square to $-4$ but neither of them is "the" square root. Similarly, $1-i$ and $-1+i$ each square to $-2i$ but neither of them is "the" square root.

Answer 2

Adding to Ethan's answer: Beginners sometimes write things like "$\sqrt{-4}=\pm2i$", but this isn't actually an equation; it's shorthand for the sentence "If $z^2=-4$, then $z=2i$ or $z=-2i$".

You shouldn't do algebra with an expression like $\sqrt{-4}$ that doesn't denote exactly one number, because you'll quickly lose track of the "or".

For example, say we know that $a^2=-1$ and $b^2=-1$.

Is it possible that $a\times b=1$? Yes, because we could have $a=i$ and $b=-i$.

Is it possible that $a\times a=1$? No, because this is $a^2$, which we've assumed is $-1$.

But if you write $a=\sqrt{-1}$ and $b=\sqrt{-1}$ and substitute these "values" into the expressions above, then you get "$\sqrt{-1}\times\sqrt{-1}$" for both, and you've lost information. The best way to avoid this kind of confusion is to explicitly keep track of the cases instead of trying to use "$\sqrt{}$" (or "$\pm$") expressions to denote multiple values at once.