Why doesnt √-4 equal 2?

Question

Why doesnt √-4 equal 2 if using the principle x to the power of m/n equals the nth root of x to the power of m causes √-4 = ∜(-4)^2 = ∜16 = 2?

Answer 1

$\sqrt x$ is, by definition, the non-negative real number which when multiplied by itself becomes $x$.

So, $\sqrt{-4}$ would be a positive number which when multiplied by itself becomes $-4$. But there is no such number. Therefore $\sqrt{-4}$ is undefined.

If you look carefully in your math book, you will see that the manipulations you do are specifically defined only when whatever is under the square root sign is $\geq0$. If they haven't specified that in your book then it's a bad book.

To those who know about complex numbers, I personally do not like to mix square roots symbols and complex numbers, because they lead to all sorts of problems (the manipulation in the question post included), and do not help much. So to me, $\sqrt{-4}$ is still undefined in that context. Thus, for instance, $i$ is not really defined as $i=\sqrt{-1}$, but as $i^2=-1$.