Why can't we take a square root of a negative number?

It depends in which context you are.

Taking a square root of a real number it's impossible beacuse any real number if squared gives a positive value.

But, if you are in the context of complex numbers, then the picture changes. Actually complex numbers where first invented in order to overcome this problem. The imaginary unit ##i## is in fact defined as:

##i=sqrt(-1)##

Every complex number ##z## is made by the sum of a real part ##a## and an imaginary part ##b##:

##z = a+ib##

Real numbers can then be thought as complex numbers with no imaginary part.

So in the context of complex numbers the square root of a negative real number is well defined and the result is pure imaginary.

To make a simple example let's take the number ##-9##:

## sqrt(-9)=sqrt(9)sqrt(-1)=3i ##