It's come to my attention that I don't actually understand what a square root really is (the operation). The only way I know of to take square roots (or nth root, for that matter) it to know the an...
Powers are multiplying a value by itself. What is a square root? To be clear, I'm asking what function a square root actually performs? I can't seem to come up with an answer without the concept of powers to refer to. For example; how would you go about solving the square root of $121$ without prior knowledge that $11^2$ is $121$?
For convenience, the square root of non-negative real numbers is usually taken to be the non-negative real value, but there is nothing other than practicality to stop you from taking some other pattern. Such arbitrary choices can raise significant issues when considering, for example, cube-root functions defined on the real and complex numbers.
The square root of i is (1 + i)/sqrt (2). [Try it out my multiplying it by itself.] It has no special notation beyond other complex numbers; in my discipline, at least, it comes up about half as often as the square root of 2 does --- that is, it isn't rare, but it arises only because of our prejudice for things which can be expressed using small integers.
When researching square roots I found that $\sqrt{x}$ is the principal square root and $\pm\sqrt{x}$ is the square roots, with the reason for why being given through an example equation by user9464...
It is impossible to find the square root of negative one, or the square root of any negative number, because no number times itself can equal a negative number.