The square root of a variable is negative? If the square root of a variable is negative, as shown below:
$$\sqrt x = -1$$
Then what is $x$ equal to? The closest answer I can think of is $i^4$.
$$\sqrt{i^4}=i^{\frac42}=i^2=-1$$
But if $i^4$ is evaluated first, then it doesn't work:
$$\sqrt{i^4}=\sqrt{(-1)(-1)}=\sqrt1=1$$
 A: If a negative number is a square root of $x$, then $x$ is the square of that negative number, and thus $x$ is a positive number.  Every nonzero complex number has two square roots.  However, when you talk about the square root of a positive number, you generally mean the positive square root, not the negative square root.   
A: As others have pointed out, every complex number (except $0$) has two distinct square roots.  There's an old-fashioned expression: "multiple-valued function".  The idea is that the square root function has two values rather than just one: the square root of $4$ is "$\pm2$".  The term is simply a misnomer according to the definition of "function" that's been used for probably almost a century now, and I know of at least one professor who was quite offended by the expression for that reason.
But there's also the idea of "branches".  Suppose $z=1$ so that $\sqrt{z}=1$.  Then let $z$ move in the positive direction (counterclockwise) around the unit circle centered at $0$.  As $z$ moves around the circle, $\sqrt{z}$ moves around the circle half as fast.  By the time $z$ reaches $-1$, $\sqrt{z}$ has moved half as far along the circle and reached $i$.  $z$ keeps going, so that by the time $z$ has gone all the way around the circle, $\sqrt{z}$ has moved half-way around and reached $-1$.  There you have a square root of $1$, and it is $-1$.  As $z$ continues to move around the circle, $\sqrt{z}$ continues to move half as fast, and when $z$ reaches $-1$ for the second time, $\sqrt{z}$ reaches $-i$.  So that's the second square root of $-1$.  $z$ then keeps going, and when it reaches $1$ for the second time, $\sqrt{z}$ then comes back to $1$.  So one says that this function has two "branches".  When you return to $1$ for the first time, you're in the other branch than the one you start in.  But when you return to $1$ again, you're back in the same branch you started in.  In the same way, the cube-root function has three branches.  And some functions have infinitely many branches.  The arctangent function is one of those.
A: Using the Quaternion definition, $ x $ can be any of $i^2j^2$, $j^2k^2$ or $i^2k^2$, without using any arbitrairy operations order.
If $\sqrt{x} = -1$, $ x = -1 * -1$ and $-1$ can be defined by $i^2$, $j^2$ and $k^2$
