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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:

$$\sqrt{i^4}=\sqrt{(-1)(-1)}=\sqrt1=1$$

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    The usual definition of $\sqrt{x} = a$ is "the positive solution of the equation $x^2 = a$", so $\sqrt{x}$ is always positive *by definition*.2012-02-24
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    Most references to the square root function are talking about the principal or positive square root. Both $3$ and $-3$ are square roots of $x = 9$, that is, I can square either of them and get $x$. You seem to be refering to the opposite problem where you are looking for solutions to $x^2 = -1$.2012-02-24
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    $i^4 = (i^2)(i^2) = (-1)(-1) = 1$. So you are saying that $\sqrt{1}=-1$; this is sort-of-true in the complex numbers (it's one of the two possible branches of the square root function) and positively false in the real numbers (where the square root function is always nonnegative).2012-02-24

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