Simplifying $\left|\left|\sqrt{-x^2}-1\right|-2\right|$ How do we simplify the expression $\left|\left|\sqrt{-x^2}-1\right|-2\right|$?
This is very confusing.  Do they cancel out and become just simply $\sqrt{-x^2}-1-2$?
 A: Assuming you are working with real numbers only, your expression is only defined if $x = 0$. Following on from that the answer would be $||0 - 1| - 2| = ||-1| - 2| = |1 - 2| = |-1| = 1$.
If we allow x to be complex or imaginary, it gets a bit more, well, complex. Let x be the complex number $a + bi$. Squaring this gives $a^2 + 2abi - b^2$. The negative sign changes this to $b^2 - 2abi - a^2$, and the expression has now become $||ai - b - 1| - 2|$.
If we suppose that the real part of this complex number is 0 (meaning it is a pure imaginary number), then we can get $||-b - 1| - 2|$. The answer would then depend on the value of b. Otherwise, there is no way to simplify this expression; although the absolute value of $a + bi$ is $\sqrt{a^2 + b^2}$ (can you figure out why?) we have flipped the real and imaginary parts, and also have a pesky 1 in the mix.
Note also that the absolute value of the absolute value is indeed the absolute value, but you cannot just "cancel out" the absolute value brackets. As you can see, you must take into account the operations in between; if you had evaluated the brackets as you say the answer would be -3, which is ridiculous, because absolute value gives only positive answers.
