sqrt question

[Han]

(04-07-2017 01:23 PM)Claudio L. Wrote:  Both answers are correct!
Every power in the form a^(1/n) has 'n' answers in the complex plane. sqrt is the particular case n=2, so it has 2 answers:
sqrt(-4) = +2i and -2i
sqrt(-9) = +3i and -3i
doing all 4 possible combinations for the product you arrive at the 2 possible answers: +6 and -6.

sqrt( (-4)*(-9) ) = sqrt(36) = +6 and -6

By doing the operations in one order or the other, you are simply choosing (or accidentally falling into) one branch or the other, nothing wrong with that, but don't ever forget that there's 2 sides of that coin.

There are some nuances here that perhaps we are glossing over. Using the notation \( \sqrt{x} \) (which has a standard mathematical convention of meaning the positive root) to denote an extension into the complex plane means certain properties that were true only for non-negative real values of \(x \) no longer hold for more general \( x \). Ordinary \( \sqrt{x} \) is a function, whereas complex \( \sqrt{x} \) is a 1-to-2 map (not a function). So if we want the operation to behave as a function, there cannot be multiple answers. On the other hand, if this is not an issue, then both 6 and -6 are "correct" answers.

Another question (regarding consistency) is why one would take \( \sqrt{-4} \) to be 2i and \( \sqrt{-9} \) to be -3i (i.e. one positive and the other negative, leading to inconsistency in the signs)? For functions, they should both be positive or both be negative. For more general maps, this is not a concern.