The question $\sqrt{\frac{5-12i}{5+12i}}$ can be simplified after rationalising the denominator to $\frac{5}{13}-\frac{12}{13}i$. However, the answers have $\pm\frac{1}{13}\left(5-12i\right)$ and I don't know where the plus minus comes from. I know this happens in the case: $\sqrt{a+ib}=x+iy$, but I did not have to use that in my working out.
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3If $z$ is square root of $w$ then $-z$ is also a square root of $w$. – Kavi Rama Murthy Apr 01 '21 at 10:04
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Your solution is also correct if you demand that the square root symbol denotes a square root function, usually with the branch cut along the negative real half-axis, with values largely having positive real part. But you have to be careful with this as the composition laws of roots may contain a sign change. – Lutz Lehmann Apr 01 '21 at 10:09
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The question is: what exactly does $\sqrt z$ mean in the problem? If one assumes that this is an analytic branch of the function then only one of two possible values is correct. If any branch is allowed then both values are correct. – user Apr 01 '21 at 11:22
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Ok yes, @KaviRamaMurthy that's right, you must take $\pm$ for square roots. But then... why does Lutz Lehmann say my answer is still correct? – user71207 Apr 11 '21 at 13:11