What is the appropriate way to compute $z=\frac{1}{\sqrt{7+24i}}?$

Question

What is the appropriate way to compute $$z=\frac{1}{\sqrt{7+24i}}?$$

I'm asking because in 2 different books I found 2 different answers for the problem:

Book A: $(4-3i)/25$ and $(-4+3i)/25$;

Book B: $(-4+3i)/25$.

To add to the confusion Wolfram Alpha gives the single answer $(4-3i)/25$, which I believe is correct by my own development.

The difficulty, I think, is on how to deal with the signs in the positive root of a complex number. This is not, I believe, the same thing as solving $z^2=1/(7+24i)$. Like, on reals, the problem of solving $x^2=4$ or computing $x=\sqrt{4}$.

I would appreciate some clarification, if possible.

Answer 1

Note that $$(4+3i)^2=16-9+24i$$ and $$\frac{1}{4+3i}=\frac{4-3i}{25}$$

Answer 2

Book A just gives the square root and its negative (both valid).

For a complex nonzero $w$ there are always two distinct solutions $z$ to $z^2=w.$ Though there may be some way to single out one of them as "the" square root, it's not that simple since complex numbers aren't ordered.

Answer 3

Correct answer is $ (4-3i)/25 $ as given by Wolfram Alpha. The real part of the principal square root is always nonnegative. ( Source: https://en.wikipedia.org/wiki/Square_root#Algebraic_formula )