<s>[INST] Finding the square roots of a complex number. 
Express $z=4\sqrt2(1+i)$ in modulus/argument form. Hence find the two
  square roots of $z$ and mark their representations on an Argand
  Diagram.

So far I've worked out the mod/arg form of the complex number which is just
$$
z = 8(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}) \\
$$
Then used $\alpha = \frac{\theta +360k}{n}$ where $k$ is $0,1,2,...,n-1$ and got two results for the argument. The first being $\frac{-7\pi}{8}$ and the other being $\frac{\pi}{8}$.
My final answers were
$$
z_1 = 2\sqrt2(\cos\frac{-7\pi}{8} + i\sin\frac{-7\pi}{8}) \\
z_2 = 2\sqrt2(\cos\frac{\pi}{8} + i\sin\frac{\pi}{8})
$$
For some reason, the answers from the book were 
$$
z_1 = 2\sqrt2(\cos\frac{\pi}{8} + i\sin\frac{\pi}{8}) \\
z_2 = -2\sqrt2(\cos\frac{\pi}{8} + i\sin\frac{\pi}{8})
$$
What was it that I did wrong in my calculations? Thanks in advance!
 [/INST] Both are correct. In fact, they are the same.
$$ 2\sqrt 2 (\cos\frac{-7\pi}{8} + i\sin\frac{-7\pi}{8}) = -2 \sqrt 2 (\cos\frac{\pi}{8} + i\sin\frac{\pi}{8}) $$ 
Moreover, $-2\sqrt 2 (\cos\frac{\pi}{8} + i\sin\frac{\pi}{8})$ looks better.
</s>