<s>[INST] Complex Numbers Questions... I am having trouble with these problems:


*

*Find all complex numbers $z$ satisfying the equation
$$\frac{z+1}{z-1} = i.$$

*The value
$$\left(\frac{1+\sqrt 3}{2\sqrt 2}+\frac{\sqrt 3-1}{2\sqrt 2}i\right)^{72}$$
is a positive real number. What real number is it?
On 1) , I was thinking about substituting $z$ for $a+bi$ and then solving. Is this correct?
 [/INST] For 1 just use simple arithmetic: it's a degree one equation.
For 2, compute first the square:
\begin{align}
\left(\frac{1+\sqrt 3}{2\sqrt 2}+\frac{\sqrt 3-1}{2\sqrt 2}i\right)^{2}
&=
\left(\frac{1+\sqrt 3}{2\sqrt 2}\right)^2-
\left(\frac{\sqrt 3-1}{2\sqrt 2}\right)^2+
2\frac{1+\sqrt 3}{2\sqrt 2}\frac{\sqrt 3-1}{2\sqrt 2}i
\\[6px]
&=\frac{1+3+2\sqrt{3}}{8}-\frac{3+1-2\sqrt{3}}{8}+
2\frac{3-1}{8}i
\\[6px]
&=
\frac{\sqrt{3}}{2}+\frac{1}{2}i
\\[6px]
&=\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}
\end{align}
</s>