Any hint for this product of complex numbers I have to calculate the modulus of $z$. I've already tried to find a general formula for $\left(\frac{1+i}{2}\right)^{2^n}$, which seems to be $\frac{1}{2^{2^{n-1}}}, \forall \geq3$, using the trigonometric form and de Moivre's formula.
$z=\left[1+\frac{1+i}{2}\right] \left[1+\left(\frac{1+i}{2}\right)^2\right] \left[1+\left(\frac{1+i}{2}\right)^4\right] \cdots \left[1+\left(\frac{1+i}{2}\right)^{2^n}\right]$
How should I keep solving this?
Thanks!

Use $\frac {1+i} 2=\frac 1 {\sqrt 2} e^{i\pi/4}$. Since $|e^{i\theta}|=1$ for all real $\theta$ we get  $|\frac {1+i} 2|^{2^{n}}=\frac 1 {2^{2^{n-1}}}$. 
