Question

The solutions of the equation $z^4+4 z^3 i-6 z^2-4 z i-i=0$ are the vertices of a convex polygon in the complex plane. If the area of the convex polygon is $2^{ m / n }$ where $m , n$ are coprime, then find the value of $( m + n )$.

Answer 1

$z^4+4 z^3 i+6 z^2 i^2+4 z i^3+i^4=1+i$
$(z+i)^4=1+i \Rightarrow \quad|z+i|^4=\sqrt{2} \Rightarrow|z+i|=2^{1 / 8} $
$|z+i|=2^{1 / 8} $
$\text { Area }=\frac{d^2}{2}=\frac{4\left|z_1+i\right|^2}{2} $
$=2 \cdot 2^{1 / 8} \cdot 2^{1 / 8}=2^{5 / 4}$