Question

The solutions of the equation $z^4+4z^{3}i-6z^2-4zi-i=0$ 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

Answer: 9

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