2021 MECC Mock AMC 12

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Problem 25

25. All the solution of $(z-2-\sqrt{2})^{24}=4096$ are vertices of a polygon. The smallest solution that when express in polar form, has a $y$ value greater than $0$ but smaller than $\frac{\pi}{2}$ can be expressed as $\frac{\sqrt{a}+\sqrt{b}+i\sqrt{c}-i\sqrt{d}+\sqrt{e}+f+i\sqrt{g}-hi}{j}$ which $a,b,c,d,e,f,g,h,j$ are all not necessarily distinct positive integers, $i$ in this case represents the imagenary number, $i$, and the fraction is in the most simplified form. Find $a+b+c+d+e+f+g+h+j$.