Equivalent forms of expressions with complex numbers

Question

Which expressions are equivalent to $ {1\over{(9i+z)^4}} + {1\over{(9i-z)^4}}$

Select all that apply.

$ {18i\over{(81−z)^8}}$

$ {−18i\over{(81+z)^8}}$

$ {18i\over{(81+z)^8}}$

$ {−18i\over{(81−z)^8}}$

Could someone walk me through this thoroughly? I would be incredibly grateful.

Answer 1

$$ {1\over{(9i+z)^4}} + {1\over{(9i-z)^4}}=$$ $$\frac{1}{(z+9i)^4}-\left(\frac{959}{88529281}+\frac{9360}{88529281}i\right)=$$ $$\frac{1}{(z+9i)^4}-\left(\frac{1}{9409}e^{tan^{-1}\left(\frac{9360}{959}\right)}\right)$$ $$\left(\left(z^2-81\right)^2e^{4arg\left(z+9i\right)}\right)-\left(\frac{1}{9409}e^{tan^{-1}\left(\frac{9360}{959}\right)}\right)=$$ $$-\frac{e^{tan^{-1}\left(\frac{9360}{959}\right)}}{9409}+\left(z^2-81\right)^2e^{4arg\left(z+9i\right)}=$$