Question

Комплексные числа
((1+i)^3 -1)/((1-i)^3 +1)​

Answer 1

$frac{(1+i)^{3}-1 }{(1-i)^{3} +1}=frac{1^{3}+3*1^{2}*i+3*1*i^{2}+i^{3}-1}{1^{3}-3*1^{2}*i+3*1*i^{2}-i^{3}+1}=frac{1+3i-3-i-1}{1-3i-3+i+1}=frac{2i-3}{-2i-1}=frac{3-2i}{1+2i}=frac{(3-2i)(1-2i)}{(1+2i)(1-2i)}=frac{3-6i-2i+4i^{2} }{1-4i^{2} }=frac{3-8i-4}{1+4}=-frac{1+8i}{5}$

Answer 2

$frac{(1+i)^{3}-1 }{(1-i)^{3} +1}=frac{1^3+3cdot1^{2}cdoti+3cdot1cdoti^2+i^3-1}{1^3-3cdot1^2cdot i+3cdot1cdo ti^2-i^{3}+1}=frac{1+3i-3-i-1}{1-3i-3+i+1}=\\frac{2i-3}{-2i-1}=frac{3-2i}{1+2i}=frac{(3-2i)(1-2i)}{(1+2i)(1-2i)}=frac{3-6i-2i+4cdot i^2}{1-4i^2}=\\frac{-8i+3+4 cdot(-1)}{1-4cdot(-1)}=frac{-8i+3-4}{1+4}=frac{-8i-1}{5}$