1) (1-i)(1+i)^= (1-i)(1 - 2i + i^2= (1 - i - 2i + 2= 1 - 3i + = 3 - 3i
2) (1-i)^3/(1+i)^= (1 - i)^3 / (1 + i)^= (1 - i)^3 / (1 - 5i - 10 + 10i - 20i^2 + 40 - 40i - 80i^2= (1 - 3i + 3i^2 - i^3) / (-39 - 45i= (1 - 3i + 3(-1) - (-i)) / (-39 - 45i= (1 - 3i - 3 + i) / (-39 - 45i= (-2 - 2i) / (-39 - 45i)
3) (1-i)^3/(1+i)^= (1 - i)^3 / (1 + i)^= (1 - i)^3 / (1 - 7i - 21 + 35i - 35i^2 + 35 - 105i - 105i^2= (1 - 3i + 3i^2 - i^3) / (-91 - 139i - 140i^2= (1 - 3i + 3(-1) - (-i)) / (-91 - 139i - 140(-1)= (1 - 3i - 3 + i) / (-91 - 139i + 140= (-2 - 2i) / (-91 - 139i)
4) (1-2i)^2-(1+i)^3/(3-2i)^3-(2-i)^= (1 - 2i)(1 - 2i) - (1 + i)^3 / (3 - 2i)(3 - 2i)(3 - 2i) - (2 - i)(2 - i= (1 - 2i - 2i + 4i^2) - (1 + 3i + 3i^2 + i^3) / (3 - 2i)(3 - 2i)(3 - 2i) - (4 - 4i - i + i^2= (1 - 4i - 1) - (1 + 3i + 3(-1) - i) / (3 - 6i - 6i + 12i^2)(3 - 6i - 6i + 12i^2)(3 - 6i - 6i + 12i^2) - (4 - 5i + 1= (-4i + 0) - (1 + 3i - 3 - i) / (9 - 18i - 18i + 36i^2)(9 - 18i - 18i + 36i^2)(9 - 18i - 18i + 36i^2) - (5i - 3= (-4i) - (1 - i) / (9 - 36i + 36i^2)(9 - 36i + 36i^2)(9 - 36i + 36i^2) - (5i - 3= 0 - (1 - i) / (9 - 36i - 36 + 36) - (5i - 3= -(1 - i) / (9 - 36i - 36 + 36) - (5i - 3= -(1 - i) / (9 - 36i) - (5i - 3)
1) (1-i)(1+i)^
= (1-i)(1 - 2i + i^2
= (1 - i - 2i + 2
= 1 - 3i +
= 3 - 3i
2) (1-i)^3/(1+i)^
= (1 - i)^3 / (1 + i)^
= (1 - i)^3 / (1 - 5i - 10 + 10i - 20i^2 + 40 - 40i - 80i^2
= (1 - 3i + 3i^2 - i^3) / (-39 - 45i
= (1 - 3i + 3(-1) - (-i)) / (-39 - 45i
= (1 - 3i - 3 + i) / (-39 - 45i
= (-2 - 2i) / (-39 - 45i)
3) (1-i)^3/(1+i)^
= (1 - i)^3 / (1 + i)^
= (1 - i)^3 / (1 - 7i - 21 + 35i - 35i^2 + 35 - 105i - 105i^2
= (1 - 3i + 3i^2 - i^3) / (-91 - 139i - 140i^2
= (1 - 3i + 3(-1) - (-i)) / (-91 - 139i - 140(-1)
= (1 - 3i - 3 + i) / (-91 - 139i + 140
= (-2 - 2i) / (-91 - 139i)
4) (1-2i)^2-(1+i)^3/(3-2i)^3-(2-i)^
= (1 - 2i)(1 - 2i) - (1 + i)^3 / (3 - 2i)(3 - 2i)(3 - 2i) - (2 - i)(2 - i
= (1 - 2i - 2i + 4i^2) - (1 + 3i + 3i^2 + i^3) / (3 - 2i)(3 - 2i)(3 - 2i) - (4 - 4i - i + i^2
= (1 - 4i - 1) - (1 + 3i + 3(-1) - i) / (3 - 6i - 6i + 12i^2)(3 - 6i - 6i + 12i^2)(3 - 6i - 6i + 12i^2) - (4 - 5i + 1
= (-4i + 0) - (1 + 3i - 3 - i) / (9 - 18i - 18i + 36i^2)(9 - 18i - 18i + 36i^2)(9 - 18i - 18i + 36i^2) - (5i - 3
= (-4i) - (1 - i) / (9 - 36i + 36i^2)(9 - 36i + 36i^2)(9 - 36i + 36i^2) - (5i - 3
= 0 - (1 - i) / (9 - 36i - 36 + 36) - (5i - 3
= -(1 - i) / (9 - 36i - 36 + 36) - (5i - 3
= -(1 - i) / (9 - 36i) - (5i - 3)