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Question
Find the value of the following expression:
(1 + i)6 + (1 − i)3
Solution
(1 + i)6 + (1 − i)3
= [(1 + i)2]3 + (1 − i)3
= [12 + i2 + 2i]3 + (13 − i3 + 3i2 − 3i)
= [1 − 1 + 2i]3 + (1 + i − 3 − 3i) [∵ i2 = −1, i3 = −i]
= (2i)3 + (−2 − 2i)
= 8i3 − 2 − 2i
= −8i − 2 − 2i [∵ i3 = −i]
= −10i − 2
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