Find the value of following expression: (1 + i)6 + (1 – i)3

1.	Find the value of following expression: (1 + i)6 + (1 – i)3
Answer» (1 + i)⁶ + (1 – i)³ = {(1 + i)² }³ + (1 – i)²(1 – i) = {1 + i² + 2i}³ + (1 + i² – 2i)(1 – i) = {1 – 1 + 2i} ³ + (1 – 1 – 2i)(1 – i) = (2i)³ + (– 2i)(1 – i) = 8i³ + (– 2i) + 2i² [since i³ = – i, i² = – 1] = – 8i – 2i – 2 = – 10 i – 2 = – 2(1 + 5i)

Answer»

(1 + i)⁶ + (1 – i)³ = {(1 + i)² }³ + (1 – i)²(1 – i)

= {1 + i² + 2i}³ + (1 + i² – 2i)(1 – i)

= {1 – 1 + 2i} ³ + (1 – 1 – 2i)(1 – i)

= (2i)³ + (– 2i)(1 – i)

= 8i³ + (– 2i) + 2i²

[since i³ = – i, i² = – 1]

= – 8i – 2i – 2

= – 10 i – 2

= – 2(1 + 5i)

Discussion