Simplify: (1 + i)³ / (1 + i³)

Solution

Simplify numerator: (1 + i)³ Use the identity: (a + b)³ = a³ + 3a²b + 3ab² + b³ So: (1 + i)³ = = 1³ + 3(1²)(i) + 3(1)(i²) + i³ = 1 + 3i + 3(i²) + i³ = 1 + 3i + 3(–1) + i³ = 1 + 3i – 3 + i³ = –2 + 3i + i³ Now calculate i³: i³ = i² × i = (–1) × i = –i So: Numerator = –2 + 3i – i = –2 + 2i Simplify denominator: 1 + i³ = 1 – i (–2 + 2i) / (1 – i) Multiply numerator and denominator by the conjugate of the denominator: Multiply by (1 + i)/(1 + i): = [(–2 + 2i)(1 + i)] / [(1 – i)(1 + i)] Denominator: (1 – i)(1 + i) = 1² – i² = 1 – (–1) = 2 Numerator: (–2 + 2i)(1 + i) = –2(1 + i) + 2i(1 + i) = (–2 – 2i) + (2i + 2i²) = (–2 – 2i + 2i – 2) = (–2 – 2) = –4 So: Final result = –4 / 2 = –2