Prove that (1 + i)5 (1 + 1/i) 5 = 32 - Maths - Complex Numbers and Quadratic Equations

Question

Prove that (1 + i)5 (1 + 1/i) 5 = 32

Shridhar Kaushik · Accepted Answer

Taking L H S $(1 + i)^{5} {(1 + \frac{1}{i})}^{5} = (1 + i)^{4} (1 + i) (1 + \frac{1}{i})^{4} (1 + \frac{1}{i}) = {(1 + i)^{2}}^{2} (1 + i) {(1 + \frac{1}{i})^{2}}^{2} (1 + \frac{1}{i}) = {1 + i^{2} + 2 i}^{2} (1 + i) {1 + \frac{1}{i^{2}} + \frac{2}{i}}^{2} (1 + \frac{1}{i}) = {1 - 1 + 2 i}^{2} (1 + i) {1 - 1 + \frac{2}{i}}^{2} (1 + \frac{1}{i})$ $= 4 i^{2} (1 + i) \frac{4}{i^{2}} (1 + \frac{1}{i}) = 4 (- 1) (1 + i) \frac{4}{- 1} (1 + \frac{1}{i}) = 16 (1 + i) \frac{(i + 1)}{i} = 16 \frac{(1 + i)^{2}}{i} = 16 \frac{(1 + i^{2} + 2 i)}{i} = 16 \frac{(1 - 1 + 2 i)}{i} = 16 \times \frac{2 i}{i} = 32 = R H S$