Question

The value of $(1+i{)}^4+(1-i{)}^4$ is
(a) 8
(b) 4
(c) −8
(d) −4

Answer 1

(c) $-$8

$$\begin{gathered} \mathrm{Using}{a}^4+b^4={\left(a^2+b^2\right)}^2-2a^2{b}^2 \\ (1+i{)}^4+(1-i{)}^4 \\ ={\left({\left(1+i\right)}^2+{\left(1-i\right)}^2\right)}^2-2{\left(1+i\right)}^2{\left(1-i\right)}^2 \\ ={\left(1+i^2+2i+1+i^2-2i\right)}^2-2\left(1+i^2+2i\right)\left(1+i^2-2i\right) \\ ={\left(1-1+2i+1-1-2i\right)}^2-2\left(1-1+2i\right)\left(1-1-2i\right) \\ =\left(0\right)-2\left(2i\right)\left(-2i\right)\left(\because i^2=-1\right) \\ =8i^2 \\ =-8 \end{gathered}$$