The value of
Question:

The value of $(1+i)^{4}+(1-i)^{4}$ is

(a) 8

(b) 4

(c) −8

(d) −4

Solution:

(c) −8

Using $a^{4}+b^{4}=\left(a^{2}+b^{2}\right)^{2}-2 a^{2} b^{2}$

$(1+i)^{4}+(1-i)^{4}$

$=\left((1+i)^{2}+(1-i)^{2}\right)^{2}-2(1+i)^{2}(1-i)^{2}$

$=\left(1+i^{2}+2 i+1+i^{2}-2 i\right)^{2}-2\left(1+i^{2}+2 i\right)\left(1+i^{2}-2 i\right)$

$=(1-1+2 i+1-1-2 i)^{2}-2(1-1+2 i)(1-1-2 i)$

$=(0)-2(2 i)(-2 i) \quad\left(\because i^{2}=-1\right)$

$=8 i^{2}$

$=-8$

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