Question

Find the value of
$(1+i{)}^6+(1-i{)}^6$

Answer 1

Simplifying ${\left(1+i\right)}^6+{\left(1-i\right)}^6$.

${\left(1+i\right)}^6+{\left(1-i\right)}^6={\left({\left(1+i\right)}^2\right)}^3+{\left({\left(1-i\right)}^2\right)}^3$

$={\left(1+i^2+2i\right)}^3+{\left(1+i^2-2i\right)}^3$

$={\left(1-1+2i\right)}^3+{\left(1-1-2i\right)}^3$

$=8i^3-8i^3$

$=8\left(-i\right)-8\left(-i\right)$

$=-8i+8i$

$=0$