Question

Evaluate : $(1+i{)}^6+(1-i{)}^3$.

Answer 1

Simplifying ${\left(1+i\right)}^6+{\left(1-i\right)}^3$.

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

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

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

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

$=8i^3-2-2i$

$=-8i-2-2i$

$=-2-10i$