What is the value of $${\left( { - \sqrt { - 1} } \right)^{4n + 3}} + {\left( {{i^{41}} + {i^{ - 257}}} \right)^9},{\text{where }}n \in N?$$

Solution :
Consider,
$$\eqalign{ & {\left( { - \sqrt { - 1} } \right)^{4n + 3}} + {\left( {{i^{41}} + {i^{ - 257}}} \right)^9} \cr & = \,{\left( { - i} \right)^{4n + 3}} + {\left[ {{{\left( {{i^4}} \right)}^{10}}.{i^1} + {{\left( {{i^3}} \right)}^{ - 85}}.{i^{ - 2}}} \right]^9} \cr & = \,{\left( { - i} \right)^{4n + 3}} + {\left[ {i + \frac{1}{{{{\left( {{i^3}} \right)}^{85}}}}.\frac{1}{{{i^2}}}} \right]^9} \cr & = \,{\left( { - i} \right)^{4n + 3}} + {\left( {i + \frac{1}{i}} \right)^9} \cr & = \, - {\left( { - 1} \right)^{4n + 3}}{\left( i \right)^{4n}}{\left( i \right)^3} + {\left( {i - i} \right)^9} \cr & = - \left( 1 \right)\left( { - i} \right) + 0 \cr & = i \cr} $$