The unreal world -'3'

${ i }^{ 37 }={ \quad i }^{ \left( 4\times 9 \right) }\times i\quad =\quad \left( { i }^{ 4 } \right) ^{ 9 }\times i\quad =\quad \left( 1\times i \right) \quad =\quad i.\\ \frac { 1 }{ { i }^{ 67 } } \quad =\quad \frac { 1 }{ { i }^{ 67 } } \times \frac { i }{ i } \quad =\quad \frac { i }{ { i }^{ 68 } } \quad =\quad \frac { i }{ { \left( { i }^{ 4 } \right) }^{ 17 } } \quad =\quad \frac { i }{ 1 } \quad =\quad i.\\ \therefore \quad \left( { i }^{ 37 }+\frac { 1 }{ { i }^{ 67 } } \right) \quad =\quad \left( i+i \right) =\quad 2i.$

$37 \equiv 1 \pmod {4}, ~~-67 \equiv 1 \pmod {4} \\\therefore ~i^{37}=i^1 ,~~\dfrac 1 {i^{67}}=i^{-67}=i^1. ~~\implies Exp.= 2i. \\~~\\$

Dinesh Nath Goswami has nicely solved the problem. I give here only to show the mod. way. I have up voted for him.

nice approach