Question

find the value of i^57 + 1/ i^125

Answer 1

 i^57 + 1/ i^125

= I * I^56 + 1/ I * i^124

= I * {(I)^4}^14  + 1 / I * {(I)^4}^31

Now as we know that-          ( I)^4 =1


so, I + 1/I =( I^2+1)/I
   since I ^2 =  -1

hence ( I^2+1)/I  = 0 ans

Answer 2

we know (i)^4n=1 where n is natural number
now i^57=i^(4*14+1)=(i^4*14).i=i
i^125=i^(4.31+1)=(i^4*31).i=i
also we know 1/i =-i
so, i^57+1/i^125=0