Question

$i^5+i^{-25}+i^{45}$
`$i^5+i^(-25)+i^(45)$`

Answer 1

1/i +2i

Explanation:

i^5+i^-25+i^45 i^2=-1

i^4*i +i^-24*i^-1 +i^44 *i

(i²)² i+ (i²)^-12*i^-1+(i²)^22 *i since i²=1

i+i^-1+i=

i+1/i +i=1/i +2i

Answer 2

`i^5+i^(-25)+i^(45)`

Rewrite the term
`i^(-25)`

`=i^5+(1)/(i^(25))+i^(45)`

Expand each term so we have

`=i(i^2)^2+(1)/(i(i^2)^(12))+i(i^2)^(22)`

Use the fact that
`i^2=-1`

`=i(-1)^2+(1)/(i(-1)^(12))+i(-1)^(22)`

Use the fact that
`(-1)^(a)=1` when a is an even number

`=i+(1)/(i)+i`

Simplify

`=i-i+i`

`=i`

Let me know if you need any clarifications, thanks!