[tex]i^5+i^{-25}+i^{45}[/tex]
Rewrite the term [tex]i^{-25}[/tex]
[tex]=i^5+\dfrac{1}{i^{25}}+i^{45}[/tex]
Expand each term so we have
[tex]=i(i^2)^2+\dfrac{1}{i(i^2)^{12}}+i(i^2)^{22}[/tex]
Use the fact that [tex]i^2=-1[/tex]
[tex]=i(-1)^2+\dfrac{1}{i(-1)^{12}}+i(-1)^{22}[/tex]
Use the fact that [tex](-1)^{a}=1[/tex] when a is an even number
[tex]=i+\dfrac{1}{i}+i[/tex]
Simplify
[tex]=i-i+i[/tex]
[tex]=i[/tex]
Let me know if you need any clarifications, thanks!
