Consider the remainder of the division of the power exponent of i by 4.
For a remainder r equal to 0, 1, 2 or 3, we have a power equal to 1, i, -1 or -i respectively, therefore:
[tex]8i^7+6i^5-5i^3-3i^2-7i-9 =[/tex]
[tex]8*(-i) + 6*i - 5*(-i) - 3*(-1) - 7i - 9 =[/tex]
[tex]- 8i + 6i + 5i + 3 - 7i - 9 =[/tex]
[tex]3 - 9 - 8i + 6i + 5i- 7i =[/tex]
[tex]=\boxed{- 6 - 4i}[/tex]
Answer:
[tex]\boxed{- 6 - 4i}[/tex]
