Simplify $8i^6+6i^5-5i^3-3i^2-7i-9$

Question

Simplify $8i^6+6i^5-5i^3-3i^2-7i-9$.

Answer choices:

1. $-14 +4i$
2. $-4+4i$
3. $-10i$
4. $-14-18i$

When I solved this, none of the choices matched. First, I assumed if $i$ had an even exponent generally the answer is $-1$ and simplified my equation into the following steps.

1. $-8+6i^5-5i^3+3-7i-9$

2. $-14+6i^5-5i^3-7i$

I do not know if there is another to further simplify or if these are the correct first steps. How would you answer this problem?

Answer 1

Observe that $i^6= -1,i^5=i,i^4 =1,i^3 = -i,i^2= -1.$ So the answer to your question is $-14 + 4i.$

Answer 2

Hint:

$$i^{4k+n}=i^n$$

$$i^1=i,\ i^2=-1,\ i^3=-i,\ i^4=1$$