Question

How do you simplify `5i^32 + 7i^11 + 3i^2 - i^17` in `a + bi` form?

Answer 1

`5i^32+7i^11+3i^2-i^17=2-8i`

Explanation:

First, you need to know that the imaginary number `i=sqrt(-1)`.
Let's write out some powers of `i`:
`i=sqrt(-1)`
`i^2=i*i=sqrt(-1)*sqrt(-1)=-1`
`i^3=i^2*i=-1i=-i`
`i^4=i^2*i^2=-1*-1=1`
`i^5=i^4*i=1i=i`
This pattern continues...
I encourage you to try it out. It's much easier with a pencil and paper than here with all the codes.
Essentially, for each power you raise `i` to, find the number closest to that power that divides evenly by 4 and use it to break the exponent down into smaller powers of `i`.

In this problem, we have:
`i^32`, which may be written as `(i^4)^8=(1)^8=1`
`i^11`, which is `(i^4)^2*i^3=1*i^3=-i`
`i^2`, which is just `-1`,
and
`i^17`, which is `(i^4)^4*i=1*i=i`

Plug these simplified values in your original expression:
`5(1)+7(-i)+3(-1)-i`
and treat `i` like a variable to simplify:
`5-7i-3+i=2-8i`

If you're confused about those middle steps, watch a video:
https://goo.gl/a3K93B

Hope that helps! Imaginary numbers are cool!