algebra precalculus - Why doesn't simplifying $i$ using division not seem to work?

I was wondering why simplifying $i$ doesn't seem to work with division the same way it does multiplication.

For example, the following works:

$i^4 = 1$

$i^{31} = (i^4)^7 \cdot i^3 = 1 \cdot (-1) \cdot i = -i$

But not when applying the rules of exponents to this concept with division:

$i^{31} = \cfrac{(i^4)^8}{i^1} = \cfrac{1}{i} = \cfrac{1}{\sqrt{-1}} \ne -i$

Have I messed things up with the rules?