I am lost on how to divide this problem and why they won't offer examples in the book is mind boggling.
8x^4y^2/4x^7y^2
I will assume you meant
8x^4y^2/(4x^7y^2)
remember you can only multiply or divide powers if the have the same base, so do all like "factors"
8 divides by 4 leaving 2 on top
x^4 divides by x^7 leaving x^3 at the bottom
y^2 divides by y^2 giving you 1, in effect the y^2 cancels
so you are left with
2/x^3
if you understand negative exponents then the answer could be written as 2x^(-3)