x=4X10^m and y=2X10^n
xy=8X10^2
hence (4X10^m)(2X10^n) = 8X10^2
so 8X10^(m+n) = 8X10^2
simplified: 10^(m+n) = 10^2 (divide by 8)
hence m+n = 2
x/y=2X10^8
hence (4X10^m)/(2X10^n) = 2X10^8
so 2X10^(m-n) = 2X10^8
simplified: 10^(m-n) = 10^8 (divide by 2)
hence m-n = 8
now m+n = 2
+ m-n = 8
_____________
2m = 10 (since n-n = 0)
hence m = 5 (since 2X5 = 10)
substitute into m+n = 2
5+n = 2
n = 2 - 5
n = -3
so: m=5 and n=-3
Lovely question. Hope this helps. Cheers.