The binomial expansion of the Left Hand Side
contains two terms, among many, which are
a^a*n and b^a*n which appear to make the LHS greater than the RHS,
but when we assign arbitrary values,
say a=10, b=1,000,000,000 and n=100
the LHS is (1,000,000,010)^1000, which would contain 9,001 digits;
the RHS becomes
100 x (100^1,000,000,000) which would contain more than 2 billion digits!
This happened because we assumed b>>n.
Otherwise, the LHS may be greater.
Say, when a=10, b=100, n=1000.
LHS would be 110^10,000 containing 20,414 digits and the RHS would be much smaller, viz. 100*(1000^100), containing approximately 300 digits!