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!