  In quantum mechanics, bosonic operators are mathematical objects that are
used to represent the creation ($a^\dagger$) and annihilation ($a$) of bosonic
particles. The natural power of a linear combination of bosonic operators
represents an operator $(a^\dagger x+ay)^n$ with $n$ as the exponent and
$x,\,y$ are the variables free from bosonic operators. The normal ordering of
these operators is a mathematical technique that arranges the operators so that
all the creation operators are to the left of the annihilation operators,
reducing the number of terms in the expression. In this paper, we present a
general expansion of the natural power of a linear combination of bosonic
operators in normal order. We show that the expansion can be expressed in terms
of binomial coefficients and the product of the normal-ordered operators using
the direct method and than prove it using the fundamental principle of
mathematical induction. We also derive a formula for the coefficients of the
expansion in terms of the number of bosons and the commutation relation between
the creation and annihilation operators. Our results have important
applications in the study of many-body systems in quantum mechanics, such as in
the calculation of correlation functions and the evaluation of the partition
function. The general expansion presented in this paper provides a powerful
tool for analyzing and understanding the behavior of bosonic systems, and can
be applied to a wide range of physical problems.
