Operator square roots are ubiquitous in theoretical physics. They appear, for example, in the Holstein-
Primakoff representation of spin operators and in the Klein-Gordon equation. Often the use of a perturbative
expansion is the only recourse when dealing with them. In this paper, we show that under certain conditions,
differential equations can be derived which can be used to ﬁnd perturbatively inaccessible approximations to
operator square roots. Speciﬁcally, for the number operator nˆ = a
†a we show that the square root
√nˆ near
nˆ = 0 can be approximated by a polynomial in nˆ. This result is unexpected because a Taylor expansion fails. A
polynomial expression in nˆ is possible because nˆ is an operator, and its constituents a and a
† have a non trivial
commutator [a, a
†] = 1 and do not behave as scalars. We apply our approach to the zero-mass Klein-Gordon
Hamiltonian in a constant magnetic ﬁeld and, as a main application, the Holstein-Primakoff representation of
spin operators, where we are able to ﬁnd new expressions that are polynomial in bosonic operators. We prove
that these new expressions exactly reproduce spin operators. Our expressions are manifestly Hermitian, which
offers an advantage over other methods, such as the Dyson-Maleev representation.