  In this paper we consider the Waring rank of monomials over the real and the
rational numbers. We give a new upper bound for it by establishing a way in
which one can take a structured apolar set for any given monomial
$X_0^{a_0}X_1^{a_1}\cdots X_n^{a_n}$ ($a_i>0$). This bound coincides with the
real Waring rank in the case $n=1$ and in the case $\min(a_i)=1$, which are all
the known cases for the real rank of monomials. Our bound is also lower than
any other known general bounds for the real Waring rank. Since all of the
constructions are still valid over the rational numbers, this provides a new
result for the rational Waring rank of any monomial as well. Some examples and
computational implementation for potential use are presented in the end.
