  The first two terms in the large $N$ asymptotic expansion of the $\beta$
moment of the characteristic polynomial for the Gaussian and Laguerre
$\beta$-ensembles are calculated. This is used to compute the asymptotic
expansion of the spectral density in these ensembles, in the exponentially
small region outside the leading support, up to terms $o(1)$ . The leading form
of the right tail of the distribution of the largest eigenvalue is given by the
density in this regime. It is demonstrated that there is a scaling from this,
to the right tail asymptotics for the distribution of the largest eigenvalue at
the soft edge.
