  In this paper, we study polar quotients and \L ojasiewicz exponents of plane
curve singularities, which are {\em not necessarily reduced}. We first show
that the polar quotients is a topological invariant. We next prove that the \L
ojasiewicz gradient exponent can be computed in terms of the polar quotients,
and so it is also a topological invariant. As an application, we give effective
estimates of the \L ojasiewicz exponents in the gradient and classical
inequalities of polynomials in two (real or complex) variables.
