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The purpose of this paper is to introduce and study new extension ofRako\v{c}evi\'{c}'s property $(w)$ and property $(b)$ introduced byBerkani--Zariouh in \cite{berkani-zariouh1}, in connection with other Weyl typetheorems and recent properties. We prove in particular, the two followingresults: 1. A bounded linear operator $T$ satisfies property $(w_{\pi_{00}})$if and only if $T$ satisfies property $(w)$ and$\sigma_{uf}(T)=\sigma_{uw}(T).$ 2. $T$ satisfies property $(gw_{\pi_{00}})$ ifand only if $T$ satisfies property $(w_{\pi_{00}})$ and$\pi_{0}(T)=p_{0}^a(T).$ Classes of operators are considered as illustratingexamples.

Extended Rako\v{c}evi\'{c}'s property