In this paper we prove a blow-up result for the semi linear system of weakly coupled effectively damped waves with different power nonlinearities \begin{align*} \begin{split} &u_{tt}-\Delta u+b(t)u_{t}=|v|^{p}, \quad v_{tt}-\Delta v+b(t)v_{t}= |u|^{q}, &u(0,x)=u_{0}(x),\quad u_{t}(0,x)=u_{1}(x),\quad v(0,x)=v_{0}(x),\quad v_{t}(0,x)=v_{1}(x), \end{split} \end{align*} where $b(t)$ will be explained in detail in the next sections. We apply the so called ``test function method'' to determine the range for the exponents $p,q>0$ in the nonlinear terms in which local in time existence may not globally prolonged with respect to the $t$ variable under suitable integral sign assumptions for the Cauchy data $u_0,u_1,v_0,v_1$. Since we prove the blow-up in a complementary range for powers of the nonlinear terms to that for the global existence of small data solutions (see \cite{DjaoutiReissig}), the main blow-up of this paper is optimal.