Summary: Let \(H_k\) denote the space of primitive holomorphic cusp forms of even integral weight \(k\) for the full modular group \(\Gamma = \mathrm{SL} (2, \mathbb{Z})\). Denote by \(\lambda_{\mathrm{sym}^m f}(n)\) the \(n\)th normalized coefficient of the Dirichlet expansion of the \(m\)th symmetric power \(L\)-function associated to \(f\). In this paper, we establish the asymptotic formulas of sums of pairwise maxima concerning the normalized coefficients of symmetric power \(L\)-functions. We also establish similar results for the normalized coefficients of Rankin-Selberg \(L\)-functions \(L(\mathrm{sym}^i f \times \mathrm{sym}^j f,s)\) and \(L(\mathrm{sym}^i f \times \mathrm{sym}^j g,s)\) attached to \(f\) and \(g\), respecitvely. As applications, we also consider the proportion of the sign changes among the difference of normalized coefficients associated with symmetric power \(L\)-functions and Rankin-Selberg \(L\)-functions attached to \(f\) and \(g\), respectively.