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Precision measurements of the semileptonic decays $D_s^+ \to \eta e^+ \nu_e$and $D_s^+ \to \eta^\prime e^+ \nu_e$ are performed with 7.33\,fb$^{-1}$ of$e^+e^-$ collision data collected at center-of-mass energies between 4.128 and4.226 GeV with the BESIII detector. The branching fractions obtained are$\mathcal{B}(D_s^+ \to \eta e^{+} \nu_e)$ = $(2.255\pm0.039_{\rm stat}\pm0.051_{\rm syst})\%$ and $\mathcal{B}(D_s^+ \to \eta^{\prime} e^{+} \nu_e)$ =$(0.810\pm0.038_{\rm stat}\pm 0.024_{\rm syst})\%$. Combining these resultswith the $\mathcal{B}(D^+\to\eta e^+ \nu_e)$ and $\mathcal{B}(D^+\to\eta^\primee^+ \nu_e)$ obtained from previous BESIII measurements, the $\eta-\eta^\prime$mixing angle in the quark flavor basis is determined to be $\phi_{\rm P} =(40.0\pm2.0_{\rm stat}\pm0.6_{\rm syst})^\circ$. Moreover, from the fits to thepartial decay rates of $D_s^+ \to \eta e^+ \nu_e$ and $D_s^+ \to \eta^\primee^+ \nu_e$, the products of the hadronic transition form factors$f_+^{\eta^{(\prime)}}(0)$ and the modulus of the $c\to s$Cabibbo-Kobayashi-Maskawa matrix element $|V_{cs}|$ are determined by usingdifferent hadronic transition form factor parametrizations. Based on thetwo-parameter series expansion, the products $f^\eta_+(0)|V_{cs}| =0.4519\pm0.0071_{\rm stat}\pm0.0065_{\rm syst}$ and$f^{\eta^\prime}_+(0)|V_{cs}| = 0.525\pm0.024_{\rm stat}\pm0.009_{\rm syst}$are extracted. All results determined in this work supersede those measured inthe previous BESIII analyses based on the 3.19 fb$^{-1}$ subsample of data at4.178 GeV.

Precision Measurements of $D_s^+ \to \eta e^+ \nu_e$ and $D_s^+ \to \eta^\prime e^+ \nu_e$