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By analyzing 482 pb$^{-1}$ of $e^+e^-$ collision data collected at $\sqrts=4.009$ GeV with the BESIII detector at the BEPCII collider, we measure theabsolute branching fractions for the semileptonic decays $D_{s}^{+}\to\etae^{+}\nu_{e}$ and $D_{s}^{+}\to \eta'e^{+}\nu_{e}$ to be${B}(D_{s}^{+}\rightarrow\eta e^{+}\nu_{e})=(2.30\pm0.31\pm0.08)$\% and${B}(D_{s}^{+}\rightarrow\eta'e^{+} \nu_{e}) = (0.93\pm0.30\pm0.05)$\%,respectively, and their ratio$\frac{{B}(D_{s}^{+}\rightarrow\eta'e^{+}\nu_{e})}{{B}(D_{s}^{+}\rightarrow\eta e^{+}\nu_{e})}=0.40\pm0.14\pm0.02$, where thefirst uncertainties are statistical and the second ones are systematic. Theresults are in good agreement with previous measurements within uncertainties;they can be used to determine the $\eta-\eta'$ mixing angle and improve uponthe $D_s^+$ semileptonic branching ratio precision.

Measurements of the absolute branching fractions for $D_{s}^{+}\rightarrow\eta e^{+}\nu_{e}$ and $D_{s}^{+}\rightarrow\eta^{\prime} e^{+}\nu_{e}$