  By analyzing 482 pb$^{-1}$ of $e^+e^-$ collision data collected at $\sqrt
s=4.009$ GeV with the BESIII detector at the BEPCII collider, we measure the
absolute branching fractions for the semileptonic decays $D_{s}^{+}\to\eta
e^{+}\nu_{e}$ 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 the
first uncertainties are statistical and the second ones are systematic. The
results are in good agreement with previous measurements within uncertainties;
they can be used to determine the $\eta-\eta'$ mixing angle and improve upon
the $D_s^+$ semileptonic branching ratio precision.
