  We present measurements of the $E$-mode polarization angular auto-power
spectrum ($EE$) and temperature-$E$-mode cross-power spectrum ($TE$) of the
cosmic microwave background (CMB) using 150 GHz data from three seasons of
SPTpol observations. We report the power spectra over the spherical harmonic
multipole range $50 < \ell \leq 8000$, and detect nine acoustic peaks in the
$EE$ spectrum with high signal-to-noise ratio. These measurements are the most
sensitive to date of the $EE$ and $TE$ power spectra at $\ell > 1050$ and $\ell
> 1475$, respectively. The observations cover 500 deg$^2$, a fivefold increase
in area compared to previous SPTpol analyses, which increases our sensitivity
to the photon diffusion damping tail of the CMB power spectra enabling tighter
constraints on \LCDM model extensions. After masking all sources with
unpolarized flux $>50$ mJy we place a 95% confidence upper limit on residual
polarized point-source power of $D_\ell = \ell(\ell+1)C_\ell/2\pi
<0.107\,\mu{\rm K}^2$ at $\ell=3000$, suggesting that the $EE$ damping tail
dominates foregrounds to at least $\ell = 4050$ with modest source masking. We
find that the SPTpol dataset is in mild tension with the $\Lambda CDM$ model
($2.1\,\sigma$), and different data splits prefer parameter values that differ
at the $\sim 1\,\sigma$ level. When fitting SPTpol data at $\ell < 1000$ we
find cosmological parameter constraints consistent with those for $Planck$
temperature. Including SPTpol data at $\ell > 1000$ results in a preference for
a higher value of the expansion rate ($H_0 = 71.3 \pm
2.1\,\mbox{km}\,s^{-1}\mbox{Mpc}^{-1}$ ) and a lower value for present-day
density fluctuations ($\sigma_8 = 0.77 \pm 0.02$).
