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We present measurements of the $E$-mode polarization angular auto-powerspectrum ($EE$) and temperature-$E$-mode cross-power spectrum ($TE$) of thecosmic microwave background (CMB) using 150 GHz data from three seasons ofSPTpol observations. We report the power spectra over the spherical harmonicmultipole range $50 < \ell \leq 8000$, and detect nine acoustic peaks in the$EE$ spectrum with high signal-to-noise ratio. These measurements are the mostsensitive to date of the $EE$ and $TE$ power spectra at $\ell > 1050$ and $\ell> 1475$, respectively. The observations cover 500 deg$^2$, a fivefold increasein area compared to previous SPTpol analyses, which increases our sensitivityto the photon diffusion damping tail of the CMB power spectra enabling tighterconstraints on \LCDM model extensions. After masking all sources withunpolarized flux $>50$ mJy we place a 95% confidence upper limit on residualpolarized 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 taildominates foregrounds to at least $\ell = 4050$ with modest source masking. Wefind that the SPTpol dataset is in mild tension with the $\Lambda CDM$ model($2.1\,\sigma$), and different data splits prefer parameter values that differat the $\sim 1\,\sigma$ level. When fitting SPTpol data at $\ell < 1000$ wefind cosmological parameter constraints consistent with those for $Planck$temperature. Including SPTpol data at $\ell > 1000$ results in a preference fora higher value of the expansion rate ($H_0 = 71.3 \pm2.1\,\mbox{km}\,s^{-1}\mbox{Mpc}^{-1}$ ) and a lower value for present-daydensity fluctuations ($\sigma_8 = 0.77 \pm 0.02$).

Measurements of the Temperature and E-Mode Polarization of the CMB from 500 Square Degrees of SPTpol Data