This paper analyzes a sample of 489 Spitzer/IRAC sources in the Extended Groth Strip whose spectral energy distributions fit a red power law from 3.6 to 8.0 micron. The median for sources with known redshift is <z>=1.6. Though all or nearly all of the sample are likely to be active galactic nuclei, only 33% were detected in the EGS X-ray survey (AEGIS-X) using 200 ks Chandra observations. The detected sources are X-ray luminous with L_X > 10^43 erg/s and moderately to heavily obscured with N_H > 10^22 cm^-2. Stacking the X-ray-undetected sample members yields a statistically significant X-ray signal, suggesting that they are on average more distant or more obscured than sources with X-ray detections. The ratio of X-ray to mid-infrared fluxes suggests that a substantial fraction of the sources undetected in X-rays are obscured at the Compton-thick level, in contrast to the X-ray-detected sources, all of which appear to be Compton-thin. For the X-ray-detected power-law sources with redshifts, an X-ray luminosity L_X ~ 10^44 erg/s marks a transition between low-luminosity, blue sources dominated by the host galaxy to high-luminosity, red power-law sources dominated by nuclear activity. X-ray-to-optical ratios, infrared variability, and 24 micron properties of the sample are consistent with the identification of infrared power-law sources as active nuclei, but a rough estimate is that only 22% of AGNs are selected by the power law criteria. Comparison of the power-law selection technique and various IRAC color criteria for identifying AGNs confirms that high-redshift samples selected via simple IRAC colors may be heavily contaminated by starlight-dominated objects.
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