We present the results of a pilot study search for Fast Radio Bursts (FRBs) using the Murchison Widefield Array (MWA) at low frequencies (139 - 170 MHz). We utilised MWA data obtained in a routine imaging mode from observations where the primary target was a field being studied for Epoch of Reionisation detection. We formed images with 2 second time resolution and 1.28~MHz frequency resolution for 10.5 hours of observations, over 400 square degrees of the sky. We de-dispersed the dynamic spectrum in each of 372,100 resolution elements of 2$times$2 arcmin$^{2}$, between dispersion measures of 170 and 675~pc~cm$^{-3}$. Based on the event rate calculations in Trott, Tingay & Wayth (2013), which assumes a standard candle luminosity of $8times10^{37}$ Js$^{-1}$, we predict that with this choice of observational parameters, the MWA should detect ($sim10$,$sim2$,$sim0$) FRBs with spectral indices corresponding to ($-$2, $-$1, 0), based on a 7$sigma$ detection threshold. We find no FRB candidates above this threshold from our search, placing an event rate limit of $<700$ above 700 Jy.ms per day per sky and providing evidence against spectral indices $alpha<-1.2$ ($Spropto u^{alpha}$). We compare our event rate and spectral index limits with others from the literature. We briefly discuss these limits in light of recent suggestions that supergiant pulses from young neutron stars could explain FRBs. We find that such supergiant pulses would have to have much flatter spectra between 150 and 1400 MHz than have been observed from Crab giant pulses to be consistent with the FRB spectral index limit we derive.
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