We study fluctuation effects in a two species reaction-diffusion system, with three competing reactions $A+A→\emptyset$, $B+B→\emptyset$, and $A+B→\emptyset$. Asymptotic density decay rates are calculated for $d≤ 2$ using two separate methods - the Smoluchowski approximation, and also field theoretic/renormalisation group (RG) techniques. Both approaches predict power law decays, with exponents which asymptotically depend only on the ratio of diffusion constants, and not on the reaction rates. Furthermore, we find that, for $d<2$, the Smoluchowski approximation and the RG improved tree level give identical exponents. However, whereas the Smoluchowski approach cannot easily be improved, we show that the RG provides a systematic method for incorporating additional fluctuation effects. We demonstrate this advantage by evaluating one loop corrections for the exponents in $d<2$, and find good agreement with simulations and exact results. Comment: LaTeX file (41 pages) + 13 postscript figures, uuencoded