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Let $r_n*\in {ℛ}_{nn}$ be the best rational approximant to $f\left(x\right)=x^{\alpha }$, 1 > α > 0, on [0,1] in the uniform norm. It is well known that all poles and zeros of $r_n*$ lie on the negative axis ${ℝ}_{<0}$. In the present paper we investigate the asymptotic distribution of these poles and zeros as n → ∞. In addition we determine the asymptotic distribution of the extreme points of the error function $e_n=f-r_n*$ on [0,1], and survey related convergence results.