= Solution
Write $Z=U^a$. Since $a>0$, the map is increasing on $[0,1]$, and for $0\leqslant z\leqslant1$,
$$
\mathbb P(Z\leqslant z)=\mathbb P(U\leqslant z^{1/a})=z^{1/a}.
$$
<Differentiation> gives
$$
f_Z(z)=\frac1a z^{1/a-1},\qquad0<z<1,
$$
and zero outside this interval. Since $B(1/a,1)=a$, this is exactly the <beta distribution> <probability density function> with \b[parameters $(1/a,1)$]. The <power transform of a uniform random variable> therefore satisfies $U^a\sim\operatorname{Beta}(1/a,1)$; the second parameter does not depend on $a$.
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