 Power transform of a uniform random variable

The cumulative distribution function of $U^a$ on $[0,1]$ is $z^{1/a}$. Differentiating gives density $z^{1/a-1}/a$ on $(0,1)$, which is the beta distribution probability density function with parameters $(1/a,1)$. This also gives a simple instance of inverse transform sampling: to sample a density $\alpha z^{\alpha -1}$ on $(0,1)$, raise a uniform draw to the power $1/\alpha$.