I am completely stuck with the following problem, because I do not know how to start:
Let $X_1,...,X_n$ be independent and exponentially distributed with unknown parameter , and let $Y=min(X_1,...,X_n)$. It can be shown that $Y$ is exponentially distributed with parameter $\frac \beta n$. Consider testing $H_0: \beta=\beta_{0}$ versus $H_1: \beta>\beta_{0}$. Suppose the test is performed by rejecting $H_0$ when $Y>c$. Find the power function for this test.
The power function of a test with rejection region $R$ is defined by $\beta(\theta)=\mathbb P_\theta(X \in R)$.
Now I would start off by defining $\beta \left(\frac \beta n \right)=\mathbb P_\theta(Y>c)$, but I don't know what to do from there. My intuition would be to find the density of $Y$, if possible, and work towards $=\mathbb P_\theta(Z>...)$ where $Z$ denotes a standard Normal rv, and ... is the result of applying the same transformations to $c$ that I applied to $Y$ to get $Z$.
However, I might be thinking in the wrong direction. I would be thankful for any hints to help me get started on this problem!
Transforming to standard normal is not necessary nor helpful. The distribution of $Y$ is actually given in the question! Look up how to compute $P(Y>c)$ when $Y$ follows that distribution, and you are done.
If the distribution of $Y$ wasn't given, you could find the distribution of $Y$ and/or the probability of the event $Y>c$ based on the definitions of $Y$ and the $X_i$s, but that seems to be not required in this exercise.
Note also that the argument of the power function should be the parameter of interest ($\beta$), not the parameter of $Y$ ($\beta/n$) as you have written.