are identically exponentially distributed.
find the likelihood ratio test of size alpha of
find the expression for the power function.
BE very careful here.
When you say gamma or exponential you NEED to be more precise.
DO you mean...
or...
Most people use the first one.
BUT the solution just posted used the second.
HENCE the answers to these two are either the sample mean
or the reciprocal of the sample mean, since one theta is the reciprocal of the other.