Variation on inverse gamma: $1/X^r \sim \textrm{InvGamma}({}~~,{}~~)$ if $X \sim \textrm{Gamma}(a, b). $ If $X \sim \textrm{Gamma}(a, b), $ then $1/X \sim \textrm{InvGamma}(a, b). $
Therefore if
$X \sim \textrm{Gamma}(a, b), $ then $1/X^r \sim \textrm{InvGamma}({}~~,{}~~)? $
My brain is exploding I don't have a great math skills unfortunately.
 A: If $X \sim G \left( a,b \right)$ is a gamma random variable with shape parameter $a$ and scale parameter $b$ with mean $ab,$ then $X^r$ (where $r>0$) is a generalized gamma random variable with three parameters. The first shape parameter is $a,$ the scale parameter will be $b^r,$ and the second shape parameter is $1 \over r$ .
We write it as $Y=X^r \sim GG \left(a,b^r,{1 \over r} \right)$
There is more detail given here: http://mathscience.kiau.ac.ir/Content/Vol4No1/2.pdf 
The reciprocal of $Y$ has a generalized inverse gamma distribution. A reference for that is here: https://arxiv.org/pdf/1005.3274.pdf 
A: Did you check Wikipedia? Answers can be found there, but must be put together from various pages. First, https://en.wikipedia.org/wiki/Inverse-gamma_distribution  the inverse gamma distribution is the distribution of the inverse $1/X$ when $X$ has a gamma distribution, so by reading that definition in reverse, the inverse of an inverse gamma random variable must have a gamma distribution.  You ask for the distribution of the $r$th power of that, that is, the $r$th power of a gamma random variable. According to https://en.wikipedia.org/wiki/Gamma_distribution#Related_distributions_and_properties  that have a generalized gamma distribution, which you can read about here:  https://en.wikipedia.org/wiki/Generalized_gamma_distribution
