Operator acting on the function

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Homework Statement


Calculate the action of the operator on the function f(x)



Homework Equations


Operator - exp(a*x^2*(d/dx))


The Attempt at a Solution


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I don't really see what you need the commutator expansion for, to be honest.
Why don't you just expand the exponential in a power series?
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

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