In your solution, the "a=b" part is correct.

In the second part,  you have only find the solutions in the form a=b^r with r being 1, 3, 4 or any real bigger than 4.  You have not deal with the other r's, (eg. r=1.5)

Finally, what about solutions that is not in the form a=b^r?  You need to show no such solution exist before making your conclusion.