Representing an integer and its powers in two unrelated number systems

We prove that under a necessary independence assumption, there are only finitely many integers having bounded sum-of-digits in base b1 while also having some integral power with bounded sum-of-digits in base b2 where b1 and b2 are coprime. These powers can also be effectively listed in principle. Furthermore, analogous results involving both radix and Ostrowski numeration systems have been established.