Digits of powers of $2$ and $3$

Question

Does there exist a positive power of $2$ whose digits in decimal representation can be rearranged to form a power of $3$?

How do we deal with the last digits of powers of $3$? We know that a power of $3$ ends in $1,3,9,7,$ but how do we deal with the other digits?

Answer 1

The answe is no :

If you rearrange the digits of a power of $2$, the digit-sum will not be divisble by $3$, hence the number will not be divisible by $3$. In particular, it will not be a power of $3$.

Answer 2

$512 = 2^9$

seems to work fine compared with $125=5^3$