It is possible for an irrational base to represent the same number using two finite strings. For example, there is a number $\varphi>1$ such that $\varphi^2-\varphi-1=0$ and therefore $100_\varphi=11_\varphi$.
Is this also possible for a rational base?
(This is a follow-up to the comments on the question "Are exact representations of numbers in fractional bases unique?")
No.
Suppose there is a rational $b>1$ such that $\sum_{k=0}^na_kb^k=0$ and the integer coefficients $a_k$ are not all zero. Without loss of generality, $a_0$ and $a_n$ are nonzero. And since this polynomial came from comparing two $b$-expansions, suppose that each $|a_k|<b$.
Let $b=p/q$ in lowest terms. By the rational root theorem, $p$ divides $a_0$, so $p\leq |a_0|<b=p/q$, which is absurd.