How can I tell if a fraction like $13/3125$ will have a terminating or non-terminating decimal expansion without actually dividing it?

You can determine this by examining the prime factorization of the denominator of the fraction, provided the fraction is in its simplest form (i.e., the numerator and denominator are coprime). A rational number $p/q$ will have a terminating decimal expansion if and only if the prime factorization of its denominator $q$ contains only powers of $2$ and/or $5$. That is, $q$ must be of the form $2^m \cdot 5^n$, where $m$ and $n$ are non-negative integers. If the denominator has any other prime factor besides $2$ or $5$, then the decimal expansion will be non-terminating repeating. For your example, $13/3125$, first check if it's in simplest form (yes, $13$ is prime and $3125$ is not divisible by $13$). Then, prime factorize the denominator: $3125 = 5^5$. Since the denominator only has $5$ as a prime factor (which fits the $2^m \cdot 5^n$ form with $m=0, n=5$), it will have a terminating decimal expansion.