While the exact solution depends on the actual problem, there are various approaches you can try to take before simply calculating in brute force using an arbitrary or multiple precision math library (BigInt, GMP, MPFR, ARPREC, etc.).
First, use some math. Can the expressions be simplified? You said that the source of these tasks involves puzzles, so I would be very inclined to look at this approach for a solution, as this may be a factor in the puzzles' aha moment. Equations with factorials, such as binomial probabilities can typically be simplified or calculated (indirectly) using mathematical techniques rather than brute force.
Factoring the numbers and cancelling common factors would be one of the first things I would try, by hand if need be. A multiple precision calculator can helpful.
Would re-framing the question or its values in either a different base (e.g. binary, hexadecimal) or a difference scale (e.g. logarithmic base 2, 10, or e -- natural) make the values easier to deal with? One example of a logarithmic scale, is the decibel, used for RF, and audio levels.
Using techniques not as commonly taught nowadays, but well known amongst programmers, engineers, mathematicians who are familiar with the slide rule can sometime be helpful.
Depending on the question, doing an approximation first can sometimes lead you towards the correct answer by preventing the minutiae from distracting you from attacking the problem creatively.
For your example; calculate a related (approximate), but simplified equation.
which is very close to the correct or exact answer
Another "trick" is to use modulo (
a \bmod n ) which is one of my favourite ways to reduce numbers, so if you know some basic abstract algebra you can sometimes work with modular arithmetic.
That is an off-the-cuff, very rough guide to how I would approach a "puzzle" equation or programming problem that involves large numbers.