Binary representation of the inverse of a big number
The first $k$ digits on the right of the decimal point of the representation of $1/p$ in base $b$ also are the representation in base $b$ of $\left\lfloor b^k/p\right\rfloor$ left-padded to $k$ digits with zeroes.

If we want to perform computations to $k$ places after the decimal point for quantity $x\in\Bbb R$, we can use the integer quantity $\left\lfloor b^k\,x\right\rfloor$, or better $\left\lfloor b^k\,x+\frac12\right\rfloor$.

Any language (e.g. python) or library (e.g. GMP) capable of handling arbitrary precision integers let you make the necessary computations.

E.g. in python 3,

p = 3**161
print("{0:b}".format(2**700//p))


yields a 445-bit bitstring which, padded with 255 0 on the left, and then 0. on the left, is $1/p$ in binary to 700 binary places.