The product of two rational numbers is always rational
Proof:
Let a = p/q and b = r/s be two rational numbers. We will have p,q,r,s be integers where denominators q and s are nonzero. We cannot divide by zero.
Multiplying a and b gives
a*b = (p/q)*(r/s) = (p*r)/(q*s)
Due to the closure property of integer multiplication, the expressions p*r and q*s are integers. The result we get is in the form of integer/integer. Basically we have two integers being divided. Furthermore, q*s is nonzero as neither q nor s is zero.
Therefore, if a and b are rational, then a*b is also rational as well.