The product will be some (other) irrational number.
If the rational number is 1, then 1 times the irrational number is itself. Nothing changes.
If the rational number is not 1 (and also nonzero), and you multiply it with some irrational number, then the result will be some other irrational number.
Rule:
x*y = z
x is rational, y is irrational, z is irrational