Answer: The claim is true
The proof is below
=====================================================
A = some nonzero rational number
B = some irrational number
Because value A is rational, this means A = p/q where p,q are nonzero integers.
Let's do a proof by contradiction. The claim is that A*B is irrational, but let's do the opposite and assume A*B is rational. We should get to a contradiction somewhere based on this assumption.
If A*B is rational, then A*B = r/s for some integers r and s
Then we can have the following steps
A*B = r/s
(p/q)*B = r/s
B = (r/s)*(q/p)
B = (rq)/(sp)
B = (some integer)/(some other nonzero integer)
B = some rational number
Recall we made B irrational. There's no way to have B rational and irrational at the same time. This is where the contradiction happens. This causes the original assumption of "A*B is rational" to be false. Therefore, the opposite must be the case and A*B has been proven to be irrational.
Therefore, the product of a rational number and an irrational number is irrational.
------------------------
Example:
5 is rational
sqrt(3) is irrational
5*sqrt(3) is irrational