Assume that the product of a rational and an irrational can be rational. In other words:
$$\frac{a}{b} * k = \frac{c}{d}$$
Rearranging, we get:
$$\begin{align} k &= \frac{c}{d} * \frac{b}{a} \\ &= \frac{cb}{da} \end{align}$$
This shows that \(k\) is a rational, which is a contradiction.
Therefore, a product of a rational and an irrational is always irrational.