There are no two irrational numbers whose product is a rational number. This can be proven by contradiction.
Suppose that there exist two irrational numbers a and b such that the product ab is rational. Then we can write ab = p/q, where p and q are integers and q is not equal to zero.
Since a is irrational, it cannot be expressed as a ratio of two integers. Similarly, since b is irrational, it cannot be expressed as a ratio of two integers. However, if we multiply both sides of the equation ab = p/q by q, we get:
a = p/(bq)
Since p and q are integers, and b is irrational, the denominator bq is not equal to zero and is also irrational. Therefore, we have expressed a as a ratio of two numbers, one of which is irrational, which contradicts the definition of a irrational number.
Thus, we have shown that it is not possible for the product of two irrational numbers to be rational.