Actually the product of a nonzero rational number and another rational number will always be rational.The product of a nonzero rational number and an IRrational number will always be irrational. (You have to include the "nonzero" caveat because zero times an irrational number is zero, which is rational)
It is always irrational.
It is irrational.
The product will be irrational.
The product of an irrational number and a rational number, both nonzero, is always irrational
An irrational number.
The product of a rational and irrational number can be rational if the rational is 0. Otherwise it is always irrational.
Suppose a is rational (and non-zero) and x is irrational. Suppose ax is rational;write ax = b where b is rational.Then x = b/a, and x would be rational, contradiction.
Such a product is always irrational - unless the rational number happens to be zero.