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Not if the rational number is zero. In all other cases, the product is irrational.
The product of 0 and an irrational is 0 (a rational), the product of a non-zero rational and any irrational is always irrational.
No, but the only exception is if the rational number is zero.
It is usually irrational but it can be rational if the ration number in the pair is zero. So the correct answer is "either".
Unless the rational number is zero, the answer is irrational.