Yes, unless the rational number is 0.
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Only if the rational number is 0.
It can be a rational number or an irrational number. For example, sqrt(2)*sqrt(50) = 10 is rational. sqrt(2)*sqrt(51) = sqrt(102) is irrational.
Any and every rational number.
Irrational.
The product of a rational and irrational number can be rational if the rational is 0. Otherwise it is always irrational.