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.
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Only if the rational number is 0.
No.
The sum 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.
Yes, it will always be irrational.