No
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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.
Sqrt(18/2) =Sqrt(9) =3 which is rational.
No. sqrt(2)/pi is not rational.
Yes
sqrt(2) / x is rational only if x is a multiple of sqrt(2). sqrt(2/x) is rational only if x is two times the square of some number (not necessarily a whole number).