Question

If x is a non-zero rational and √y is irrational, then show that x√y is irrational.

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Answer 1

let us assume, to the contrary, that x√y is rational.

so, it can be written in the form a/b , where a,b are coprimes.

x√y=a/b

=:√y = a/bx

this implies that √y is rational( a,b and x are integers and a/bx is rational);which is a contradiction.

so our assumption is incorrect.

hence , x√y is irrational.