Prove that square root of 'n' is not a rational number if 'n' is not a perfect square.

If n is not a perfect square then  is irrational

Let on the contrary say it is rational .

Then

  where p and q are coprime integers.

so n =p²/q²

 

p² =nq²

This shows p divides q

which is a contradiction.

Hence  is irrational if n is not a perfect square.