question_answer

  Prove that  is irrational. OR Prove that is irrational.

Answer:

  Suppose,  is a rational number. Let where a is rational number,                                                                                                      (1)       Therefore, On squaring both sides, we get                                                                (1) which is a contradiction as the right hand side is a rational number whileis irrational. Hence,   is irrational.                      (1) OR Suppose, is rational. Then, there exist cop rime positive integers p and q such that                                               (1)                                     (1) is irrational while is rational, but irrational number can never be equal to a rational number. Thus, our assumption is wrong. Hence, is irrational.               (1)