If a is a nonzero rational and √b is irrational number then show that a√b is irrational number.


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Answer:


Step by Step Explanation:
  1. Let a be a nonzero rational and let √b be irrational.
    Then, we have to show that a√b is irrational.
  2. If possible, let a√b be rational number.
    Then, a√b =  
    x
    y
     , where x and y are non-zero integers,having no common factor other than 1.
  3. Now, a√b =  
    x
    y
      ⇒ √b =  
    x
    ay
     ....(i)
  4. But, p and aq are both rational and aq ≠0
    Therefore,  
    x
    ay
      is rational.
  5. Thus, from (i), it follows that √b is rational number.
    Where, this contradict the fact that √b is irrational.
    However, this contradiction arises by assuming that a√b is rational.
  6. Hence, a√b is irrational.

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