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If a is a nonzero rational and √b is irrational number then show that a√b is irrational number.
Answer:
- Let a be a nonzero rational and let √b be irrational.
Then, we have to show that a√b is irrational. - If possible, let a√b be rational number.
Then, a√b =
, where x and y are non-zero integers,having no common factor other than 1.x y - Now, a√b =
⇒ √b =x y
....(i)x ay - But, p and aq are both rational and aq ≠0
Therefore,
is rational.x ay - 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. - Hence, a√b is irrational.
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