If a is a nonzero rational and √b is irrational number then show that a√b is irrational number.
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Step by Step Explanation:
- 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.
- Now, a√b = ⇒ √b = ....(i)
- But, p and aq are both rational and aq ≠0
Therefore, is rational.
- 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.